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# $(1+1/n)^{n}$ is an increasing sequence
###### Theorem 1.
The sequence $(1+1/n)^{n}$ is increasing.
###### Proof.
To see this, rewrite $1+(1/n)=(1+n)/n$ and divide two consecutive terms of the sequence:
$\displaystyle{\left(1+{1\over n}\right)^{n}\over\left(1+{1\over n-1}\right)^{n% -1}}$ $\displaystyle=$ $\displaystyle{\left({n+1\over n}\right)^{n}\over\left({n\over n-1}\right)^{n-1}}$ $\displaystyle=$ $\displaystyle\left({(n-1)(n+1)\over n^{2}}\right)^{n-1}{n+1\over n}$ $\displaystyle=$ $\displaystyle\left(1-{1\over n^{2}}\right)^{n-1}\left(1+{1\over n}\right)$
Since $(1-x)^{n}\geq 1-nx$, we have
$\displaystyle{\left(1+{1\over n}\right)^{n}\over\left(1+{1\over n-1}\right)^{n% -1}}$ $\displaystyle\geq$ $\displaystyle\left(1-{n-1\over n^{2}}\right)\left(1+{1\over n}\right)$ $\displaystyle=$ $\displaystyle 1+{1\over n^{3}}$ $\displaystyle>$ $\displaystyle 1,$
hence the sequence is increasing. ∎
###### Theorem 2.
The sequence $(1+1/n)^{n+1}$ is decreasing.
###### Proof.
As before, rewrite $1+(1/n)=(1+n)/n$ and divide two consecutive terms of the sequence:
$\displaystyle{\left(1+{1\over n}\right)^{n+1}\over\left(1+{1\over n-1}\right)^% {n}}$ $\displaystyle=$ $\displaystyle{\left({n+1\over n}\right)^{n+1}\over\left({n\over n-1}\right)^{n}}$ $\displaystyle=$ $\displaystyle\left({(n-1)(n+1)\over n^{2}}\right)^{n}{n+1\over n}$ $\displaystyle=$ $\displaystyle\left(1-{1\over n^{2}}\right)^{n}\left(1+{1\over n}\right)$
Writing $1+1/n$ as $1+n/n^{2}$ and applying the inequality $1+n/n^{2}\leq(1+1/n^{2})^{n}$, we obtain
$\displaystyle{\left(1+{1\over n}\right)^{n+1}\over\left(1+{1\over n-1}\right)^% {n}}$ $\displaystyle\leq$ $\displaystyle\left(1-{1\over n^{2}}\right)^{n}\left(1+{1\over n^{2}}\right)^{n}$ $\displaystyle=$ $\displaystyle\left(1-{1\over n^{4}}\right)^{n}$ $\displaystyle<$ $\displaystyle 1,$
hence the sequence is decreasing.
###### Theorem 3.
For all positive integers $m$ and $n$, we have $(1+1/m)^{m}<(1+1/n)^{n+1}$.
###### Proof.
We consider three cases.
Suppose that $m=n$. Since $n>0$, we have $1/n>0$ and $1<1+1/n$. Hence, $(1+1/n)^{n}<(1+1/n)^{n+1}$.
Suppose that $m. By the previous case, $(1+1/n)^{n}<(1+1/n)^{n+1}$. By theorem 1, $(1+1/m)^{m}<(1+1/n)^{n}$. Combining, $(1+1/m)^{m}<(1+1/n)^{n+1}$.
Suppose that $m>n$. By the first case, $(1+1/m)^{m}<(1+1/m)^{m+1}$ By theorem 2, $(1+1/m)^{m+1}<(1+1/n)^{n+1}$. Combining, $(1+1/m)^{m}<(1+1/n)^{n+1}$. ∎
Title $(1+1/n)^{n}$ is an increasing sequence 11nnIsAnIncreasingSequence 2013-03-22 15:48:39 2013-03-22 15:48:39 rspuzio (6075) rspuzio (6075) 14 rspuzio (6075) Theorem msc 33B99 |
# Factoring Quadratic Equations - GMAT Math Study Guide
## Definitions
• Quadratic Equation - An equation that can be written in the form ax2 + bx + c = 0.
For example, 6x2 + 2x + 1 = 0 is a quadratic equation while 6x + 2 is not a quadratic equation.
• Factoring - The process of breaking apart of an equation into factors (or separate terms) such that when the separate terms are multiplied together, they produce the original equation.
For example, x2 - x - 2 = (x+1)(x-2). In this case, the equation x2 - x - 2 = 0 can be broken apart into two factors [i.e., (x+1)(x-2) = 0] such that when these two separate terms (i.e., factors) are multiplied together, the result is the original equation.
## Basic Factoring
In factoring a basic quadratic equation such as x2 + 6x + 8 = 0, you must find two numbers that add to b (i.e., +6 in this case) and multiply to c (+8 in this case). The numbers +4 and +2 have the properties necessary. Consequently, (x + 4) and (x + 2) are the two factors.
To summarize:
In factoring a quadratic of the form x2 + bx + c, look for two numbers that add to b and multiply to c
### Examples of Basic Factoring
If you have not done factoring in years or it is entirely new, you may be confused at this point. However, the following examples and explanation of going between factored and quadratic form should clarify most confusion.
x2 + x - 12 = 0
Find two numbers that add to +1 and multiply to -12.
Two such numbers are +4 and -3.
(x + 4)(x - 3) = 0
x = -4 or x = +3 since each value satisfies the equation (x + 4)(x - 3) = 0.
Another example:
x2 - 3x - 10 = 0
Find two numbers that add to -3 and multiply to -10.
Two such numbers are -5 and +2.
(x - 5)(x + 2) = 0
x = +5 or x = -2 since each value satisfies the equation (x - 5)(x + 2) = 0.
Another example:
x2 + 7x + 6 = 0
Find two numbers that add to +7 and multiply to +6.
Two such numbers are +6 and +1.
(x + 6)(x + 1) = 0
x = -6 or x = -1 since each value satisfies the equation (x + 6)(x + 1) = 0.
## Reverse Factoring
The reverse of factoring is called FOIL, which stands for first, outer, inner, last. To acquire the quadratic form (ax2 + bx + c = 0) from the factored form [(x - a)(x - b) = 0]: (1) multiply the first terms, then the outer terms, then the inner terms, and finally the last terms (2) add each of the terms together and simplify. For example:
(x - 4)(x + 2) = ?
First: x(x) = x2
Outer: x(2) = 2x
Inner: (-4)(x) = -4x
Last: -4(2) = -8
(x - 4)(x + 2) = x2 + 2x - 4x - 8 = x2 - 2x - 8
### Translating Between Factored and Quadratic Form
Factoring, as defined above, is the process of breaking apart of an equation into factors (or separate terms) such that when the separate terms are multiplied together, they produce the original equation. Factoring works on the following fundamental relationship:
(x - r1)(x - r2) = 0 = x2 + bx + c where r1 and r2 are the roots, or solutions, of the quadratic equation
As an example:
Roots: +6, -4
(x - 6)(x + 4) = ?
x2 + 4x - 6x - 24
x2 - 2x - 24
Consequently, if you saw x2 - 2x - 24 = 0 as a question, you could quickly solve it by factoring it as follows:
x2 - 2x - 24 = 0
(x - 6)(x + 4) = 0
x = 6, -4 since both of these values make the equation (x - 6)(x + 4) = 0 true.
## Three Common Forms
There are three common forms that are easily factored. It is essential that you can recognize these three factored forms and quickly work with them:
### Difference of Squares
a2 - b2 = (a + b)(a - b)
For example:
x2 - 4 = (x + 2)(x - 2)
a = x, b = +2
### A Plus B Squared
a2 + 2ab + b2 = (a + b)2
For example:
x2 + 4x + 4 = (x + 2)2
a = x, b = +2
### A Minus B Squared
a2 - 2ab + b2 = (a - b)2
For example:
x2 - 4x + 4 = (x - 2)2
a = x, b = +2
## Dividing By Zero: Undefined
The rules of mathematics and division in particular state that you cannot divide by zero. Consequently, x divided by zero is undefined just as 1 divided by zero is undefined and 0 divided by 0 is undefined. Further, if you are factoring an equation with a variable in the denominator, any value of that variable that makes the denominator zero is not a legitimate solution. This is best explained and understood with examples.
Begin by factoring the equation as much as possible.
Top: Two numbers that add to +4 and multiply to -12 are +6 and -2.
Bottom: Two numbers that add to +5 and multiply to +4 are +1 and +4.
The solutions (or roots) are x = -6 or x = +2. Since the denominator cannot equal zero, x = -4 or x = -1 [both values that cause the entire denominator to be equal to zero] are not solutions but are instead values of x that cause the entire expression to be undefined.
Another Example
Begin by factoring the equation as much as possible.
Top: Two numbers that add to -10 and multiply to +21 are -7 and -3.
Bottom: Two numbers that add to -6 and multiply to +8 are -4 and -2.
The solutions (or roots) are x = +3 or x = +7. Since the denominator cannot equal zero, x = +2 or x = +4 [both values that cause the entire denominator to be equal to zero] are not solutions but are instead values of x that cause the entire expression to be undefined.
## Solutions Summary
Quadratic equations can have zero, one, or two real solutions.
x2 + 9 = 0; No real solution
x2 + 6x + 9 = 0; One real solution: x = -3
x2 - 4 = 0; Two real solutions: x = 2 or x = -2 |
Bertrand’s Chord Problem
The history of probability theory has been influenced strongly by paradoxes, results that seem to defy intuition. Many of these have been reviewed in a recent book by Prakash Gorroochurn [2012]. We will have a look at Bertrand’s Paradox (1889), a simple result in geometric probability.
Let’s start with an equilateral triangle and add an inscribed circle and a circumscribed circle. It is a simple geometric result that the radius of the outer circle is twice that of the inner one. Bertrand’s problem may be stated thus:
Problem: Given a circle, a chord is drawn at random. What is the probability that the chord length is greater than the side of an equilateral triangle inscribed in the circle?
We will consider three ways of drawing a chord in the outer circle:
1. Fix the end-points of the chord.
2. Choose the chord centre on a fixed diameter.
3. Fix the mid-point of the chord.
We will find that the probability differs for the three methods of choice.
(1) Fix the end-points of the chord.
On the basis of symmetry, we may choose one endpoint of the chord to coincide with a vertex of the equilateral triangle. Since the other endpoint (e.g., A or B) must be on one of three arcs, all of equal length, we may argue that each arc is equally probable. The chord length is less than the side of the triangle for the arcs adjacent to the first endpoint (e.g. point A) and exceeds it for points on the centre arc (e.g., point B). Thus, the probability of this event is .
(2) Choose the chord centre-point on a fixed diameter.
We first choose a diameter. Then we select a point at random on it, and draw the chord through the point and normal to the diameter. Since the inner circle has diameter half that of the outer circle, there is a 50% chance that the point falls within the inner circle (e.g., point B). Thus, the probability of the chord length exceeding the triangle side is ½.
(3) Fix the mid-point of the chord.
We fix the chord by choosing its midpoint. Excepting the centre point of the circle, only one chord has a given midpoint. If the point falls outside the inner circle (e.g., point A), the chord length is less than the side of the triangle. Otherwise (e.g., point B) it exceeds it.
But the area of the inner circle is one quarter that of the outer circle, since the radius is half. Therefore, the chance of the point being within the inner circle is 25% and the probability of the event is ¼.
Discussion
We have found three answers for the probability P of the event:
1. Fixed end-points of the chord: P = .
2. Chord centre on a fixed diameter: P = ½.
3. Fixed mid-point of the chord: P = ¼.
Which is the correct result? We cannot say: the question is not well-posed and the answer depends on the method used to choose the chord.
Difficulties with results such as Bertrand’s chords problem gave rise to dissatisfaction with classical probability theory and acted as an impetus for the later development of an axiomatic foundation of the subject. David Hilbert recognized this need and identified an axiomatic basis for probability theory as one of his famous 23 problems for the twentieth century.
Poincaré, and later Edwin T. Jaynes, argued that the solution should be insensitive to changes in the position, orientation and size of the circle and triangle. Jaynes used this invariance to obtain a probability of ½. He also found that this agreed with actual experiments dropping sticks onto circles. However, this was not universally accepted and the debate continues today.
Final Remark
Imagine the plane covered with all possible lines. Drop a circular form, such as a hula-hoop, on the plane. A set of lines intersect the circle. What proportion of them exceed the length of an inscribed equilateral triangle? We hope to use this approach to develop another solution to Bertrand’s Problem. If successful, details will follow.
Sources
• Bertrand, Joseph, 1889: Calculs de Probabilités. Gaultier-Villars, Paris.
• Gorroochurn, Prakash, 2012: Classic Problems of Probability. Wiley, ISBN: 978-1-118-06325-5 |
# Chapter 9 * Quadratic Equations and Functions
Today we will be learning how to:
Use quadratic equations to model real-life
problems
Evaluate the radical expressions √ab and √a √b for
the given values of a and b.
a = 4, b = 9
a = 25, b = 4
a = 36, b = 16
given values of a and b.
a = 4, b = 49
a = 16, b = 64
a = 144, b = 100
a
and
b
a
b
for the
Product Property
The square root of a product equals the product of
the square roots of the factors.
√ab = √a √b where a ≥ 0 and b ≥ 0
Ex. √4 100 = √4 √100
Quotient Property
The square root of a quotient equals the quotient
of the square roots of the numerator and
denominator.
a
a
b
b
Ex.
where a ≥ 0 and b > 0
9
9
25
25
An expression is in simplest form if:
No perfect square factors other than 1 are in the radicand
√8 = √42 = 2√2
No fractions are in the radicand
5
5
5
16
4
16
No radicals appear in the denominator of a fraction
1
1
4 2
Example 1
Simplify the expression √48
Example 2
Simplify the expression
7
16
18
3
80
45
Example 3
The distance d you can see to the horizon depends on your
height h. A model is d2 = 1.5h, with d in miles and h in
feet.
Find the exact distance you can see from the top of a 400 ft
building.
Example 3
The distance d you can see to the horizon depends on your
height h. A model is d2 = 1.5h, with d in miles and h in
feet.
Find the distance in part 1 to the nearest tenth.
Example 3
The distance d you can see to the horizon depends on your
height h. A model is d2 = 1.5h, with d in miles and h in
feet.
If you were 3 × 400 = 1200 ft up in a skyscraper, how far could you
see to the nearest tenth?
HOMEWORK
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What is FOIL?
The FOIL Method — First Outer Inner Last — is the best system to use when multiplying two binomials. You can think of it as a slightly more advanced version of the distributive property of multiplication, which tells us that a(b + c) = ab + ac.
When multiplying (a+b)(c+d), for example, FOIL reminds us that we must distribute both the a and the b across the c and the d. Each part of the first binomial must be multiplied by each of the parts of the second binomial. Then, we add all of the partial products together.
What is a binomial?
A binomial is an algebraic expression that consists of the sum or the difference of two terms, for example:
• 2 + x
• x + y
• 3a + 5b
• xy – ab
• 3x3 – 2y4
When to use the FOIL method
Whenever you need to multiply two binomials!
How to use the FOIL method
Example 1
F O I L
(a+b)(c+d) = ac + ad + bc + bd
There are four partial products that must be added together:
1. The product of the First terms of each binomial (a and c)
2. The product of the “Outer” terms (a and d)
3. The product of the “Inner” terms (b and c)
4. The product of the Last terms (b and d)
Example 2
(x-4)(x+3)
• First: x × x = x2
• Outer: x × 3 = 3x
• Inner: -4 × x = -4x
• Last: -4 × 3 = -12
x2 + 3x – 4x – 12
x2 -1x -12
Example 3
(a-b)(c-d)
• First: a× c = ac
• Outer: a× -d = -ad
• Inner: -b × c = -bc
• Last: -b × -d = +bd
ac – ad – bc + bd
Example 4
(x2-y)(y2+x)
• First: x2 × y2 = x2y2
• Outer: x2 × x = x3
• Inner: -y × y2 = -y3
• Last: -y × x = -yx
x2y2 + x3 – y3 – yx
TRY THESE!
1. (6 + a)(a – 2)
2. (5x2-a)(5x2+a)
3. (d + e/2)(-a – 1)
4. (3s4-t2)(2s3+t3)
1. 6a -12 + a2 – 2a Simplify: a2 + 4a - 12
2. 25x4 + 5ax2 - 5ax2 – a2 Simplify: 25x4 – a2
3. –ad – d – ea/2 – e/2
4. 6s7 + 3s4t3 – 2s3t2 – t5
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# How do you simplify the expression (cottsec^2t-cott)/(sint tant+cost)?
$\sin t$
#### Explanation:
(I'm writing this bit - the intro - last. I've just worked the problem and I had no idea where it would go or how to simplify it, beyond the fact that I took it apart piece by piece and took substitution opportunities where I could. It's easy to look at something complicated and shrug and think "I can't do this", but if you can start by doing something - anything - you can start to make sense of it. This is not to say that the first few things you do will make the problem simple right away - it can get rather messy. But stick with it and it should resolve itself into something simpler!)
$\frac{\cot t {\sec}^{2} t - \cot t}{\sin t \tan t + \cos t}$
In the numerator, let's factor out $\cot t$ to get to a $\left({\sec}^{2} t - 1\right)$ term and in the denominator let's look at multiplying the sin and tan so that we can work with the cos:
$\frac{\cot t \left({\sec}^{2} t - 1\right)}{\sin t \left(\sin \frac{t}{\cos} t\right) + \cos t}$
In the numerator, we can use the trig identity ${\tan}^{2} t = {\sec}^{2} t - 1$ and substitute in. In the denominator, I'm going to work on getting the terms to add:
$\frac{\cot t \left({\tan}^{2} t\right)}{\left({\sin}^{2} \frac{t}{\cos} t\right) + \cos t \left(\cos \frac{t}{\cos} t\right)}$
$\frac{\left(\frac{1}{\tan} t\right) \left({\tan}^{2} t\right)}{\left({\sin}^{2} \frac{t}{\cos} t\right) + \left({\cos}^{2} \frac{t}{\cos} t\right)}$
$\tan \frac{t}{\frac{{\sin}^{2} t + {\cos}^{2} t}{\cos} t}$
$\frac{\tan t \cos t}{{\sin}^{2} t + {\cos}^{2} t}$
We can substitute in the denominator, using the trig identity ${\sin}^{2} t + {\cos}^{2} t = 1$
$\frac{\left(\sin \frac{t}{\cancel{\cos}} t\right) \cancel{\cos} t}{1}$
$\sin t$ |
# Eureka Math Algebra 1 Module 4 Lesson 11 Answer Key
## Engage NY Eureka Math Algebra 1 Module 4 Lesson 11 Answer Key
### Eureka Math Algebra 1 Module 4 Lesson 11 Example Answer Key
Example
Now try working backward. Rewrite the following standard form quadratic expressions as perfect squares.
### Eureka Math Algebra 1 Module 4 Lesson 11 Exploratory Challenge Answer Key
Exploratory Challenge
Find an expression equivalent to x2 + 8x + 3 that includes a perfect square binomial.
(x + 4)2 – 13
### Eureka Math Algebra 1 Module 4 Lesson 11 Exercise Answer Key
Opening Exercise
Rewrite the following perfect square quadratic expressions in standard form. Describe patterns in the coefficients for the factored form, (x + A)2, and the standard form, x2 + bx + c.
For each row, the factored form and standard form are equivalent expressions, so (x + A)2=x2 + bx + c. A, the constant in factored form of the equation, is always half of b, the coefficient of the linear term in the standard form. c, the constant term in the standard form of the quadratic equation, is always the square of the constant in the factored form, A.
Exercises
Rewrite each expression by completing the square.
Exercise 1.
a2 – 4a + 15
(a – 2)2 + 11
(Note: Since the constant term required to complete the square is less than the constant term, + 15, students may notice that they just need to split the + 15 strategically.)
Exercise 2.
n2 – 2n – 15
(n – 1)2 – 16
Exercise 3.
c2 + 20c – 40
(c + 10)2 – 140
Exercise 4.
x2 – 1000x + 60 000
(x – 500)2 – 190 000
Exercise 5.
y2 – 3y + 10
(y – $$\frac{3}{2}$$)2 + $$\frac{31}{4}$$
Exercise 6.
k2 + 7k + 6
(k + $$\frac{7}{2}$$)2 – $$\frac{25}{4}$$
Exercise 7.
z2 – 0.2z + 1.5
(z – 0.1)2 + 1.49
Exercise 8.
p2 + 0.5p + 0.1
(p + 0.25)2 + 0.0375
Exercise 9.
j2 – $$\frac{3}{4}$$ j + $$\frac{3}{4}$$
(j – $$\frac{3}{8}$$)2 + $$\frac{39}{64}$$
Exercise 10.
x2 – bx + c
(x – $$\frac{b}{2}$$)2 + c – $$\frac{b^{2}}{4}$$
### Eureka Math Algebra 1 Module 4 Lesson 11 Problem Set Answer Key
Rewrite each expression by completing the square.
Question 1.
q2 + 12q + 32
(q + 6)2 – 4
Question 2.
m2 – 4m – 5
(m – 2)2 – 9
Question 3.
x2 – 7x + 6.5
(x – $$\frac{7}{2}$$)2 – 5.75
Question 4.
a2 + 70a + 1225
(a + 35)2
Question 5.
z2 – 0.3z + 0.1
(z – 0.15)2 + 0.0775
Question 6.
y2 – 6by + 20
(y – 3b)2 + 20 – 9b2
Question 7.
Which of these expressions would be most easily rewritten by factoring? Justify your answer. |
# Multiplying roots - solving for $x$
If I have one square root that involves a coefficient, and one square root that doesn't, how do I solve for $x$?
$$3\sqrt3 = \sqrt{3x}$$
I tried to simplify it like:
$$\frac{3\sqrt3} {\sqrt3} = \sqrt x$$
But ran into a dead end...what do I do next?
-
After seeing it happen many times, I can't help reminding you to be sure you're not confusing $\sqrt{3x}$ with $\sqrt{3}x$, which are two quite different things. – Michael Hardy Apr 12 '13 at 15:42
Do you see that $\dfrac {3 \sqrt 3}{\sqrt 3} = 3 \sqrt {\dfrac{3}{3}} = 3 \sqrt 1 = 3$? – amWhy Apr 12 '13 at 15:47
I would immediately square both sides and get rid of the square root. – Stefan Smith Apr 12 '13 at 23:56
Note that: $$\dfrac {3 \sqrt 3}{\sqrt 3} = 3 \sqrt {\dfrac{3}{3}} = 3 \sqrt 1 = 3$$
But we can think of approaching the original equation by first noting: $$3\sqrt3 = \sqrt{3x} \iff 3\sqrt 3 = \sqrt 3 \times \sqrt x$$
and "canceling" $\;\sqrt 3\;$ from each side of the equation.
$$3\sqrt 3 = \sqrt 3 \sqrt x \quad \iff \quad 3 = \sqrt x$$
Now simply square both sides of the equation: $$(3)^2 = \left(\sqrt x\right)^2 \quad \iff \quad 9 = x$$
-
You have $$\sqrt{x} = \frac{3\sqrt{3}}{\sqrt{3}}.$$ Note here that $x$ must be positive. You can now simply square both sides: $$x = (\sqrt{x})^2 = \left(\frac{3\sqrt{3}}{\sqrt{3}}\right)^2.$$ Left for you to do is to simplify things.
- |
# 12.1.2 Measures of Central Tendencies
12.1.2 Mode, Median and Mean
1. The mode of a set of data is the value of item which occurs most frequently.
Example:
3, 7, 6, 9, 7, 1, 5, 7, 2
Mode = 7
2. When a set of data is given in a frequency table, the value or item which has the highest frequency is the mode.
3. The median of a set of data is the value located in the middle of the set when the data is arranged in numerical order.
- If the total number of data is odd, then the median is the value in the middle of the set.
- If the total number of data is even, then the median is the average of the two middle values of the set
Example 1:
Find the medians of the following sets of data:
(a) 10, 9, 11, 6, 5, 8, 7
(b) 10, 9, 11, 6, 5, 8, 7,12
Solution:
(a) Number of data values = 7 ← (Odd number)
Rearranging the data in order of magnitude:
5, 6, 7, 8, 9, 10, 11
Therefore, median = 8
(b) Number of data values = 8 ← (Even number)
Rearranging the data in order of magnitude:
5, 6, 7, 8, 9, 10, 11, 12
$\begin{array}{l}\therefore \text{Median}=\frac{8+9}{2}\\ \text{}=8.5\end{array}$
4. When a set of data is given in a frequency table, the value situated in the middle position of the data is the median.
5. The mean of a set of data is obtained by using formula below.
$\text{Mean =}\frac{\text{sum of all values of data}}{\text{total number of data}}$
Example:
Find the mean of the following sets of data items:
-5, -2, -1, 7, 4, 9
Solution:
$\begin{array}{l}\text{Mean}=\frac{\left(-5\right)+\left(-2\right)+\left(-1\right)+7+4+9}{6}\\ \text{}=\frac{12}{6}\\ \text{}=2\end{array}$
6. When data is given in a frequency table, the mean can be found by using the formula below.
$\text{Mean =}\frac{\text{sum of}\left(\text{value}×\text{frequency}\right)}{\text{total frequency}}$
Example:
The table below shows the scores obtained by a group of players in a game.
Score 1 2 3 4 5 Frequency 5 12 8 15 10
Find the mean of the scores.
Solution:
$\begin{array}{l}\text{Mean}\\ =\text{}\frac{\text{sum of}\left(\text{score}×\text{frequency}\right)}{\text{total frequency}}\\ =\frac{\left(1×5\right)+\left(2×12\right)+\left(3×8\right)+\left(4×15\right)+\left(5×10\right)}{5+12+8+15+10}\\ =\frac{163}{50}\\ =3.26\end{array}$ |
The fraction 12/100 is same to 0.12 when converted come a decimal. See listed below detalis on how to convert the portion 12/100 to a decimal value.
You are watching: 12/100 as a decimal
### Fraction to Decimal Converter
Enter a portion value:Ex.: 1/2, 2 1/2, 5/3, etc. Note that 2 1/2 means two and half = 2 + 1/2 = 2.5
Fraction come decimal explained:
## How to convert from portion to decimal?
To easily transform a portion to a decimal, divide the molecule (top number) through the denominator (bottom number).
### Example 1: exactly how to transform 4/8 come a decimal?
Step 1:Divide 4 by 8: 4 ÷ 8 = 1 ÷ 2 = 0.5Step 2:Multiply the an outcome by 100 and add the decimal sign: 0.5 × 100%Answer: 4/8 = 50%
### Example 2: just how to transform 1 1/3 come a decimal?
Step 1:Divide 1 by 3: 1 ÷ 3 = 0.3333Step 2:Add this worth to the the essence part: 1 + 0.3333 = 1.3333Step 3:Multiply the result by 100 and include the decimal sign: 1.3333 × 100%Answer: 1 1/3 = 133.33%
Note: the an outcome was rounded to 2 decimal places.
fractioninchesmm
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3/640.04691.1906
1/160.06251.5875
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33/640.515613.0969
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5/80.625015.8750
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3/40.750019.0500
See more: 1 Inch Ginger Equals How Much Powder ? How Much Ground Ginger Is Equal To Fresh
fractioninchesmm
49/640.765619.4469
25/320.781319.8438
51/640.796920.2406
13/160.812520.6375
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27/320.843821.4313
55/640.859421.8281
7/80.875022.2250
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11.000025.4000 |
3 Tutor System
Starting just at 265/hour
# Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts.
Construction according to the required data is done as follows:
Step1: Draw a line segment AB=7.6 cm and at point A draw a ray AC making acute angle with AB.
Step 2: As shown in figure, start locating 13 marks (5+8) as $$A_1,A_2,A_3,.....A_{13}$$, such that $$AA_1=AA_2=AA_3$$..and so on.
Step 3: Now join the points $$A_{13}$$ and B.
Step 4: From point $$A_5$$, draw a line parallel to $$A_{13}B$$ and mark the intersection point as P.
Step 5: Measure AP and PB. we will find AP= 2.9cm and PB =4.7cm(approx) which are in approx ration of 5:8.
Justification:
In $$ΔAA_5P and △AA_13B$$ we have,
$$A_5P ∥ A_{13}B$$
$$\frac{AP}{BP}=\frac{AA_5}{A_5A_{13}}$$
(By the Basic proportionality theorem)
$$\frac{AP}{BP}=\frac{5}8$$ {∵ $$\frac{AA_5}{A_5A_{13}}=\frac{5}8$$}
$$\therefore$$ , AP:BP=5:8 |
## The Best College mathematics answers
Math home work can be a tricky thing for some students. Math is a difficult subject for some, so doing homework on it can be frustrating. Some tips to help with math homework are to get a tutor, practice at home, and try to understand the concepts. A tutor can help go over the material and help with any confusion. Also, practicing math problems at home can be helpful. Doing a few problems each night can help solidify the material. Lastly, trying to understand the concepts can be very helpful. If a student understands why they are doing a certain math problem, it can make the problem much easier. Math homework can be tough, but these tips can make it a little bit easier.
A rational function is any function which can be expressed as the quotient of two polynomials. In other words, it is a fraction whose numerator and denominator are both polynomials. The simplest example of a rational function is a linear function, which has the form f(x)=mx+b. More generally, a rational function can have any degree; that is, the highest power of x in the numerator and denominator can be any number. To solve a rational function, we must first determine its roots. A root is a value of x for which the numerator equals zero. Therefore, to solve a rational function, we set the numerator equal to zero and solve for x. Once we have determined the roots of the function, we can use them to find its asymptotes. An asymptote is a line which the graph of the function approaches but never crosses. A rational function can have horizontal, vertical, or slant asymptotes, depending on its roots. To find a horizontal asymptote, we take the limit of the function as x approaches infinity; that is, we let x get very large and see what happens to the value of the function. Similarly, to find a vertical asymptote, we take the limit of the function as x approaches zero. Finally, to find a slant asymptote, we take the limit of the function as x approaches one of its roots. Once we have determined all of these features of the graph, we can sketch it on a coordinate plane.
When dealing with data, there are typically three different types of averages that can be used in order to summarize the information: the mean, the median, and the mode. Of these, the mode is often the most difficult to calculate. However, once you understand the definition of mode and how it is used, solving for it becomes a relatively straightforward process. Mode is simply the value that appears most frequently in a data set. In order to calculate it, first identify all of the unique values in your data set and then count how many times each one occurs. The value that occurs most often is the mode. In some cases, there may be more than one mode, or no mode at all. When this happens, it is said to be bimodal or multimodal if there are two or more modes, respectively, and unimodal if there is only one.
## Help with math
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# Inverse Tan
Inverse tan is the inverse function of the trigonometric function ‘tangent’. It is used to calculate the angle by applying the tangent ratio of the angle, which is the opposite side divided by the adjacent side of the right triangle. Based on this function, the value of tan 1 or arctan 1 or tan 10, etc. can be determined. It is naming convention for all inverse trigonometric functions to use the prefix ‘arc’ and hence inverse tangent is denoted by arctan. Although it is not uncommon to use tan-1, we will use arctan throughout this article.
## Inverse Tangent Formula
The formula for adding two inverse tangent function is derived from tan addition formula.
In this formula, by putting a = arctan x and b = arctan y, we get
For Integration:
Some of the important formulae for calculating integrals of expressions involving the arctan function are:
∫ arctan (x)dx = {x arctan (x)} – {In (x2+1)/2} + C
∫ arctan (ax)dx = {x arctan (ax)}- {In (a2x2+1)/2a} + C
∫ x arctan (ax)dx = {x2 arctan (ax)/2} – {arctan(ax)/2a2 } – {x / 2a} + C
∫ x2 arctan (ax)dx = {x3 arctan (ax)/3} – {In (a2x+ 1)/6a3 } – {x2/6a} + C
## Calculus of Arctan Function
This section gives the formula to calculate the derivative and integral of the arctan function.
Derivative of tan inverse x
The differentiation of tan inverse x is given below. The derivative of arctan x is denoted by d/dx(arctan(x)) and for complex values of x , the derivative is equal to 1/1+x2 for x ≠ -i, +i.
Therefore, the differentiation of tan inverse x is given by:
$$\frac{d}{dx}tan^{-1}x = \frac{1}{1 + x^2}$$
Integral of inverse tan
For obtaining an expression for the definite integral of the inverse tan function, the derivative is integrated and the value at one point is fixed. The expression is:
$$arctan (x) = \int_{0}^{x}\frac{1}{y^{2}+1}dy$$
## Inverse Tan Graph
### Relationship Between Inverse Tangent Function and Other Trigonometric Functions
Consider a triangle whose length of adjacent and opposite are 1 and x respectively. Therefore, the length of the hypotenuse is $$\sqrt{1+x^{2}}$$. For this triangle, if the angle $$\theta$$ is the arctan, then the following relationships hold true for the three basic trigonometric functions:
Sin (arctan(x)) = $$\frac{x}{\sqrt{1+x^{2}}}$$
Cos (arctan(x)) = $$\frac{1}{\sqrt{1+x^{2}}}$$
Tan (arctan(x)) = x
## Inverse Tangent Properties
The basic properties of the inverse tan function (arctan) are listed below:
Notation: y = arctan (x)
Defined as: x = tan (y)
Domain of the ratio: all real numbers
Range of the principal value in radians: -π/2 < y < π/2
Range of the principal value in degrees: -90° < y < 90°
### What is the Value of tan-1 Infinity?
To calculate the value of the tan inverse of infinity(∞), we have to check the trigonometry table. From the table we know, the tangent of angle π/2 or 90° is equal to infinity, i.e.,
tan 90° = ∞ or tan π/2 = ∞
Therefore, tan-1 (∞) = π/2 or tan-1 (∞) = 90°
## Solved Examples On Inverse Tan
Example 1:
Prove that $$4\left(2 \tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{7}\right)=\pi$$
Solution:
Consider $$2 \tan ^{-1} \frac{1}{3}$$
This can also be written as:
$$tan ^{-1} \frac{1}{3}+tan ^{-1} \frac{1}{3}\\=tan ^{-1}\left(\frac{\frac{1}{3}+\frac{1}{3}}{1-\frac{1}{3} . \frac{1}{3}}\right) \\ =tan ^{-1} \frac{3}{4}$$
Now, LHS =$$4\left(2 tan ^{-1} \frac{1}{3}+tan ^{-1} \frac{1}{7}\right) \\ =4\left(tan ^{-1} \frac{3}{4}+tan ^{-1} \frac{1}{7}\right) \\ =4\ tan ^{-1}\left(\frac{\frac{3}{4}+\frac{1}{7}}{1-\frac{3}{4} . \frac{1}{7}}\right) \\ =4\ tan ^{-1}\left(\frac{25}{28} \times \frac{28}{25}\right) \\ =4\ tan ^{-1} 1 \\ =4 \times \frac{\pi}{4} \\ =\pi$$
= RHS
Hence proved.
Example 2:
Show that $$\tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{2}{9}+\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}=\pi / 4$$
Solution:
LHS =
$$\begin{array}{l} =tan ^{-1} \frac{1}{4}+tan ^{-1} \frac{2}{9}+tan ^{-1} \frac{1}{5}+tan ^{-1} \frac{1}{8} \\ =tan ^{-1} \frac{\frac{1}{4}+\frac{2}{9}}{1-\frac{1}{4} . \frac{2}{9}}+tan ^{-1} \frac{\frac{1}{5}+\frac{1}{8}}{1-\frac{1}{5} .\frac{1}{8}} \\ =tan ^{-1}\left(\frac{17}{36} \times \frac{36}{34}\right)+tan ^{-1}\left(\frac{13}{40} \times \frac{40}{39}\right) \\ =tan ^{-1} \frac{1}{2}+tan ^{-1} \frac{1}{3} \\ =tan ^{-1} \left ( \frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2} .\frac{1}{3}} \right ) \\ =tan ^{-1}(1) \\ =\frac{\pi}{4} \end{array}$$
= RHS
Example 3:
Find the value of x, for which $$3 \tan ^{-1} \frac{1} {2+\sqrt{3}}-\tan ^{-1}\frac{1}{x}=\tan ^{-1}\frac{1}{3}$$
Solution:
Consider, tan 15° = tan(45° – 30°)
= (tan 45° – tan 30°)/(1 – tan 45° tan 30°)
$$\begin{array}{l} \Rightarrow \tan 15^{\circ}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}} \\ \Rightarrow \tan 15^{\circ}=\frac{\sqrt{3}-1}{\sqrt{3}+1} \\ \Rightarrow \tan 15^{\circ}=\frac{(\sqrt{3}-1)(\sqrt{3}+1)}{(\sqrt{3}+1)(\sqrt{3}+1)} \\ \Rightarrow \tan 15^{\circ}=\frac{3-1}{4+2 \sqrt{3}} \\ \Rightarrow \tan 15^{\circ}=\frac{1}{2+\sqrt{3}} \\ \Rightarrow \tan ^{-1}\left(\frac{1}{2+\sqrt{3}}\right)=15^{\circ} \\ \Rightarrow \tan ^{-1}\left(\frac{1}{2+\sqrt{3}}\right)=\frac{\pi}{12}\end{array}$$
From the given,
$$\begin{array}{l}3 \tan ^{-1} \frac{1}{2+\sqrt{3}}-\tan ^{-1} \frac{1}{x}=\tan ^{-1} \frac{1}{3} \\ \Rightarrow 3 \cdot \frac{\pi}{12}-\tan ^{-1} \frac{1}{x}=\tan ^{-1} \frac{1}{3} \\ \Rightarrow-\tan ^{-1} \frac{1}{x}=\tan ^{-1} \frac{1}{3}-\frac{\pi}{4} \\ \Rightarrow \tan ^{-1} \frac{1}{x}=\tan ^{-1} 1-\tan ^{-1} \frac{1}{3}\ [ since, \frac{\pi}{4}=\tan ^{-1} 1] \\ \Rightarrow \tan ^{-1} \frac{1}{x}=\tan ^{-1} \frac{1-\frac{1}{3}}{1+1 . \frac{1}{3}} \end{array}$$
⇒ tan-1(1/x) = tan-1(1/2)
⇒ 1/x = 1/2
⇒ x = 2
Therefore, the value of x is 2.
### Practice Problems
1. Show that $$4 tan ^{-1} \frac{1}{5}-tan ^{-1} \frac{1}{70}+tan ^{-1} \frac{1}{99}=\frac{\pi}{4}$$
2. If y = tan-1(sec x + tan x), then find dy/dx.
3. Find the value of x if tan-1[(x – 1)/(x – 2)] + tan-1[(x + 1)/(x + 2)] = π/4. |
# How do you solve w+19=49?
Mar 3, 2018
For this you will have to do basic algebra, and isolate the variable.
#### Explanation:
To isolate the variable, you will have to get it alone on one side of the equation. In order to do this, you will have to take away the numbers that is on the same side as the variable in the equation.
Since it is a positive 19 that is being added to the "w" variable, you will have to subtract 19 to get "w" by itself. However, be careful because whatever you do to one side you must do to the other. Because of this rule, you will have to subtract 19 from the 49 as well. This means that the right side of the equation will equal 30, and the left side will equal to "w", meaning the value of "w" is 30.
$w + 19 = 49$
$w + \left(19 - 19\right) = \left(49 - 19\right)$
$w + 0 = \left(49 - 19\right)$
$w = 30$
Mar 3, 2018
$w = 30$
#### Explanation:
$w + 19 = 49$
$w = 49 - 19$
$w = 30$
~Hope this helps! :) |
# Positive Function / Negative Function
## What is a Positive Function?
A positive function has function values greater than zero (i.e., f(x) > 0). The domain (inputs) of the function can be negative, but the outputs (y-values) must be greater than zero. In other words, a positive function has values that are positive for all arguments of its domain.
A non-negative function is similar, except that it includes zero in its range.
Graphically, if a function’s output values are all above the x-axis, then the function is positive. Conversely, if the output values are all below the x-axis, then the function is negative. A function can also be positive for certain function intervals. For example, the function f(x) = x3 is positive on the open interval (0, ∞) but negative on the interval (-∞, 0).
A caution: a positive function isn’t necessarily an increasing function (although it can be). The function f(x) = 4x2 + 2, shown on the above graph, is completely above the x-axis, which means it is a positive function. However, notice that it is only increasing for function values on the right-hand side of the vertical axis; the function is decreasing for values to the left of the y-axis. In other words, positive functions can have derivatives that are negative or positive.
A couple of interesting properties:
• A positive function f(x) is log-convex if log f(x) is convex [1].
• A linear combination of positive functions is a positive function.
## What is a Negative Function?
A negative function has values that are all negative (i.e., f(x) < 0). The domain (inputs) of the function can be positive, but every output (y-value) must be less than zero. In other words, a negative function has values that are negative for all arguments of its domain. Graphically, all output (y) values are below the horizontal axis.
## Positive Function and Integrals
The definite integral of a positive function represents area under the graph of the function from a to b.
A positive function is integrable if it is a measurable function and if the integral is less than infinity [2].
## References
Image created with Desmos.com.
[1] Ni, L. Additional Problems-Set 5. Retrieved March 6, 2021 from: https://mathweb.ucsd.edu/~lni/math220/Pre-pr5.pdf
[2] Hunter, J. Chapter 4: Integration. Retrieved March 6, 2022 from: https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch4.pdf |
Equivalent Ratios
Equivalent ratios are the ratios that are the same when we compare them. Two or more ratios can be compared with each other to check whether they are equivalent or not. For example, 1:2 and 2:4 are equivalent ratios.
In other words, we can say, two ratios are equivalent to each other if one of them can be expressed as the multiple of the other. Hence, to get the equivalent ratio of another ratio, we have to multiply the two quantities (antecedent and consequent) by the same number. This method is similar to the method of finding equivalent fractions.
Let us learn in this article how to find the equivalent ratios with examples. But before we proceed, first we need to understand about ratios and their quantities.
What is Ratio?
In Mathematics, a ratio compares two quantities named as antecedent and consequent, by the means of division. For example, when we cook food, then each ingredient has to be added in a ratio. Thus, we can say, a ratio is used to express one quantity as a fraction of another quantity.
A ratio is usually expressed with the symbol ‘:’. The comparison or simplified form of two quantities of the same kind is referred to as ratio.
Note: The quantities of any ratio should be of the same kind The units of two quantities of the ratio should be same
Numerator and Denominator
We can also express the ratio as a fraction. If a:b, is a ratio, then a/b is its fraction form. Thus, we can easily compare two or more equivalent ratios in the form of equivalent fractions.
Standard Form of Ratio
The standard form of the ratio is given below:
Ratio = a : b = Numerator : Denominator Or Ratio = a / b = Numerator / Denominator
How to Find Equivalent Ratios?
As we know, two or more ratios are equivalent if their simplified forms are the same. Thus, to find a ratio equivalent to another we have to multiply the two quantities, by the same number.
Another way to find equivalent ratios is to convert the given ratio into fraction form and then multiply the numerator and denominator by the same number to get equivalent fractions. Then again we can write the resulting fraction as an equivalent ratio.
Also, if we have to compare any two equivalent ratios, then we can divide the two quantities by the highest common factor and get the simplest form of ratio. Hence, we can compare them.
The examples of equivalent ratios are:
• 2 : 4 :: 4 : 8
• 10 : 20 :: 20 :40
• 1 : 2 :: 2 : 4
• 0.5 : 1 :: 2:4
Solved Examples
Q.1: Find the equivalent ratios of 8 : 18.
Solution: Let us first write the given ratio as a fraction.
8:18 ⇒ 8/18
Now multiply the numerator and denominator by 2
= (8 × 2)/(18 × 2)
= 16/36
Or we can write, the above fraction as a ratio;
= 16 : 36
So, 16 : 36 is an equivalent ratio of 8 : 18.
Q.2. Find any two equivalent ratios of 4 : 5.
Solution: Let us first write the given ratio as a fraction.
4:5 ⇒ 4/5
Now multiply the numerator and denominator by 2, to get the first equivalent fraction.
= 4/5
= (4 × 2)/(5 × 2)
= 8/10
Or
4:5 = 8:10
Again, multiply and divide ⅘ by another natural number, such as 3, as given below:
= 4/5
= (4 × 3)/(5 × 3)
= 12/15
Or
4:5 = 12:15
Hence, the two equivalent ratios of 4 : 5 are 8 : 10 and 12 : 15.
Q.3. Compare the given ratios if they are equivalent or not.
14:21, 2:3, 1:1.5, 6:9
Solution: Let us write the given ratios as fractions.
14:21 ⇒ 14/21
2:3 ⇒ ⅔
1:1.5 ⇒ 1/1.5
6:9 ⇒ 6/9
Now, we have to find the common factors that divide the numerator and denominator evenly and hence we get the simplified form of fractions.
14/21 = ⅔ (HCF = 7)
⅔ = ⅔ (Already simplified form)
1/1.5 = 10/15 = ⅔ (HCF = 5)
6/9 = ⅔ (HCF = 3)
Thus, we can see all the above fractions are equivalent since their simplified forms are the same. Therefore, the given ratios are also equivalent to each other.
Q.4: A bag contains 4 red balls and 9 white balls. What is the ratio of red balls to the white balls?
Solution: Number of red balls = 4
Number of white balls = 9
Therefore, the ratio of red balls to the white balls is 4:9.
Practice Questions
1. Find the equivalent ratio of 5:20.
2. What is the equivalent ratio of ¾?
3. Are the ratios 1:2 and 2:3 equivalent?
4. 5 bowls cost Rs.100. What would be the cost of 8 such bowls?
Frequently Asked Questions on Equivalent Ratios
What are equivalent ratios?
When the comparison of two different ratios is same, the such ratios are called equivalent ratios. For example, 1:2 and 3:6 are equivalent.
How can we find the equivalent ratio of 6:4?
To find the equivalent ratio of 6:4, convert the ratio into fraction and then multiply and divide the fraction by a common factor.
6:4 = 6/4 x (2/2) = 12/8
Thus, 12/8 is equivalent to 6:4.
Are 30 : 20 and 24 : 16 equivalent ratios?
30:20 and 24:16 are equivalent ratios, since the lowest form of both ratios is 3:2.
What is the simplest form of 14:21?
The simplest form of 14:21 is ⅔. |
# (Fractions) How many cookies did Meiling bake
• MHB
• Johnx1
In summary, Meiling baked some cookies. She gave away 3/4 of the butter cookies and 1/4 of the green tea cookies. After she ate 9 green tea cookies, she had an equal number of butter cookies and green tea cookies left.
Johnx1
My work:
Number of butter cookies = B
Number of Green Tea cookies = G
We know there are 2/7G of the total cookies because 5/7B
Then I did 3/4B = 1/4G - 9.
I tried different ways, but I couldn't find a correct way to do it.
Johnx said:
My work:
Number of butter cookies = B
Number of Green Tea cookies = G
We know there are 2/7G of the total cookies because 5/7B
Then I did 3/4B = 1/4G - 9.
I tried different ways, but I couldn't find a correct way to do it.
let x = total number of cookies
number of butter cookies = 5x/7
number of tea cookies = 2x/7
number of butter cookies left = (1/4)(5x/7) = 5x/28
number of tea cookies left = (3/4)(2x/7) - 9 = 3x/14 - 9
5x/28 = 3x/14 - 9
5x = 6x - 252
x = 252
check ...
B = (5/7)252 = 180
T = (2/7)252 = 72
gave away (3/4)180 = 135 ... number of butter cookies left = 45
gave away (1/4)72 + ate 9 = 27 ... number of tea cookies left = 45
skeeter said:
number of butter cookies left = (1/4)(5x/7) = 5x/28
number of tea cookies left = (3/4)(2x/7) - 9 = 3x/14 - 9
I see where I made the mistake. I had to times the number of cookies that were left.
Thank you.
## 1. How do you calculate fractions?
In order to calculate fractions, you need to divide the top number (numerator) by the bottom number (denominator). For example, if Meiling baked 3/4 of a batch of cookies, you would divide the total number of cookies by 4 and then multiply by 3 to get the number of cookies Meiling baked.
## 2. What is the difference between a proper and improper fraction?
A proper fraction is one where the numerator is smaller than the denominator, for example 1/2. An improper fraction is one where the numerator is equal to or larger than the denominator, for example 5/4.
## 3. How can I simplify fractions?
To simplify fractions, you need to find the greatest common factor (GCF) of the numerator and denominator. Then, divide both numbers by the GCF to get the simplified fraction. For example, if Meiling baked 12/24 of a batch of cookies, the GCF is 12 and the simplified fraction would be 1/2.
## 4. What is the relationship between fractions and decimals?
Fractions and decimals are two different ways of representing the same value. Fractions use a numerator and denominator, while decimals use a decimal point and digits after the decimal point. You can convert between fractions and decimals by dividing the numerator by the denominator.
## 5. How many cookies did Meiling bake if she used 1/3 of the batch of cookie dough?
If Meiling used 1/3 of the batch of cookie dough, she baked 1/3 of the total number of cookies. To find the number of cookies she baked, you would multiply the total number of cookies by 1/3. For example, if there were 36 cookies in the batch, Meiling would have baked 12 cookies (36 x 1/3 = 12).
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Paul's Online Notes
Home / Calculus I / Review / Trig Equations with Calculators, Part I
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### Section 1.5 : Solving Trig Equations with Calculators, Part I
2. Find the solution(s) to $$\displaystyle 6 + 5\cos \left( {\frac{x}{3}} \right) = 10$$ that are in $$\left[ {0,38} \right]$$. Use at least 4 decimal places in your work.
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Hint : Find all the solutions to the equation without regard to the given interval. The first step in this process is to isolate the cosine (with a coefficient of one) on one side of the equation.
Start Solution
Isolating the cosine (with a coefficient of one) on one side of the equation gives,
$\cos \left( {\frac{x}{3}} \right) = \frac{4}{5}$
Hint : Using a calculator and your knowledge of the unit circle to determine all the angles in the range $$\left[ {0,2\pi } \right]$$ for which cosine will have this value.
Show Step 2
First, using our calculator we can see that,
$\frac{x}{3} = {\cos ^{ - 1}}\left( {\frac{4}{5}} \right) = 0.6435$
Now we’re dealing with cosine in this problem and we know that the $$x$$-axis represents cosine on a unit circle and so we’re looking for angles that will have a $$x$$ coordinate of $$\frac{4}{5}$$. This means that we’ll have angles in the first (this is the one our calculator gave us) and fourth quadrant. Here is a unit circle for this situation.
From the symmetry of the unit circle we can see that we can either use –0.6435 or $$2\pi - 0.6435 = 5.6397$$ for the second angle. Each will give the same set of solutions. However, because it is easy to lose track of minus signs we will use the positive angle for our second solution.
Hint : Using the two angles above write down all the angles for which cosine will have this value and use these to write down all the solutions to the equation.
Show Step 3
From the discussion in the notes for this section we know that once we have these two angles we can get all possible angles by simply adding “$$+ \,2\pi n$$ for $$n = 0, \pm 1, \pm 2, \ldots$$” onto each of these.
This then means that we must have,
$\frac{x}{3} = 0.6435 + 2\pi n\hspace{0.25in}{\mbox{OR }}\hspace{0.25in}\frac{x}{3} = 5.6397 + 2\pi n\hspace{0.25in}n = 0, \pm 1, \pm 2, \ldots$
Finally, to get all the solutions to the equation all we need to do is multiply both sides by 3 and we’ll convert everything to decimals to help with the final step.
\begin{align*}x & = 1.9305 + 6\pi n & \hspace{0.25in}{\mbox{OR }}\hspace{0.25in} x & = 16.9191 + 6\pi n\hspace{0.25in} & n = 0, \pm 1, \pm 2, \ldots \\ & = 1.9305 + 18.8496n & \hspace{0.25in}{\mbox{OR }}\hspace{0.35in} & = 16.9191 + 18.8496n\hspace{0.25in} & n = 0, \pm 1, \pm 2, \ldots \end{align*}
Hint : Now all we need to do is plug in values of $$n$$ to determine which solutions will actually fall in the given interval.
Show Step 4
Now let’s find all the solutions. First notice that, in this case, if we plug in negative values of $$n$$ we will get negative solutions and these will not be in the interval and so there is no reason to even try these. So, let’s start at $$n = 0$$ and see what we get.
$\begin{array}{lclcl}{n = 0:} & \hspace{0.25in} & {x = 1.9305\,}& \hspace{0.25in} {{\mbox{OR}}} \hspace{0.25in} &{x = 16.9191}\\{n = 1:} & \hspace{0.25in} & {x = 20.7801\,}& \hspace{0.25in} {{\mbox{OR}}} \hspace{0.25in} &{x = 35.7687}\end{array}$
Notice that with each increase in $$n$$ we were really just adding another 18.8496 onto the previous results and by doing this to the results from the $$n = 1$$ step we will get solutions that are outside of the interval and so there is no reason to even plug in $$n = 2$$.
So, it looks like we have the four solutions to this equation in the given interval.
$\require{bbox} \bbox[2pt,border:1px solid black]{{x = 1.9305,\,\,16.9191,\,\,20.7801,\,\,35.7687}}$
Note that depending upon the amount of decimals you used here your answers may vary slightly from these due to round off error. Any differences should be slight and only appear around the 4th decimal place or so however. |
We come across different mathematical operations while dealing with numbers. We say that basic four operations are as follows: Addition, Subtraction, Multiplication and division.
By Addition we mean to find the sum of given two numbers. Now let us look at inverse operation of addition. Subtraction is the inverse operation of Addition, which simply means that if we add two Integers 'a' and 'b', and get the result as “c”, then we also can write above relation in reverse order as follows:
c – a = b and c – b = a,
Thus we come to the conclusion that Subtraction is the inverse operation of Addition. It also indicates that if same number is once added and then same number is subtracted from the given number, then result will remain exactly same.
In the same way we also say that if any of the number is added to its additive inverse, then result we get will be zero. It will be clear by following example:
Let us consider the number +5, so additive inverse of the number 5 will be – 5. Now if we will add original number and its additive inverse we get always zero. It can be written as follows: +5 – 5 = 0.
On the other hand we say that inverse of any of negative number will be a positive number, so if we have a number say -4, then its additive inverse will be – (-4) = 4, which is a positive number.
So we say that zero is the additive identity of any given number. It simply means that if we will add zero to any number, it always results in the same number itself. Mathematically it is represented as follows:
If “a” is any number, then we write:
a + 0 = a. |
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
3.2: Function Notation
Difficulty Level: At Grade Created by: CK-12
What is Function Notation?
Objectives
The lesson objective for Function Notation is:
• Understanding function notation.
Introduction
If you think of the relationship between two quantities, you can think of this relationship in terms of a ‘function machine’. The function machine shows how the function responds to various functions. If I triple the input and subtract one, the machine will respond to the function in such a way that it will convert x\begin{align*}x\end{align*} into 3x1\begin{align*}3x - 1\end{align*}. If the function is f\begin{align*}f\end{align*}, and 3 is fed into the machine, 3(3)1=8\begin{align*}3(3) - 1 = 8\end{align*} comes out.
Watch This
Guidance
To represent functions as equations, the symbol f(x)\begin{align*}f(x)\end{align*} is often used. The symbol f(x)\begin{align*}f(x)\end{align*} is pronounced as “f\begin{align*}f\end{align*} of x\begin{align*}x\end{align*}.” This means that the equation is a function that is written in the terms of the variable x\begin{align*}x\end{align*}. An example of such a function is f(x)=3x+4\begin{align*}f(x) = 3x+4\end{align*}. Functions can also be written using a letter other than f\begin{align*}f\end{align*} and a variable other than x\begin{align*}x\end{align*}. An example of such a case is v(t)=2t25\begin{align*}v(t) = 2t^2 - 5\end{align*} and d(h)=4h3\begin{align*}d(h) = 4h-3\end{align*}. In addition to representing functions as equations, you have been shown a variety of ways by which a function can be represented. These ways included:
• As a graph
• As ordered pairs
• As a table of values
• As an arrow or mapping diagram
• As mapping notation
When a function is represented as an equation, an ordered pair can be determined by evaluating various values of the assigned variable. If f(x)=3x4\begin{align*}f(x)=3x-4\end{align*}, then f(4)=?\begin{align*}f(4)=?\end{align*}
f(x)f(x)f(4)f(4)=3x4=3(4)4=124=8\begin{align*}f(x) &= 3x-4\\ f(x) & = 3(4) - 4\\ f(4) & = 12-4\\ f(4) & = 8\end{align*}
Graphically, if f(4)=8\begin{align*}f(4) = 8\end{align*}, this means that the point (4, 8) is a point on the graph of the line.
Example A
If f(x)=x2+2x+5\begin{align*}f(x) = x^2 + 2x +5\end{align*} find.
a) f(2)\begin{align*}f(2)\end{align*}
b) f(7)\begin{align*}f(-7)\end{align*}
c) f(1.4)\begin{align*}f(1.4)\end{align*}
To determine the value of the function for the assigned values of the variable, substitute the values into the function.
\begin{align*}& f(x) = x^2 + 2x+5 && \quad f(x) = x^2+2x+5 && \quad f(x)=x^2+2x+5\\ & {\color{red}\downarrow} \qquad \ \ {\color{red}\downarrow} \qquad {\color{red}\searrow} && \quad \ {\color{red}\downarrow} \qquad \ \ {\color{red}\downarrow} \qquad \ {\color{red}\searrow} && \quad \ \ {\color{red}\downarrow} \qquad \ {\color{red}\downarrow} \qquad \ {\color{red}\searrow}\\ & f(2) =(2)^2 +2(2) + 5 && \ f(-7) = (-7)^2+2(-7)+5 && \ f(1.4) = (1.4)^2+2(1.4) + 5\\ & f(2) = 4 + 4 + 5 && \ f(-7) = 49 - 14 +5 && \ f(1.4)=1.96 +2.8+5\\ & \boxed{f(2)=13} && \boxed{f(-7)=40} && \boxed{f(1.4) = 9.76}\end{align*}
Example B
Functions can also be represented as mapping rules. If \begin{align*}g(x)\rightarrow 5-2x\end{align*} find the following in simplest form:
a) \begin{align*}g(y)\end{align*}
b) \begin{align*}g(y-3)\end{align*}
c) \begin{align*}g(2y)\end{align*}
\begin{align*}& \ g(x) \rightarrow 5 -2x && \qquad g(x) \rightarrow 5-2x \mapsto\\ & \ {\color{red}\downarrow} \qquad \qquad \quad {\color{red}\downarrow} && \qquad \ \ {\color{red}\downarrow} \qquad \qquad \ {\color{red}\downarrow}\\ & \ g(y) \rightarrow 5 - 2(y) && \ g(y-3) \rightarrow 5-2 (y-3) && \text{Apply the distributive property}\\ & \boxed{g(y) \rightarrow 5 - 2y} && \ g(y-3) \rightarrow 5-2y+6 && \text{Combine like terms}\\ & && \boxed{g(y-3) \rightarrow 11-2y}\\ \\ & \ \ g(x) \rightarrow 5-2x\\ & \ g(2y) \rightarrow 5-2(2y)\\ & \boxed{g(2y) \rightarrow 5-4y}\end{align*}
Example C
If \begin{align*}P(a)=\frac{2a-3}{a+2}\end{align*}
a) evaluate
i) \begin{align*}P(0)\end{align*}
ii) \begin{align*}P(1)\end{align*}
iii) \begin{align*}P \left ( -\frac{1}{2} \right )\end{align*}
b) Find a value of ‘\begin{align*}a\end{align*}’ where \begin{align*}P(a)\end{align*} does not exist.
c) Find \begin{align*}P(a-2)\end{align*} in simplest form
d) Find ‘\begin{align*}a\end{align*}’ if \begin{align*}P(a)=-5\end{align*}
a) \begin{align*}& \ P(a) = \frac{2a-3}{a+2} && \ P(a) =\frac{2a-3}{a+2} && \qquad \ P(a)=\frac{2a-3}{a+2}\\ & \ P(0) =\frac{2(0)-3}{(0)+2} && \ P(1) = \frac{2(1)-3}{(1)+2} && \ P\left ( -\frac{1}{2} \right ) = \frac{2\left( -\frac{1}{2} \right )-3}{\left ( -\frac{1}{2} \right ) + 2}\\ & \boxed{P(0) = \frac{-3}{+2}} && \ P(1) = \frac{2-3}{1+2} && \ P \left ( -\frac{1}{2} \right ) = \frac{^1\cancel{2}\left ( -\frac{1}{\cancel{2}} \right )-3}{-\frac{1}{2} + \frac{4}{2}}\\ & && \boxed{P(1)=\frac{-1}{3}} && \ \ P \left ( -\frac{1}{2} \right ) = \frac{-1-3}{\frac{3}{2}}\\ & && && \ P\left ( -\frac{1}{2} \right ) = -4 \div \frac{3}{2}\\ & && && \ P \left ( -\frac{1}{2} \right ) = -4\left ( \frac{2}{3} \right )\\ & && && \boxed{P\left ( -\frac{1}{2} \right )} = \frac{-8}{3}\end{align*}
b) \begin{align*}P(a) = \frac{2a-3}{a+2}\end{align*} The function will not exist if the denominator equals zero.
\begin{align*}& \quad \ \ a+ 2 = 0\\ & a+2-2=0-2\\ & \qquad \quad \ \boxed{a=-2}\end{align*}
\begin{align*}& P(a) = \frac{2a-3}{(-2)+2}\\ & P(a) = \frac{2a-3}{0} && \text{Division by zero is undefined.}\end{align*}
Therefore, if \begin{align*}a=-2\end{align*}, then \begin{align*}P(a)=\frac{2a-3}{a+2}\end{align*} does not exist.
c) \begin{align*}& \qquad P(a) = \frac{2a-3}{a+2}\\ & \ P(a-2) = \frac{2(a-2)-3}{(a-2)+2} && \text{Substitue } a-2' \text{ for } a'\\ & \ P(a-2) = \frac{2a-4-3}{a-2+2} && \text{Remove brackets}\\ & \ P(a-2) = \frac{2a-7}{a} && \text{Combine like terms}\\ & \ P(a-2) = \frac{2\cancel{a}}{\cancel{a}} - \frac{7}{a} && \text{Express the fraction as two separate fractions and reduce.}\\ & \boxed{P(a-2) = 2-\frac{7}{a}}\end{align*}
d) \begin{align*}& \qquad \qquad \quad P(a) = \frac{2a-3}{a+2}\\ & \qquad \qquad \quad \ -5 = \frac{2a-3}{a+2} && \text{Let } P(a) = -5\\ & \qquad \ -5(a+2) = \left ( \frac{2a-3}{a+2} \right )(a+2) && \text{Multiply both sides by } (a+2)\\ & \qquad \ -5a -10 = \left ( \frac{2a-3}{\cancel{a+2}} \right ) (\cancel{a+2}) && \text{Simplify}\\ & \qquad \ -5a -10 = 2a-3 && \text{Solve the linear equation}\\ & -5a -10 -2a = 2a-2a-3 && \text{Move } 2a \text{ to the left by subtracting}\\ & \qquad \ -7a-10 = -3 && \text{Simplify}\\ & -7a-10+10 = -3+10 && \text{Move 10 to the right side by addition}\\ & \qquad \qquad \ -7a = 7 && \text{Simplify}\\ & \qquad \qquad \ \ \frac{-7a}{-7} = \frac{7}{-7} && \text{Divide both sides by -7 to solve for } a'.\\ & \qquad \qquad \qquad \boxed{a=-1}\end{align*}
Example D
Functions can also be used to represent and to solve word problems. The following is an example of this application.
The value \begin{align*}V\end{align*} of a digital camera \begin{align*}t\end{align*} years after it was bought is represented by the function \begin{align*}V(t) = 875 - 50t\end{align*}
a) Determine the value of \begin{align*}V(4)\end{align*} and explain what the solution mean to this problem.
b) Determine the value of \begin{align*}t\end{align*} then \begin{align*}V(t) = 525\end{align*} and explain what this represents.
c) What was the original cost of the digital camera?
a) \begin{align*}& \ V(t) = 875 - 50t\\ & \ V(4) = 875 - 50(4)\\ & \ V(4) = 875-200\\ & \boxed{V(4) = \ 675}\end{align*}
This means that the camera is valued at 675, 4 years after it was purchased. b) \begin{align*}& \qquad \ V(t) = 875 - 50t && \text{Let } V(t) = 525\\ & \qquad \ \ 525 = 875-50t && \text{Solve the equation}\\ & 525 -875 = 875 - 875 - 50t\\ & \quad \ -350 = - 50t\\ & \quad \ \ \frac{-350}{-50} = \frac{-50t}{-50}\\ & \qquad \quad \ \boxed{7 = t}\end{align*} The digital camera has a value of525, 7 years after it was purchased.
c) \begin{align*}& \ V(t) = 875 - 50t && \text{Let } t = 0.\\ & \ V(0) = 875 - 50(0)\\ & \ V(0) = 875 -0\\ & \boxed{V(0) = \875}\end{align*}
The original cost of the camera was 875. Vocabulary Function A function is a set of ordered pairs \begin{align*}(x, y)\end{align*} that shows a relationship where there is only one output for every input. In other words, for every value of \begin{align*}x\end{align*}, there is only one value for \begin{align*}y\end{align*}. Guided Practice 1. If \begin{align*}f(x)=3x^2-4x+6\end{align*} find: i) \begin{align*}f(-3)\end{align*} ii) \begin{align*}f(a-2)\end{align*} 2. If \begin{align*}f(m)=\frac{m+3}{2m-5}\end{align*} find ‘\begin{align*}m\end{align*}’ if \begin{align*}f(m) = \frac{12}{13}\end{align*} 3. The emergency brake cable in a truck parked on a steep hill breaks and the truck rolls down the hill. The distance in feet, \begin{align*}d\end{align*}, which the truck rolls, is represented by the function \begin{align*}d = f(t)=0.5t^2\end{align*}. i) How far will the truck roll after 9 seconds? ii) How long will it take the truck to hit a tree which is at the bottom of the hill 600 feet away? Round your answer to the nearest second. Answers 1. \begin{align*}f(x) = 3x^2 - 4x + 6\end{align*} i) \begin{align*}& \quad f(x) = 3x^2-4x+6 && \text{Substitute }(-3) \text{ for } x' \text{ in the function.}\\ & \ f({\color{red}-3}) = 3({\color{red} -3})^2 -4({\color{red}-3})+6 && \text{Perform the indicated operations.}\\ & \ f(-3) = 3({\color{red}9}) + 12 + 6 && \text{Simplify}\\ & \ f(-3) = 27 + 12 + 6\\ & \ f(-3) = {\color{red}45}\\ & \boxed{f(-3) = 45} \end{align*} ii) \begin{align*}& \qquad f(x) = 3x^2 - 4x +6\\ & \ f({\color{red}a-2}) = 3({\color{red}a-2})^2 -4 ({\color{red}a-2}) + 6 && \text{Write } (a-2)^2 \text{ in expanded form.}\\ & \ f({\color{red}a-2}) = 3({\color{red}a-2})({\color{red}a-2}) - 4({\color{red}a-2})+6 && \text{Perform the indicated operations.}\\ & \ f({\color{red}a-2}) = ({\color{red}3a-6})({\color{red}a-2}) - 4({\color{red}a-2})+6\\ & \ f(a-2) = {\color{red}3a^2-6a-6a+12-4a+8}+6 && \text{Simplify}\\ & \ f(a-2) = {\color{red}3a^2-16a+26}\\ & \boxed{f(a-2) = 3a^2-16a+26}\end{align*} 2. \begin{align*}& \qquad \qquad \ \ f(m) = \frac{m+3}{2m-5}\\ & \qquad \qquad \quad \ \ {\color{red}\frac{12}{13}} = \frac{m+3}{2m-5} && \text{Solve the equation for } m'.\\ & {\color{red}(13)(2m-5)} \frac{12}{13} = {\color{red}(13)(2m-5)} \frac{m+3}{2m-5}\\ & {\color{red}\cancel{(13)} (2m-5)} \frac{12}{\cancel{13}} = {\color{red}(13)\cancel{(2m-5)}} \frac{m+3}{\cancel{2m-5}}\\ & \qquad {\color{red}(2m-5)} 12 = {\color{red}(13)} m+3\\ & \qquad \ \ 24m-60 = 13m+39\\ & \ \ 24m-60 {\color{red}+60} = 13m + 39 {\color{red}+60}\\ & \qquad \qquad \ \ 24m = 13m+99\\ & \quad \quad 24m {\color{red}-13m} = 13m {\color{red}-13m} + 99\\ & \qquad \qquad \ \ 11m = 99\\ & \qquad \qquad \ \frac{11m}{{\color{red}11}} = \frac{99}{{\color{red}11}}\\ & \qquad \qquad \ \frac{\cancel{11}m}{{\color{red}\cancel{11}}} = \frac{\overset{9}{\cancel{99}}}{{\color{red}\cancel{11}}}\\ & \qquad \qquad \quad \boxed{m=9}\end{align*} 3. \begin{align*}d=f(t)=0.5^2\end{align*} i) \begin{align*}& \quad \ \ d =f(t)=0.5^2 && \text{Substitute 9 for } t'.\\ & \ f({\color{red}9}) = 0.5 ({\color{red}9})^2 && \text{Perform the indicated operations.}\\ & \ f(9) = 0.5 ({\color{red}81})\\ & \boxed{f(9)=40.5 \ feet}\end{align*} After 9 seconds, the truck will roll 40.5 feet. ii) \begin{align*}& d= f(t) = 0.5t^2 && \text{Substitute 600 for } d'.\\ & \qquad {\color{red}600} = 0.5t^2 && \text{Solve for } t'.\\ & \quad \ \ \frac{600}{{\color{red}0.5}} = \frac{0.5t^2}{{\color{red}0.5}}\\ & \quad \ \ \frac{\overset{{\color{red}1200}}{\cancel{600}}}{{\color{red}\cancel{0.5}}} = \frac{\cancel{0.5}t^2}{{\color{red}\cancel{0.5}}}\\ & \quad \ 1200 = t^2\\ & \ \sqrt{{\color{red}1200}} = \sqrt{{\color{red}t^2}}\\ & \boxed{34.64 \ seconds \approx t}\end{align*} The truck will hit the tree in approximately 35 seconds. Summary Using function notation to represent a function is similar to expressing the function as a formula. The formula is used to determine the value of a function for a given value. The formula can also be used to calculate the value of the variable when a value is given for the entire function. These concepts can then be applied to real life situations to determine the value of an indicated variable. Problem Set 1. If \begin{align*}g(x)=4x^2-3x+2\end{align*}, find expressions for the following: 1. \begin{align*}g(a)\end{align*} 2. \begin{align*}g(a-1)\end{align*} 3. \begin{align*}g(a+2)\end{align*} 4. \begin{align*}g(2a)\end{align*} 5. \begin{align*}g(-a)\end{align*} 2. If \begin{align*}f(y) = 5y-3\end{align*}, determine the value of ‘\begin{align*}y\end{align*}’ when: 1. \begin{align*}f(y) = 7\end{align*} 2. \begin{align*}f(y) = -1\end{align*} 3. \begin{align*}f(y) = -3\end{align*} 4. \begin{align*}f(y) = 6\end{align*} 5. \begin{align*}f(y) = -8\end{align*} Solve the following problem: The value of a Bobby Orr rookie card \begin{align*}n\end{align*} years after its purchase is \begin{align*}V(n)=520+28n\end{align*} i) Determine the value of \begin{align*}V(6)\end{align*} and explain what the solution means. ii) Determine the value of \begin{align*}n\end{align*} when \begin{align*}V(n)=744\end{align*} and explain what this represents. iii) Determine the original price of the card. Answers 1. If \begin{align*}g(x)=4x^2-3x+2\end{align*}, find... (a) \begin{align*}g(a)\end{align*} \begin{align*}& \ g(x) = 4x^2 - 3x +2\\ & \ g({\color{red}a}) = 4({\color{red}a})^2 - 3{\color{red}a} + 2\\ & \ g({\color{red}a}) = 4{\color{red}a}^2 - 3{\color{red}a} + 2\\ & \boxed{g(a) = 4a^2 - 3a + 2}\end{align*} (c) \begin{align*}g(a + 2)\end{align*} \begin{align*}& \qquad g(x) = 4x^2-3x+2\\ & \ g({\color{red}a+2}) = 4({\color{red}a+2})^2 - 3({\color{red}a+2}) + 2\\ & \ g({\color{red}a+2}) = 4({\color{red}a+2}) ({\color{red}a+2}) - 3 ({\color{red}a+2})+2\\ & \ g({\color{red}a+2}) = ({\color{red}4a+8}) ({\color{red}a+2}) - 3 ({\color{red}a+2})+2\\ & \ g({\color{red}a+2}) = {\color{red}4a^2+8a+8a+16-3a-6} + 2\\ & \ g({\color{red}a+2}) = {\color{red}4a^2+13a+12}\\ & \boxed{g(a+2) = 4a^2 +13a + 12}\end{align*} (e) \begin{align*}g(-a)\end{align*} \begin{align*}& \quad g(x) = 4x^2 - 3x+2\\ & \ g({\color{red}-a}) = 4({\color{red}-a})^2 - 3({\color{red}-a})+2\\ & \ g({\color{red}-a}) = 4{\color{red}a}^2 {\color{red}+} 3{\color{red}a} + 2\\ & \boxed{g(-a) = 4a^2 + 3a + 2}\end{align*} 2. If \begin{align*}f(y)=5y-3\end{align*} determine... (a) \begin{align*}f(y) = 7\end{align*} \begin{align*}& f(y) = 5y - 3\\ & \quad \ {\color{red}7} = 5y - 3\\ & 7 {\color{red}+3} = 5y - 3 {\color{red}+3}\\ & \ \ {\color{red}10} = 5y\\ & \ \frac{10}{{\color{red}5}} = \frac{5y}{{\color{red}5}}\\ & \ \frac{\overset{2}{\cancel{10}}}{{\color{red}\cancel{5}}} = \frac{\cancel{5}y}{{\color{red}\cancel{5}}}\\ & \quad {\color{red}2} = y\\ & \ \ \boxed{2 = y}\end{align*} (c) \begin{align*}f(y) = -3\end{align*} \begin{align*}& \quad f(y) = 5y - 3\\ & \quad \ \ {\color{red}-3} = 5y - 3\\ & -3 {\color{red}+3} = 5y - 3 {\color{red}+3}\\ & \qquad \ {\color{red}0} = 5y\\ & \qquad \frac{0}{{\color{red}5}} = \frac{5y}{{\color{red}5}}\\ & \qquad \frac{\overset{0}{\cancel{0}}}{{\color{red}\cancel{5}}} = \frac{\cancel{5}y}{{\color{red}\cancel{5}}}\\ & \qquad \ {\color{red}0} = y\\ & \qquad \boxed{0 = y}\end{align*} (e) \begin{align*}f(y) = -8\end{align*} \begin{align*}& \quad f(y) = 5y-3\\ & \quad \ \ {\color{red}-8} = 5y - 3\\ & -8 {\color{red}+3} = 5y -3 {\color{red}+3}\\ & \quad \ \ {\color{red}-5} = 5y\\ & \quad \ \frac{-5}{{\color{red}5}} = \frac{5y}{{\color{red}5}}\\ & \quad \ \frac{\overset{-1}{\cancel{-5}}}{{\color{red}\cancel{5}}} = \frac{\cancel{5}y}{{\color{red}\cancel{5}}}\\ & \quad \ \ {\color{red}-1} = y\\ & \quad \ \boxed{-1 = y}\end{align*} Solve the following... i) \begin{align*}& \ V(n) = 520 + 28n\\ & \ V({\color{red}6}) = 520 + 28 ({\color{red}6})\\ & \ V({\color{red}6}) = 520 + {\color{red}168}\\ & \ V({\color{red}6}) = {\color{red}688}\\ & \boxed{V({\color{red}6}) = 688}\end{align*} The solution represents the value of the card six years after it was purchased. The value of the card is688.00.
ii) \begin{align*}V(n) = 520 + 28n\end{align*}
\begin{align*}& \quad \ \ V(n) = 520 + 28n\\ & \qquad \ {\color{red}744} = 520 + 28n\\ & 744 {\color{red}-520} = 520 {\color{red}-520} + 28n\\ & \qquad \ {\color{red}224} = 28n\\ & \qquad \frac{224}{{\color{red}28}} = \frac{28n}{{\color{red}28}}\\ & \qquad \frac{\overset{8}{\cancel{224}}}{{\color{red}\cancel{28}}} = \frac{\cancel{28}n}{{\color{red}\cancel{28}}}\\ & \qquad \quad {\color{red}8} = n\\ & \qquad \ \ \boxed{8 = n}\end{align*}
\begin{align*}v(n) = 744\end{align*}. This represents the value of the card 8 years after it was purchased.
iii) \begin{align*}V(n) = 520 + 28n\end{align*}
\begin{align*}& \ V(n) = 520 + 28n\\ & \ V({\color{red}0}) = 250 + 28 ({\color{red}0})\\ & \ V({\color{red}0}) = 520 + {\color{red}0}\\ & \ V({\color{red}0}) = {\color{red}520}\\ & \boxed{V(0) = 520}\end{align*}
The original purchase price of the card was \$520.00.
Summary
In this chapter you learned about linear functions. You learned what is meant by a function, function notation and how to identify a function from a graph. You also learned how to recognize whether or not a graph represented a function or a relation by doing the vertical line test. Using function notation as a formula, you learned to determine the value for an indicated variable and how to evaluate a function for a given value.
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### Probability Fundamentals
A probability is the measurement of what chance some specified outcome has for happening. Probabilities are typically either expressed as a percentage, measured from $0\%$ to $100\%,$ or a fraction $($between $0$ to $1).$
For example, if you roll a single standard die, the probability of an even number is exactly $50\%$ or $\frac{1}{2}$ since, of the $6$ possible outcomes for what might be rolled, $3$ are even and $3$ are odd.
For the problems in this sequence, your job is to intuit if the event that's specified is less than, more than, or exactly $50\%$ likely to occur. These next five questions are meant only to test your intuition, but you can find the complete explanations if you're curious. We'll move on and introduce how to calculate probabilities numerically in the next two sets of challenges!
Anela throws a standard six-sided die $($marked from $1$ to $6)$ while Zeke throws a standard eight-sided die $($marked from $1$ to $8).$
The probability Zeke throws a larger number than Anela is $\text{\_\_\_\_\_\_\_\_\_\_}.$
You throw a pair of fair coins. The probability that the result is either both are heads or both are tails is $\text{\_\_\_\_\_\_\_\_\_\_}.$
In the region where Cormac lives, it is sunny three-quarters of the days and rains for the rest.
Every day Cormac makes a weather prediction by flipping a fair coin; heads mean predicting sun and tails mean predicting rain.
The probability Cormac will correctly predict the weather this way on a particular day is $\text{\_\_\_\_\_}.$
A standard deck with $52$ cards — $26$ red and $26$ black — is shuffled.
The probability that the top card is the same color as the bottom card is $\text{\_\_\_\_\_\_\_\_\_\_}.$
Let's modify the problem you saw earlier with Anela and Zeke slightly:
• Zeke still throws a standard eight-sided die $($marked from $1$ to $8).$
• However, Anela now throws a non-standard six-sided die that is marked from $2$ to $7$.
Which of these events are more likely to occur?
• Zeke throws a larger number than Anela.
• Zeke throws a smaller number than Anela. |
# Into Math Grade 1 Module 15 Lesson 5 Answer Key Make New Two-Dimensional Shapes
We included HMH Into Math Grade 1 Answer Key PDF Module 15 Lesson 5 Make New Two-Dimensional Shapes to make students experts in learning maths.
## HMH Into Math Grade 1 Module 15 Lesson 5 Answer Key Make New Two-Dimensional Shapes
I Can put combined shapes together to make a new shape.
How can you use combined shapes to make a new shape?
Explanation:
We can use combined shape to make new and larger shapes
I drew to show a new shape by combining shapes.
On 1-inch grid paper, have children color gridded squares to show a combined rectangle or square. Children then repeat the shape by coloring squares a different color. Have children cut out the combined squares. Children will put them together to make a new, larger composite shape.
Build Understanding
Angela combines 6 squares to make a shape.
Then she makes the same shape again.
She combines the shapes to make a new larger shape.
Draw a shape Angela could make.
How can you write an equation to model how
the squares were combined?
____ + ___ = ____
Explanation:
Angela combines 6 squares to make a shape.
Then she makes the same shape again.
She combines the shapes to make a new larger shape
I drew the larger shape she could make
It is a rectangle
6 + 6 = 12
12 squares were combined.
Turn and Talk Could the new larger shape be a square. Explain.
No, the new larger shape is a rectangle.
Step It Out
1. Samantha uses two-dimensional shapes to make larger shapes.
A. She combines 3 triangles to make a trapezoid. Draw the trapezoid.
B. Then she combines 3 triangles to make another trapezoid. Draw the trapezoid.
C. Next she combines the 2 trapezoids to make a hexagon. Draw the hexagon.
Explanation:
I drew a trapezoid by combining 3 triangles and made one more combined shape
I combined the 2 trapezoids to make a hexagon.
Check Understanding Math Board
Question 1.
Combine the 2 shapes below. Make the same shape again. Then combine the two shapes you made. Draw the shape.
Explanation:
I combined the shapes given and drew the same combined shape again
Then i combined both the combined shapes to make a new shape
The new shape is a circle.
Use Repeated Reasoning
Question 2.
• Use 2 rectangles to make a square.
• Make a second square, just like the first.
• Put the 2 squares together to make another new shape.
What is the new shape?
_______________
Explanation:
I used 2 rectangles to make a square
Then i made a second square
Next i put both the squares together to make a new shape
The new shape is a rectangle.
I’m in a Learning Mindset!
How did following directions help me combine shapes?
_______________________
_______________________
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# How to calculate per unit cost?
I need help understanding this math problem from my TEAS Mometrix Practice exam. It gives an explanation but I still don't understand. The question is:
Mandy can buy $4$ containers of yogurt and $3$ boxes of crackers for $\$9.55$. She can buy$2$containers of yogurt and$2$boxes of crackers for$\$5.90$. How much does one box of crackers cost?
The answer is $\$2.25. I understand that I have to set up the problem like this: \begin{align} 4x+3y &= 9.55 \\ 2x+2y &= 5.90 \end{align} but the explanation says that I have to multiply the bottom equation by-2$; therefore, it becomes$-4x+ -4y= -11.80and then I have to add the equations \begin{align} 4x+3y &= 9.55\\ -4x+-4y &= -11.80 \end{align} by top to bottom getting either\pm 2.25$Can you explain why this is the answer? And also why does one have multiply by -2? • First, it's the answer because it's the answer. I'm not sure what else can be said about that. Next, you don't have to multiply the second equation by$-2$. The authors just did that because they used the elimination method to solve the equations simultaneously. You can also use the substition method to solve the equations and it will give you the same answer. – Mattos Jul 17 '17 at 14:23 • Why do you think it is$\pm 2.25$? – Ennar Jul 17 '17 at 14:28 ## 3 Answers "the explanation says that I have to multiply the bottom equation by$−2$". No, you don't have to. That is just one convenient way to solve the problem. Keep in mind that your goal is "How much does one box of crackers cost?", namely, the value of$y. then I have to add the equations \begin{align} 4x+3y &= 9.55\\ -4x+-4y &= -11.80 \end{align} by top to bottom getting either\pm 2.25$. What you get should be$-y=-2.25$which implies that$y=2.25$. "Can you explain why this is the answer?"$y=\pm 2.25$is not an answer.$y=2.25is. If that is what is written, it must be a typo. One possible solution is the use of Gaussian elimination. That is the solution you suggest in your question. Note that there are basically three operations that you can apply to a system, that don't change the solution. • Type 1: Swap the positions of two equations. • Type 2: Multiply an equation by a nonzero number. • Type 3: Add to one equation to the multiple of another. That these operations don't change the solution of the system is illustrated in this nice answer. Let's start from the beginning with: \begin{align*} 4x+3y=&9.55 &&(I)\\ 2x+2y=&5.90 &&(II) \end{align*} 1. Using an operation of Type 2, will result in: \begin{align*} 4x+3y=&9.55 &&(I)\\ 4x+4y=&11.80 &&(III) = 2\cdot(I) \end{align*} 1. Using an operation of Type 3, will result in: \begin{align*} 4x+3y&=9.55 &&(I)\\ y&=2.25 &&(IV) = (III) - (I) \end{align*} 2. Now we can simply plug iny=2.25$into (I) and yield: $$4x=9.55 - 3\cdot 2.25= 2.8 \qquad ⇒ x = 0.7$$ I have to multiply the bottom equation by −2 That is not true. In the first step above, we multiplied equation (II) with 2. But we can also multiply the first one with$\frac{1}{2}: \begin{align*} 2x+1.5y=&4.775 &&(III)=\frac{1}{2}\cdot(I)\\ 2x+2y=&5.90 &&(II) \end{align*} Then you can do the second step as well, right? The reason why the answer is given like that is, that Gaussian elimination is usually taught in the way, that the resulting equations form the so called row echelon form. The idea of the row echelon form is to use the three operations above to get a system that looks like this. So the last row contains one variable, the second to last row contains two variables etc. But, as always, it does not matter what way you choose, to get the result. There are also different ways to solve it, for example you can use (II) to expressy$as a variable depending on$x$$y = \frac{5.90 - 2x}{2}= \frac{5.90}{2} - x = 2.95 - x \qquad (*)$$ and then plug that into (I): \begin{align*} &&4x+3y&= 9.55 \\ ⇔&& 4x + 3\cdot(2.95-x)&=9.55 \\ ⇔&& 4x - 3x &=9.55-8.85 \\ ⇔&& x &= 0.7 \end{align*} Then using(*)$we get$y = 2.25$. • Terrific answer, for what it's worth I upvote you. – Kevin Jul 18 '17 at 7:37 There are several ways to solve this or similar problems we have $$4 \cdot x + 3 \cdot y = 9.55 \\ 2 \cdot x + 2 \cdot y = 5.90$$ One approch as given in your answer is to multiply the second equation by 2 or -2 you then have$4 \cdot x$in both equations so can cancel for$x$by adding the two equations or subtracting the equations as appropriate. The reason this works is that by multiplying all terms by the same number you have created another true statement. This is not the only approach however, Another approach would be for example to solve one equation for say$y\$ then substituting it in the other.
$$2 \cdot x + 2 \cdot y = 5.90 \Rightarrow y = \frac{5.90 - 2 \cdot x}{2}$$
Putting that in the other equation we have
$$4 \cdot x + 3 \cdot y = 4 \cdot x + 3 \cdot \frac{5.90 - 2 \cdot x}{2} = 4 \cdot x + 8.85 - 3 \cdot x= 9.55 \Rightarrow x = 9.55 - 8.85 = 0.7$$
Putting this back in the other equation
$$2 \cdot x + 2 \cdot y = 2 \cdot 0.7 +2 \cdot y = 5.90 \Rightarrow y = \frac{5.9 - 1.4}{2} = 2.25$$ |
# How do you solve y=7x-1 and y=-x+14 using substitution?
Apr 26, 2018
The solution is $\left(\frac{15}{8} , \frac{97}{8}\right)$ or $\left(1.875 , 12.125\right)$.
#### Explanation:
Solve the system:
$\text{Equation 1:}$ $y = 7 x - 1$
$\text{Equation 2:}$ $y = - x + 14$
Both linear equations are in slope-intercept form. The solution to the system is the point that both lines have in common, the point of intersection. Substitution will be used to solve the system.
Both equations are set equal to $y$. Substitute $7 x - 1$ from Equation 1 for $y$ in Equation 2 and solve for $x$.
$y = - x + 14$
$7 x - 1 = - x + 14$
Add $x$ to both sides.
$x + 7 x - 1 = 14$
Simplify.
$8 x - 1 = 14$
Add $1$ to both sides.
$8 x = 15$
Divide both sides by $8$.
$x = \frac{15}{8}$ or $1.875$
Substitute $\frac{15}{8}$ for $x$ in Equation 1 and solve for $y$.
$y = 7 x - 1$
$y = 7 \left(\frac{15}{8}\right) - 1$
$y = \frac{105}{8} - 1$
Multiply $1$ by $\frac{8}{8}$ to get a denominator of $8$.
$y = \frac{105}{8} - 1 \times \frac{8}{8}$
$y = 105 - \frac{8}{8}$
Combine the numerators.
$y = \frac{105 - 8}{8}$
$y = \frac{97}{8}$ or $12.125$
The solution is $\left(\frac{15}{8} , \frac{97}{8}\right)$ or $\left(1.875 , 12.125\right)$.
graph{(y-7x+1)(y+x-14)=0 [-9.455, 10.545, 5.48, 15.48]}
Apr 26, 2018
$\left(x , y\right) : \left(\frac{15}{8} , \frac{97}{8}\right)$
#### Explanation:
$e {q}_{1} : y = 7 x - 1$
$e {q}_{2} : y = - x + 14$
Use $7 x - 1$ for y in $e {q}_{2}$:
$7 x - 1 = - x + 14$
$8 x = 15$
$x = \frac{15}{8}$
$y = 7 x - 1$
$y = 7 \times \frac{15}{8} - \frac{8}{8}$
$y = \frac{97}{8}$ |
## SOLVING PROBLEMS WITH PATTERNS
Example 1 :
For the following matchstick pattern, find the number of matches M required to make the
(a) 8th figure
(b) nth figure
Solution :
By observing the figures
Number of matchstick in the 1st figure = 4
Number of matchstick in the 2nd figure = 10
Number of matchstick in the 3rd figure = 16
Number of matchsticks in every figure is 2 less than the multiple of 6.
So, creating formula for this
number of matchstick in nth figure = 6n-2
(a) using the formula
n = 8
Number of match sticks in 8th figure = 6(8)-2
= 48-2
= 46
(b) Number of matchstick in nth figure = 6n-2
Example 2 :
Consider the pattern :
S1 = 1/1×2
S2 = 1/1×2 + 1/2×3
S3 = 1/1×2 + 1/2×3 + 1/3×4
………….......
.....................
a) Find the values of S1, S2, S3, and S4
b) write down the value of :
(i) S10 (ii) Sn
Solution :
S1 = 1/(1 x 2) = 1/2
S2 = 1/(1 x 2) + 1/(2 x 3) = 1/2 + 1/6 ==> 4/6 => 2/3
S3 = 1/(1x2) + 1/(2x3) + 1/(3x4)
= 1/2 + 1/6 + 1/12
= (6+2+1)/12
= 9/12
= 3/4
Observing the results it is in the form n/(n+1).
So,
Sn = n/n+1
(where n is natural number)
(a)
If n = 1S1 = 1/1+1S1 = 1/2If n = 2S2 = 2/2+1S2 = 2/3 If n = 3S3 = 3/3+1S3 = 3/4If n = 4S4 = 4/4+1S4 = 4/5
(b) (i) If n = 10
S10 = 10/10+1
S10 = 10/11
Sn = n/n+1
(ii) Sn = n/n+1
Example 3 :
Consider the pattern :
S1 = 12
S2 = 12 + 22
S2 = 12 + 22 + 32, …………
a) Check that the formula
Sn = n(n+1) (2n+1)/6
is correct for n = 1, 2, 3 and 4
b) Assuming the formula in a is always true, find the sum of
12+22+32+42+52+ ………… + 1002
which is the sum of the squares of the first one hundred integers.
Solution :
(a) When n = 1,
Sn = n(n+1) (2n+1)/6
S1 = 1(1+1) (2+1)/6
S1 = 1
It can be written as 12
when n = 2,
S2 = 2(2+1) (4+1)/6
S2 = 5
It can be written as 12 + 22
when n = 3,
S3 = 3(3+1) (6+1)/6
S2 = 84
It can be written as 12 + 22 + 32
Hence its verified.
(ii) Given :
12+22+32+42+52+ ………… + 1002
Here, n = 100
Sn = n(n+1) (2n+1)/6
S100 = 100(100+1) (200+1)/6
S100 = 338350
Example 4 :
Consider the pattern :
N1 = 13
N2 = 13 + 23
N3 = 13 + 23 + 33, …………
a) Verify that the formula
Nn = n2(n+1)2/4
is correct for n = 1, 2, 3 and 4
b) Use the above formula to find the sum of
13 + 23 + 33 + 43 + ………… + 503
c) Find the sum : 23 + 43 + 63 + 83 + ………… + 1003
Solution :
(a) When n = 1,
Nn = n2(n+1)2/4
N1 = 12(1+1)2/4
N1 = 1
It can be written as 13
when n = 2,
N2 = 22(2+1)2/4
N2 = 9
It can be written as 13 + 23
when n = 3,
N3 = 32(3+1)2/4
N3 = 36
It can be written as 13 + 23 + 33
Hence, its verified.
(b) 13 + 23 + 33 + 43 + ………… + 503
here, n = 50
Sn = n2(n+1)2/4
S50 = (50)2(50+1)2/4
S50 = 1625625
c) 23 + 43 + 63 + 83 + ………… + 1003
Factoring 23 from the series.
= 23 (13 + 23 + 33 + 43 + ………… + 503)
= 23(1625625)
= 8(1625625)
= 13005000
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WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Algebra word problems
Word problems on trains
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on unit price
Word problems on unit rate
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Double facts word problems
Trigonometry word problems
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Profit and loss word problems
Markup and markdown word problems
Decimal word problems
Word problems on fractions
Word problems on mixed fractrions
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Time and work word problems
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OTHER TOPICS
Profit and loss shortcuts
Percentage shortcuts
Times table shortcuts
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Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6 |
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Chapter – Combinatorics
[Combinatorics]
Discrete Structures
1
Discrete Structures
Chapter – Combinatorics
Discrete Structures
Elementary Combinatorics
Combinatorics is the study of arrangements or possible combination of objects. We come up with different situations where we need to identify the number of elements having similar features, number of steps required to solve the problem, amount of storage required, etc.
Basics of Counting
There are two basic counting principles that can be used to solve the counting problems. We define those two principles below:
Sum rule: The principle of disjunctive counting.
If the first task can be done in m ways and the second task can be done in n ways and if both the tasks cannot be done at a time, then there are m + n ways to do one of the task. We can generalize this rule as, if a set X is union of disjoint nonempty subsets S 1 , S 2 , …, S n , then |X| = | S 1 | + | S 2 | + … + | S n |. Remember: the set must be disjoint, for overlapping set we use different principle called inclusion exclusion principle (will be covered later). Example 1:
In how many ways we can draw a heart or a diamond from an ordinary deck of playing cards? Solution:
There are total 13 cards of heart and 13 card of diamond. So, by sum rule total number of ways of picking heart or diamond is 13 + 13 = 26. Example 2:
How many ways we can get a sum of 4 or of 8 when two distinguishable dice (say one die is red and the other is white) are rolled? Solution:
Since dice are distinguishable outcome (1, 3) is different form (3, 1) so to get 4 as sum we have the pairs (1, 3), (3, 1), (2, 2), so total of 3 ways. And similarly getting 8 can be from pairs (2, 6), (6, 2), (3, 5), (5, 3), (4, 4), so total 5 ways. Hence getting sum of 4 or 8 is 3 + 5 = 8.
2
Chapter – Combinatorics
Discrete Structures
Product Rule: Principle of sequential counting.
If a work can be done in m ways and another work can be done after the completion of
first work in n ways, then there are m × n ways to do the task that consists both the work. Generalizing the rule, if S 1 , S 2 , …, S n are non empty sets, then the number of elements in
the Cartesian product S 1 × S 2 × × S n , is the product
n
i =
1 |
S
i
|
.
n
i =
1 |
S i | i.e. | S 1 × S 2 × × S n | =
Example 1:
An office building contains 27 floors and has 37 offices on each floor. How many offices are there are in the building? Solution:
By the product rule there are 27.37 = 999 offices in the building. Example 2:
How many different three-letter initials with none of the letters can be repeated can people have? Solution:
Here the first letter can be chosen in 26 ways, since the first letter is assigned we can choose second letter in 25 ways and in the same manner we can choose third letter in 24 ways. So by product rule number of different three-letter initials are 26.25.24 = 15600.
More Examples on Basics:
Example 1:
How many strings are there of four lowercase letters that have the letter x in them? Solution:
There are total 26.26.26.26 strings of four lowercase letters, by product rule. In the same way we can say that there are 25.25.25.25 strings of four lowercase letters without x, since without x there will be a set of 25 characters only. So there are total of 26.26.26.26 - 25.25.25.25 = 66351 four lowercase letter strings with x in them. This is true because we are decrementing total numbers of strings with the number of strings that do not contain x in them so at least one x will be in the strings.
3
Chapter – Combinatorics
Discrete Structures
Example 2:
How many functions are there from the set {1, 2, …, n}, where n is a positive integer, to the set {0, 1}. Solution:
Each element from the set {1, 2, …, n}can map the set {0, 1} in 2 ways. Since there are n elements in the first set by the product rule number of possible functions are 2.2.2… n term i.e. 2 n .
th
Tree Diagrams
We can use a tree diagram to solve the counting problem (don’t worry we will study tree in detail later). Example:
Use a tree diagram to find the number of bit strings of length four with no three consecutive 0s. Solution:
0
1
0
1
0
1
1
0
1
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
From the above tree we can get that there are total number of 13 bit strings of length four with no three consecutive zeroes. For this we can explain as if a bit string start with 1 then there is only one bit string that can have three consecutive 0s (1000), the total number of bit string of length starting with 1 and have no three consecutive 0s is thus 2.2.2 –1 =7, similarly if the bit string start with 0 then there is a possibility that the next two bits may be 0 so the possible bit strings of length four with consecutive 0s starting with 0 are 0001 and 0000, so the total number of bit string of length starting with 1 and have no three consecutive 0s is thus 2.2.2 –2 =6. Using the sum rule the total number of such bit strings is 13.
4
Chapter – Combinatorics
Discrete Structures
Pigeonhole principle
The pigeonhole principle states that if there are more pigeons than pigeonholes, then there must be at least one pigeonhole with at least two pigeons. The concept of pigeons can be extended to any objects. Theorem 1: The pigeonhole principle If k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects. Proof:
We use proof by contradiction here. Suppose that k+1 or more boxes are placed into k boxes and no boxes contain more than one object in it. If there are k boxes then there must be k objects such that there are no two objects in a box. This contradicts our assumption. So there is at least one box containing two or more of the objects. Example:
Show that if there are 30 students in a class, then at least two have last names that begin with the same letter. Proof:
There are 30 students in the class and we have 26 letters in English alphabet that can be used in beginning of the last name. Since there are only 26 letters and 30 students, by pigeonhole principle at least two students have the last name that begins with the same letter. Theorem 2: The generalized pigeonhole principle If N objects are placed into k boxes, then there is at least one box containing at least
N / k
objects.
Proof:
Suppose N objects are placed into k boxes and there is no box containing more than
N / k
-1 objects. So the total number of objects is at most
k(
N / k -1) < k((N/k + 1) –1) = N. This is the contradiction that N objects are placed
into k boxes (since we showed that there are total number of objects less than N). Hence, the proof.
5
Chapter – Combinatorics
Discrete Structures
Example:
If a class has 24 students, what is the maximum number of possible grading that must be
done to ensure that there at least two students with the same grade.
Solution:
There are total 24 students and the class and at least two students must have same grade.
If the number of possible grades is k then by pigeonhole principle we have
Here the largest value that k can have is 23 since 24 = 23.1 + 1. So the maximum number
of possible grading to ensure that at least two of the students have same grading is 23.
24 / k
= 2.
Applications: Pigeonhole principles
Example 1:
How many numbers must be selected from the set {1, 3, 5, 7, 9, 11, 13, 15} to guarantee
that at least one pair of these numbers add up to 16?
Solution:
The pairs of numbers that sum 16 are (1,15), (3, 13), (5, 11), (7, 9) i.e. 4 pairs of numbers
are there that add to 16. If we select 5 numbers then by pigeonhole principle there are at
least
pair. Hence 5 numbers must be selected.
Example 2:
Find the least number of cables required to connect eight computers to four printers to
guarantee that four computers can directly access four different printers. Justify your
Solution:
If we connect first 4 computers directly to each of the 4 printers and the other 4
computers are connected to all the printers, then the number of connection required is 4 +
4.4 = 20. To verify that 20 is the least number of cables required we have if there may be
less than 20 cables then we would have 19 cables, then some printers would be connected
= 4 cables to the computers. Then the other 3 printers would have to
by at most
connect the other 4 computers here all the computers cannot simultaneously access
= 2 numbers, that are from the set of selected 5 numbers, that constitute a
5 / 4
19 / 4
6
Chapter – Combinatorics
Discrete Structures
different printer. So if we use 20 cables, then at least
printer to a computer directly. So the remaining 3 printers are required to connect only 3 computers. Hence the least number of cables required is 20. Example 3:
Among n + 1 different integral powers of an integer a, there are at least two of them that have same remainder when divided by the positive integer n. Proof:
Let a 1 , a 2 , …, a n+1 , be n+1 different integral powers of integer a. when these numbers are divided by n then the set of possible remainders is {0, 1, 2, …, n-1). Since there are n remainders and n+1 numbers by pigeonhole principle at least 2 of the reminders must be same.
20 / 4 = 5 cables connects a
Self Studies
Read chapter 4.1 and 4.2 of your textbook such that you can cover all the read materials in the class.
7 |
# 6.16. AVL Tree PerformanceΒΆ
Before we proceed any further let’s look at the result of enforcing this new balance factor requirement. Our claim is that by ensuring that a tree always has a balance factor of -1, 0, or 1 we can get better Big-O performance of key operations. Let us start by thinking about how this balance condition changes the worst-case tree. There are two possibilities to consider, a left-heavy tree and a right heavy tree. If we consider trees of heights 0, 1, 2, and 3, Figure 2 illustrates the most unbalanced left-heavy tree possible under the new rules.
Figure 2: Worst-Case Left-Heavy AVL Trees
Looking at the total number of nodes in the tree we see that for a tree of height 0 there is 1 node, for a tree of height 1 there is $$1+1 = 2$$ nodes, for a tree of height 2 there are $$1+1+2 = 4$$ and for a tree of height 3 there are $$1 + 2 + 4 = 7$$. More generally the pattern we see for the number of nodes in a tree of height h ($$N_h$$) is:
$N_h = 1 + N_{h-1} + N_{h-2}$
This recurrence may look familiar to you because it is very similar to the Fibonacci sequence. We can use this fact to derive a formula for the height of an AVL tree given the number of nodes in the tree. Recall that for the Fibonacci sequence the $$i_{th}$$ Fibonacci number is given by:
$\begin{split}F_0 = 0 \\ F_1 = 1 \\ F_i = F_{i-1} + F_{i-2} \text{ for all } i \ge 2\end{split}$
An important mathematical result is that as the numbers of the Fibonacci sequence get larger and larger the ratio of $$F_i / F_{i-1}$$ becomes closer and closer to approximating the golden ratio $$\Phi$$ which is defined as $$\Phi = \frac{1 + \sqrt{5}}{2}$$. You can consult a math text if you want to see a derivation of the previous equation. We will simply use this equation to approximate $$F_i$$ as $$F_i = \Phi^i/\sqrt{5}$$. If we make use of this approximation we can rewrite the equation for $$N_h$$ as:
$N_h = F_{h+2} - 1, h \ge 1$
By replacing the Fibonacci reference with its golden ratio approximation we get:
$N_h = \frac{\Phi^{h+2}}{\sqrt{5}} - 1$
If we rearrange the terms, and take the base 2 log of both sides and then solve for $$h$$ we get the following derivation:
$\begin{split}\log{N_h+1} = (H+2)\log{\Phi} - \frac{1}{2} \log{5} \\ h = \frac{\log{N_h+1} - 2 \log{\Phi} + \frac{1}{2} \log{5}}{\log{\Phi}} \\ h = 1.44 \log{N_h}\end{split}$
This derivation shows us that at any time the height of our AVL tree is equal to a constant(1.44) times the log of the number of nodes in the tree. This is great news for searching our AVL tree because it limits the search to $$O(\log{N})$$.
Next Section - 6.17. AVL Tree Implementation |
# How do you combine m/(20+9m+m^2)-4/(12+7m+m^2)?
Aug 18, 2016
$\frac{\left(m - 5\right)}{\left(5 + m\right) \left(3 + m\right)}$
#### Explanation:
Factorise the denominators first.
$\frac{m}{20 + 9 m + {m}^{2}} - \frac{4}{12 + 7 m + {m}^{2}}$
=$\frac{m}{\left(5 + m\right) \left(4 + m\right)} - \frac{4}{\left(3 + m\right) \left(4 + m\right)}$
$\text{Find a common denominator}$
=$\frac{m \left(3 + m\right) - 4 \left(5 + m\right)}{\left(5 + m\right) \left(4 + m\right) \left(3 + m\right)}$
=$\frac{3 m + {m}^{2} - 20 - 4 m}{\left(5 + m\right) \left(4 + m\right) \left(3 + m\right)}$
=$\frac{{m}^{2} - m - 20}{\left(5 + m\right) \left(4 + m\right) \left(3 + m\right)}$
=$\frac{\left(m - 5\right) \left(m + 4\right)}{\left(5 + m\right) \left(4 + m\right) \left(3 + m\right)}$
$\frac{\left(m - 5\right) \cancel{\left(m + 4\right)}}{\left(5 + m\right) \cancel{\left(4 + m\right)} \left(3 + m\right)}$ |
Search
## Graph Theory: Planar Graph & Subgraph and Spanning & Induced Graph
### Graph Theory
Graph Theory is a branch of mathematics that studies graphs. Graphs are abstractions of a social network, the connections between people and events, the connections between pages on a website, or any other kind of connection.
### Planar Graph
A graph is defined as a planar graph if it can be drawn in a plane and no edges intersect each other.
In other words, Planar graph edges intersect only at their endpoints. It can be drawn in such a way that no edges cross each other.
#### Regions of Plane
The representation of planar graphs is divided into connected parts/regions that are known as Regions of the Plane.
#### Degree of Region
• Degree of Interior region = Number of edges enclosing that region
• Degree of Exterior region = Number of edges exposed to that region
Here, this planar graph splits the plane into 3 regions- R1, R2, and R3 where-
Degree (R1) = 3
Degree (R2) = 3
Degree (R3) = 4
#### Planar Graph Chromatic Number
The Chromatic Number of any planar graph is always less than or equal to 4. Thus, any planar graph always requires a maximum of 4 colors for coloring its vertices.
#### Planar Graph Properties
Property 1: The sum of all the degrees of the vertices is twice the number of edges in the graph.
Property 2: The sum of all the degrees of the regions is twice the number of edges in the graph.
Property 3: If G is a Planar graph with ‘e’ edges, ‘v’ vertices and ‘r’ number of regions in the planar representation of G, then-
r = e – v + 2
This is called Euler’s Formula and remains the same in all the planar representations of the graph.
Property 4: If G is a planar graph with k components, then-
r = e – v + (k + 1)
Property 5: If a connected planar graph G has e edges and v vertices, then 3v-e≥6.
Example: Prove that complete graph K4 is planar.
The complete graph K4 contains 4 vertices and 6 edges.
We know that for a connected planar graph 3v-e≥6.Hence for K4, we have 3×4-6=6 which satisfies the property (3).
Thus K4 is a planar graph. Hence Proved.
Property 6: A complete graph Kn is a planar if and only if n<5.
Property 7: A complete bipartite graph Kmn is planar if and only if m<3 or n>3.
#### Non-planar Graph
A graph is defined as a non-planar graph if it can be drawn in a plane with edges that intersect each other.
In other words, It can be drawn in such a way that edges should cross each other.
Example: Show that K5 is non-planar.
The complete graph K5 contains 5 vertices and 10 edges.
Now, for a connected planar graph 3v-e≥6.
Hence, for K5, we have 3 x 5-10=5 (which does not satisfy property 3 because it must be greater than or equal to 6).
Thus, K5 is a non-planar graph.
### Applications of Graph Coloring
Some applications of graph coloring include:
• Register Allocation
• Map Coloring
• Bipartite Graph Checking
• Making a time table, etc.
### Handshaking Theorem
The sum of degrees of all the vertices in a graph G is equal to twice the number of edges in the graph.
Mathematically it can be stated as:
∑v∈Vdeg(v)=2e
Proof: Let G = (V, E) be a graph where V = {v1,v2, . . . . . . . . . .} be the set of vertices and E = {e1,e2 . . . . . . . . . .} be the set of edges. We know that every edge lies between two vertices so it provides degree one to each vertex. Hence each edge contributes degree two for the graph. So the sum of degrees of all vertices is equal to twice the number of edges in G.
Hence, ∑v∈Vdeg(v)=2e
### Subgraph
A graph G’ is said to be a subgraph if it is extracted from graph G where the set of vertices and edges are subsets of graph G.
In other words, if all the vertices and edges of a graph G’ belong to G, and each edge of G’ has the same end vertices in G’ as in G.
Original Subgraph
A graph by deleting the edge between V4 and V5.
A graph by deleting the edge between V4 and V5.
A graph by deleting the edge between V2 and V3.
A graph by deleting the edge from V2 to V4.
Case 1: In any planar graph, if degree of each region is K, then-
K x |R| = 2 x |E|
Case 2: In any planar graph, if degree of each region is at least K (>=K), then-
K x |R| <= 2 x |E|
Case 3: In any planar graph, if degree of each region is at most K (<=K), then-
K x |R| >= 2 x |E|
Points to remember:
• Vertex and edge sets are subsets of those of G
• a supergraph of a graph G is a graph that contains G as a subgraph.
• A graph G contains another graph H if some subgraph of G
• is H or
• is isomorphic to H.
• H is a proper subgraph if H!=G
### Spanning Tree
In graph theory, a spanning tree is a subgraph that is a tree which includes all of the vertices of G. If a vertex is missed, then it is not a spanning tree.
In other words, a spanning tree is a subgraph of G, which includes all the vertices of G with (n-1) edges. Where ‘n’ is no. of vertices. All the possible created spanning trees from the original graph would have the same no. of vertices but the no. of edges would be (n-1).
Original Spanning tree
The total number of spanning trees with n vertices that can be created from a complete graph is equal to n power (n-2).
If we have n = 5, the maximum number of possible spanning trees is equal to 5 power (5-2) = 125. Thus, 125 spanning trees can be formed from a complete graph with 5 vertices.
Fig – Spanning tree 1
Fig – Spanning tree 2
Fig – Spanning tree 3
Points to remember:
• Subgraph H has the same vertex set as G
• Possibly not all the edges
• “H spans G”
Theorem: A graph is connected if and only if it has a spanning tree.
Proof: Let G be a connected graph. Delete edges from G that are not bridges until we get a connected subgraph H in which each edge is a bridge. Then H is a spanning tree. On the other hand, if there is a spanning tree in G, there is a path between any pair of vertices in G; thus G is connected.
#### Applications of the Spanning Tree
Basically, a spanning tree is used to find a minimum path to connect all nodes of the graph. Some of the common applications of the spanning tree are listed as follows –
• Cluster Analysis
• Civil network planning
• Computer network routing protocol
#### Properties of Spanning Tree
• There can be more than one spanning tree of a connected graph G.
• A spanning tree has n-1 edges, where ‘n’ is the number of nodes.
• There can be a maximum n power (n-2) number of spanning trees that can be created from a complete graph.
• A spanning tree does not have any cycles or loop.
• A spanning tree is connected with least no. of edges, so removing one edge from the tree will make the graph disconnected.
• A spanning tree is maximally acyclic, so adding one edge to the tree will create a loop.
• If the graph is a complete graph, then the spanning tree can be constructed by removing maximum (e-n+1) edges, where ‘e’ is the number of edges and ‘n’ is the number of vertices.
So, a spanning tree is a subset of connected graph G, and there is no spanning tree of a disconnected graph.
#### Minimum Spanning Tree
A graph is said to be a minimum spanning tree, if the sum of all the weights of the edges is minimum. The weight of the spanning tree is the distance assigned to each edge connected to the vertices.
Original Minimum Spanning tree
As per the statement, Fig – 2 has the lowest sum of all the edges weight among all the Minimum Spanning tree, therefore Fig – 2 is the Minimum Spanning tree.
#### Applications of Minimum Spanning Tree
The applications of the minimum spanning tree are given as follows –
• Minimum spanning trees can be used to design water-supply networks, telecommunication networks, and electrical grids.
• Majorly, spanning trees are used to find paths in the map.
#### Algorithms for Minimum Spanning Tree
A minimum spanning tree can be found from a weighted graph by using the algorithms given below –
• Prim’s Algorithm
• Kruskal’s Algorithm
#### Prim’s algorithm
It is a greedy algorithm that starts with an empty spanning tree. It is used to find the minimum spanning tree from the graph. This algorithm finds the subset of edges that includes every vertex of the graph such that the sum of the weights of the edges can be minimized.
#### Kruskal’s algorithm
This algorithm is also used to find the minimum spanning tree for a connected weighted graph. Kruskal’s algorithm also follows a greedy approach, which finds an optimum solution at every stage instead of focusing on a global optimum.
Note: Later, in upcoming lesson, you will learn in detail about Prim’s & Kruskal’s algorithm,
### Vertex Induced Graph
An induced subgraph formed from some of the vertices and all the edges (from the original graph) connecting pairs of vertices in the original graph.
Original Vertex Induced Graph
### Edge Induced Graph
An induced subgraph formed from some of the edges (from the original graph) and the vertices that are at their endpoints.
Original Edge Induced Graph
If you find anything incorrect in the above-discussed topic and have further questions, please comment below.
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# UNITARY METHOD
Unitary method
The unitary method is a technique in mathematics for solving a problem finding the value of single unit, and then finding the necessary value by multiplying the single unit value.
Variation
This shows the relation between the two quantities
Types of variation
Direct variation:
If the increase in one quantity results in increase in other quantities and decrease in one quantity results in decrease in other quantities, then the relation between the quantities is said to be in direct variation.
Egif the cost of 2 chair is Rs 400 then, the cost of 4 chair is Rs 800
Indirect variation:
If the increase in one quantity results in decreasein other quantities and decrease in one quantity results in increasein other quantities, then the relation between the quantities is said to be in indirect variation. Time and work
Eg20 men can do a piece of the work in 40 days
1 man can do a piece of the work in 80 days
Some important formula
a. A can do a piece of the work in x days
A can do $\frac{1}{{\rm{x}}}{\rm{\: }}$of the work in 1 days${\rm{\: }}$
b. No of days =$\frac{{{\rm{total\: work\: done}}}}{{{\rm{work\: done\: in\: }}1{\rm{\: days\: }}}}$
c. Time and work (direct variation)
More work →more time
Less work → less time
d. men andtime (indirect variation)
More men↑ less time ↓
Less men ↓more time ↑
e. If A1 men can do W1 work in D1days and A2 can do W2 work in D2 days
$\frac{{{{\rm{A}}_1}{{\rm{D}}_1}}}{{{{\rm{W}}_1}{\rm{\: }}}} = \frac{{{{\rm{A}}_2}{{\rm{D}}_2}}}{{{{\rm{W}}_2}{\rm{\: }}}}{\rm{\: }}$
In case of hours
$\frac{{{{\rm{A}}_1}{{\rm{D}}_{1{\rm{\: }}}}{{\rm{H}}_1}}}{{{{\rm{W}}_1}{\rm{\: }}}} = \frac{{{{\rm{A}}_2}{{\rm{D}}_2}{{\rm{H}}_2}}}{{{{\rm{W}}_2}{\rm{\: }}}}$
f. If A can do a work in ‘x’ days and B can do the same work in ‘y’ days, then the number of days required to complete the work if A and B work together is$\frac{{{\rm{xy}}}}{{{\rm{x\: }} + {\rm{\: y}}}}$
Ratio and proportion
The term in the form of p/q is called ratio. A proportion is two ratios are equal. It can be written in two ways: two equal fractions, or, using a colon, a: b = c: d.
Problem of the allegation
In our daily life, we face many problems related to allegation. If the two quantities m and n are mixed and the cost of these object are a and b and the average cost of the mixture = ${\rm{Rs}}\frac{{{\rm{ma}} + {\rm{nb}}}}{{{\rm{m}} + {\rm{n}}}}$
Examples
1.A takes twice as much as B and thrice as much as C to finish a work; working together they can finish the work in 10 days. Find the time each will take to finish the work separately.
Soln
The working capacity of B = 2 times the working capacity of A
B = 2 A
The working capacity of C= 2 times the working capacity of A
C = 3 A
Now, A + B+ C can finish 1 work 10 days
A + 2A+ 3A can finish 1 work 10 days
6 A can finish 1 work in 10 days
A can do 1 work in 60 days
B can do 1 work in $\frac{1}{2}{\rm{\: }}$time taken by A = $\frac{1}{2}$*60 = 30 days${\rm{\: }}$
C can do 1 work in$\frac{1}{3}{\rm{\: time\: taken\: by\: A\: }} = {\rm{\: }}\frac{1}{3}{\rm{*}}60{\rm{\: }} = {\rm{\: \: }}20{\rm{\: days}}$
2. 3 duck eggs can be exchanged with 4 hen eggs .Similarly with 4 eggs of swan, 7 eggs of duck can be exchanged .If the cost of 2 hen eggs is Rs 7.5, then what is the cost of swan’s egg.
Soln
Cost of 7 eggs of duck = Cost of 4 eggs of swan
Cost of 1 egg of duck = Cost of 4/7 eggs of swan
Cost of 3 duck eggs = Cost of $\frac{4}{7}{\rm{*}}3$ = $\frac{{12}}{3} = 4{\rm{\: }}$hen eggs
Cost of 2 hen eggs = Rs 7.5
Cost of 1 hen egg = Rs 3.75
Cost of $\frac{{12}}{3}{\rm{\: }}$hen eggs = Rs 3.75 *$\frac{{12}}{3}$ = 4 *3.75 = Rs 15
i.e Cost of 3 duck eggs = Rs 15
Cost of 1 duck egg = Rs 5
Cost of 4/7 eggs of swan = Rs 5
Cost of 1 egg of swan = Rs 35/4 = 8.75
3. Two types of rice costing Rs 40 and Rs 30 per kg are mixed in the ratio of 3:4. Find the cost of rice mixture per kg.
Soln
a =40
b = 30
m : n= 3 : 4
The cost of per kg = $\frac{{{\rm{ma}} + {\rm{nb}}}}{{{\rm{m}} + {\rm{n}}}}$= $\frac{{{\rm{\: }}3{\rm{*\: }}40{\rm{\: }} + 4{\rm{*\: }}30}}{{3 + 4{\rm{\: }}}}$=$\frac{{120 + 120}}{7}$ =$\frac{{240}}{7}$ = 34.28
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Cathy Skinner REDFIELD ELEMENTARY SCHOOL, SCOTTSDALE, AZ
5th Grade Math : Unit #5 - The Value of a Whole Number - Place Value : Lesson #1
# Playing With Place Value - Manipulatives
Objective: Students will be able to see that each time a digit is moved to another place value column it is 10 times greater or 10 times less.
Standards: 5.NBT.A.1 MP4 MP5 MP7 MP8
Subject(s): Math
60 minutes
1 Introduction - 0 minutes
In the Common Core Standards, place value is no longer just the value of a digit because of the place it is in. We now need to make sure students also understand that "in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left." (5.NBT.A.1)
Your students will be using manipulatives to model with mathematics the exchange of increasing 10 times or decreasing 1/10 the value as a digit moves through place value in this lesson.
2 Engage - 30 minutes
I start my lessons with a question for students to discuss that will get them feeling included into the lesson and link their prior knowledge or experiences to the lesson. For this lesson I asked my students to talk for a minute or so about what they would like to have 10 more of. This activates students interest and gives real life examples - If I was to give my own example it may not relate to what my students know.
I overheard students talking about they would like ten times more recess, ten times more dessert, ten times more gaming time, ten times more allowance.
The next step is to review what the base 10 blocks mean. The small cube = 1 unit, the bar is 10 units, the flat is 100 units and the large cube is 1000 units. This is directly related to MP4 or modeling with mathematics and MP5, using appropriate tools strategically. I do this by holding up the manipulative and tossing a koosh ball to one student to answer. They then toss the koosh to another student who has their hands raised to tell the class what the next manipulative is.
I then start to put the blocks under my document camera and have students tell me the name of the number. If you have a Smart Board you could find these pieces in Smart Exchange. If you don't have the manipulatives or Smart Board you can have your students cut out pieces from centimeter graph paper. I would recommend gluing envelopes into Math Journals to save these for future use.
Once you feel your students are able to read the numbers from the blocks, or enough are to become your helpers, its time for a collaborative activity. Review your procedures for group work and give these instructions to your students.
1. Make a number with your blocks (MP4,5)
2. Your partner will read the number and then multiply one of the digits by 10 and change the blocks to represent the new number. (MP7 - look for and make use of structure, MP8 Look for and express regularity in repeated reasoning.)
3. Write a number sentence in your Math Journal.
3 Extend - 10 minutes
To extend this lesson I looked up Place Value in my district required textbook and assigned the pages for my students to work on independently. You can see some of their work pages in the resources below.
Because my students are 4th and 5th graders I need to meet a range of abilities - which I am sure you have in your classroom also. The fourth grade workbook did not have place value with decimals but the 5th grade did. I was okay with my 5th graders taking the information in the activity and transferring it to decimals. This way I could assess where they are and if they need extra support. The 5th grade students who were struggling I did not move them back to whole number place value - I continued to teach place value with decimals in combination. I am comfortable with the 4th grade not including decimals because it is not required in the 4th grade standards and I could really focus on place value with whole numbers for them. The ones that wanted to do the 5th grade work I found an extra book for them to work out of but did not let them write in the book or they partnered up with a 5th grader. This way I have a clean copy.
I also believe productive struggle is good - the students who struggled a little with place value read the text to give them a different input of information and then had them work with someone else. If needed (really struggling) I would have pulled a group together and work with small groups but I didn't need to this time.
When your students are done have them thank their partners. Appreciations build classroom collaboration which equals more time on task!
4 Reflect - 5 minutes
After teaching a lesson always have your students reflect on the lesson - academics and behavior. Reflection increases retention 50%.
For this lesson I wanted to make sure one of my content reflection questions specifically asked what happens to a digit when it is moved one column up or down in place value. When discussion is opened up so is MP3 - construct viable arguments and critique the reasoning of others.
1. What happens when a digit is moved one column up?
The value increases 10 times.
2. What happens when a digit is moved one column down?
The value decreases 10 times.
3. Was there anything else you noticed?
I do not always have a answer in mind for this questions because I am assessing their understanding of the lesson.
If you don't have time to do the last two questions whole group, have your students tell their answers to their partner. It is a built in appreciation!
4. What went well with you and your partner?
Arguments and Critiques
Rigor
DOVE - reflection on group collaborative strategies from the wonderful people at Tribes Learning Communities .
D efer judgement
O ffbeat ideas
V ast number
E xpand on other ideas |
## Subitising - objects: Activity 2
Activity type: Interactive Activity
## Subitising - objects: Activity 2
Course
Mathematics
Kinder
Section
Counting and Numbers
Outcome
Subitising using objects
Activity Type
Interactive Activity
Activity ID
23043
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## United States – Common Core State Standards
• ##### K.CC – Counting & Cardinality
• Mathematics
• K.CC.5 – Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
• K.CC.4 – Understand the relationship between numbers and quantities; connect counting to cardinality.
• K.CC.4.a – When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
• K.CC.4.b – Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
• ##### K.OA – Operations & Algebraic Thinking
• Mathematics
• K.OA.3 – Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
• K.OA.4 – For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
## Australia – Australian Curriculum
• ##### Number and Algebra
• Number and place value
• ACMNA002 – Connect number names, numerals and quantities, including zero, initially up to 10 and then beyond
• ACMNA003 – Subitise small collections of objects
• ##### Number and Algebra
• Number and place value
• ACMNA012 – Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count by twos, fives and tens starting from zero
## New Zealand – National Standards
• ##### 2.NA – Number and algebra
• 2.NA.1 – Apply counting-on, counting-back, skip-counting, and simple grouping strategies to combine or partition whole numbers
• 2.NA.4 – Continue number patterns based on ones, twos, fives, and tens.
## United Kingdom – National Curriculum
• ##### Year 1 programme of study
• KS1.Y1.N.NPV – Number - number and place value
• Pupils should be taught to:
• KS1.Y1.N.NPV.4 – Identify and represent numbers using objects and pictorial representations including the number line, and use the language of: equal to, more than, less than (fewer), most, least |
# Prime Numbers Definition
The prime number is the one that has only two factors i.e. 1 and the number itself. In simple words, we can say a number that can’t be divided by any other number completely is said to be Prime Number. It can have a maximum of two factors on this page.
Note:
It is also considered the opposite of Composite numbers (Numbers with at least three factors). For example, 7 is the prime number because it can’t be divided by any other number with 0 remainders left.
Note:
Prime Numbers less than 30.
3, 5, 7, 11, 13, 17, 19, 23, 29.
## Terms related to Prime Numbers
There are two main terms related to Prime Numbers that we are going to mention here.
• Co-Prime: Two numbers that have only one common factor i.e. “1”. For example, 9 and 10 are prime numbers because they have 1 as their common factor only.
• Twin Prime: The pair of prime numbers is called a Twin Prime if it has only one composite number between them. For example, (3, 5), (11, 13), and (5, 7) are Twin Primes because they have only one composite number between them.
### How to check if a number is prime or not?
You can easily find if the given number is prime or not just by dividing it by different terms. If a number is divisible by any other number except 1 and by the number itself, it is not called a Prime Number.
Example 1:
Check whether 16 is a prime number or not.
Solution:
Let us divide 16 with different prime numbers less than 16.
• 16 is divisible by 2, 4, and 8.
It shows that 16 is not a Prime number because it has 3 more factors except 1..
Example 1:
Check whether 31 is a Prime number.
Solution:
By using our division testing, we have found that 31 is not divisible by any whole number. That is why 31 is termed to be a Prime Number.
### Fun Facts of Sum
• 2 is the smallest prime number.
• The only even number that is prime too is “2”.
• 2 & 3 are only consecutive prime numbers.
• 0 & 1 is neither a prime nor a composite number.
• All numbers ending with 5 except 5 can’t be prime numbers.
• The occurrence of prime numbers reduces with the increase in value.
### FAQ's
Is 1 a Prime number?
No, 1 is neither a prime number nor a composite number. It is part of the Natural and Whole number set only.
What is the smallest prime number?
“2” is the smallest prime number.
Are all odd numbers prime?
No, odd numbers can be prime or composite. But all even numbers except 2 are composite numbers.
Can a prime number be negative?
No, a prime number can’t be negative. It is called an Integer only instead of giving any other name.
What is the difference between prime numbers and composite numbers?
Prime numbers have only two factors including 1 while Composite numbers have at least three factors including 1. |
# How to Calculate the Surface Area of a Triangular Prism
What is a Triangular Prism?
A triangular prism is a solid geometric shape that has a triangle as its base and three rectangular faces. It is also known as a tetrahedron. The triangular prism is a three-dimensional shape that has a height, a base, and three sides.
Why Calculate the Surface Area of a Triangular Prism?
Calculating the surface area of a triangular prism is essential in many applications, including architecture, construction, and engineering. Knowing the surface area of a triangular prism can help determine the amount of material needed to cover the object and the cost of the material. It is also useful in calculating the amount of paint or coating required to cover the object.
How to Calculate the Surface Area of a Triangular Prism?
Calculating the surface area of a triangular prism is a straightforward process. Here are the key steps involved:
1. Measure the base and height of the triangle: To calculate the surface area of the triangular base, measure the length of one of the sides of the base (the base), and the height of the triangle perpendicular to the base.
2. Calculate the area of the triangle: Once you have measured the base and height of the triangle, calculate its area using the formula: Area = (1/2) x base x height.
3. Calculate the area of each rectangular face: To calculate the area of each rectangular face, multiply the height of the triangular prism by the length of the rectangular face.
4. Add the areas together: Finally, add the areas of the triangular base and the three rectangular faces together to get the total surface area of the triangular prism.
Example
Let’s say we have a triangular prism with a triangular base of length 4 cm and height 5 cm. The height of the triangular prism is 10 cm, and the length of the rectangular faces is 6 cm.
Calculate the area of the triangular base: Area = (1/2) x 4 cm x 5 cm = 10 cm²
Calculate the area of each rectangular face: Area = 10 cm x 6 cm = 60 cm²
Calculate the total surface area: Total surface area = 2 x 60 cm² + 10 cm² = 130 cm²
Formula
The formula for calculating the surface area of a triangular prism is:
Surface Area = 2 x Area of the Base + Perimeter of the Base x Height of the Prism
Where:
Area of the Base = (1/2) x base x height
Perimeter of the Base = Sum of the lengths of the sides of the base
Conclusion
Calculating the surface area of a triangular prism is a simple process that involves measuring the base and height of the triangular base, calculating the area of the base, and adding the areas of each rectangular face together. The formula for the surface area of a triangular prism is straightforward and can be used to calculate the amount of material needed to cover the object or the amount of paint or coating required to cover the object. By following the key steps outlined in this article, you can easily calculate the surface area of a triangular prism.
FAQ
Q: What is a triangular prism? A: A triangular prism is a three-dimensional shape that has two congruent and parallel triangular bases and three rectangular lateral faces that connect the bases.
Q: How do you calculate the surface area of a triangular prism? A: To calculate the surface area of a triangular prism, you need to find the area of each face and add them together. Here are the steps to follow:
1. Find the area of each triangular base:
• Measure the length of the base and the height of one of the triangular bases.
• Multiply the length by the height, and then divide by 2 to get the area of one triangular base.
• Multiply this by 2 since there are two triangular bases.
2. Find the area of each rectangular face:
• Measure the length and width of one of the rectangular faces.
• Multiply the length and width to get the area of one rectangular face.
• Multiply this by 3 since there are three rectangular faces.
3. Add the areas of the triangular bases and rectangular faces together to get the total surface area of the triangular prism.
Here’s the formula to calculate the surface area of a triangular prism: Surface area = 2 x area of base + perimeter of base x height
Note that the perimeter of the base is the sum of the lengths of the three sides of the triangular base.
Q: Can you provide an example of calculating the surface area of a triangular prism? A: Sure! Let’s say we have a triangular prism with a base of length 4 cm and height 3 cm, and a height of 8 cm.
• First, find the area of one triangular base: area = (4 x 3) / 2 = 6 cm^2 area of both bases = 2 x 6 = 12 cm^2
• Second, find the area of one rectangular face: area = 4 x 8 = 32 cm^2 area of all three rectangular faces = 3 x 32 = 96 cm^2
• Finally, add the areas of the bases and faces together: surface area = 12 + 96 = 108 cm^2
Therefore, the surface area of the triangular prism is 108 cm^2. |
# What Do You Do With A Subscript In Math latest 2023
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Page Contents
## Geometry for Beginners – How to Find the Area of a Triangle
Welcome to another chapter of Geometry for Beginners. Our topic today is about finding the area of a triangle. For your best understanding of this topic, you should be familiar with the previous two topics: (1) finding the area of a rectangle and (2) understanding that area is measured with real squares with resulting labels like sq. in. or pi^2. If you don’t instantly understand any of these, you should first read the appropriate articles on geometry for beginners and get this information “rooted in your brain”. Geometry, like algebra, relies on prior knowledge. Without prior knowledge as a base, new information will have no meaning and cannot be learned effectively.
If you have the formula, in symbols and words, for the area of a rectangle in your mind, and you understand why the area is labeled in square units, then we can proceed to find the area of triangles. I want to point out that even though the title says we’re going to find the area of a triangle – and we are – there are actually many different triangles out there. Rectangles vary in some ways, but opposite sides are ALWAYS equal and angles are ALWAYS right angles, so one side is the base b and the other side is height h, and the formula A = bh ALWAYS applies to rectangles. Triangles are much more variable in shape, so we’ll need to consider a few different situations. This is, however, ONLY ONE formula for the area of a triangle in geometry, so memorization should be easy. This is the good news. What will be different is the height location. This is the part that can be annoying.
To develop the formula for the area of a triangle, we first need a diagram of a rectangle. Remember to draw it large enough to label the bottom with both the word “base” and the symbol b, and label the perpendicular side with the word “height” and the symbol h. Next to your diagram, write the area formula in words and symbols. “The area of a rectangle is the base multiplied by the height” and A = bh.
Now I want you to draw one of the diagonals of the rectangle in your diagram. In a rectangle, a diagonal connects opposite corners. Can you now see that the diagonal has just formed two identical triangles? For these triangles, finding the area is quite simple since each triangle is half of the rectangle. If the rectangle measures 6″ by 8″, then the total area is A = bh = 6 x 8 = 48 in2. The area of each triangle is then 24 in2 and brings us to the triangle area formula: A = 1/2 bh. (Note: It can be confusing to always use A for area as it doesn’t specify what the figure is. To handle this we sometimes use a small picture as a hint. The area of a triangle could then start by A but with small triangles drawn on the bottom right of the A as a subscript number would be written. I don’t have that ability here, but hope you can figure out what I mean.)
In short: the area of a triangle is half the product of the base and the height. .
Caution! Caution! Caution! Now we come to the part where you really have to be very careful. Remember that all rectangles have right angles, but not all triangles have right angles. When a triangle has a right angle, one leg of the right angle can be considered the base and the other leg is the height. But what if there is no right angle?
To deal with the “no right angle” situation, I want you to look at your rectangle diagram as if you had nailed some sticks together to make your rectangle. If you’ve ever done anything like this, you know that without a few extra support pieces, the rectangle starts to lean and loses those right angles. Your rectangle starts to look more like a parallelogram with equal opposite sides and equal opposite angles. (A rectangle is actually a “special case” of a parallelogram.) FOCUS ON THIS NOW. As we push on the top corner to make our rectangle lean more and more to the side, the base stays the same length, but the height gets SHORTER. Our rectangle which measured 6 inches by 8 inches becomes a parallelogram without a right angle which still has an 8″ base, but the 6″ side is no longer the height. The area changes as the pitch changes. If 6″ isn’t the height anymore, what is it?
I hope you remember that height is ALWAYS measured as the shortest distance to the base, and that shortest distance is measured from the top vertex to the base. There is actually no visible line yet, which represents the height. What we’re doing is “dropping a perpendicular line” from the top vertex to the base. It just means that we draw a perpendicular line segment. The length of this new line is the HEIGHT of the parallelogram. The formula for the area always remains the same: A = bh, but we have to be very careful to choose the right length as the height. Without a right angle, the height is NOT the side of the parallelogram.
Drawing diagonally on the parallelogram does the same thing as in the rectangle – divides the figure into two identical triangles; so the area of the triangles is always half of the total area. So the formula for the area of ALL triangles is the same as before: A = 1/2 bh. Again, however, we have to be careful about the number we use for the height.
On your paper, draw a non-right triangle. Label the bottom side with the word “base” and the symbol b. Locate the top vertex and draw the perpendicular line down from that vertex to the base. Label this new line as “height” and h. For now, you will only be able to find the area if you are given the height of the problem. It takes skills that we don’t yet have to determine the height if it is not given to us. For now, just remember that:
The area of a triangle is equal to half the base multiplied by the height, or A = 1/2 bh.
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Linear Programming
Maximize and minimize quantities using linear inequality systems
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Linear Programming
Suppose a craftsman can make bracelets and necklaces. A bracelet requires 10 beads and takes 10 minutes to make, while a necklace requires 20 beads and takes 40 minutes to make. The craftsman has 1000 beads to work with, and he has 1600 minutes in which to work. If a bracelet costs $5 and a necklace costs$7.50, what is the maximum revenue that the craftsman can take in? How do you know? In this Concept, you'll use systems of linear inequalities and linear programming to solve problems such as this one.
Guidance
Systems of Linear Inequalities
This lesson moves on to the concept of systems of linear inequalities. In the previous topic, you learned how to graph a linear inequality in two variables.
Step 1: Graph the equation using the most appropriate method.
• Slope-intercept form uses the \begin{align*}y-\end{align*}intercept and slope to find the line.
• Standard form uses the intercepts to graph the line.
• Point-slope uses a point and the slope to graph the line.
Step 2: If the equal sign is not included draw a dashed line. Draw a solid line if the equal sign is included.
Step 3: Test a point not on the line in the original inequality and shade the half plane that works.
In this topic, we will learn how to graph two or more linear inequalities on the same coordinate plane. The inequalities are graphed separately on the same graph and the solution for the system of inequalities is the common shaded region between all the inequalities in the system.
The common shaded region of the system of inequalities is called the feasible region.
Example A
Solve the system of inequalities \begin{align*}\begin{cases} 2x+3y\le18\\ x-4y\le12 \end{cases}\end{align*}.
Solution:
The first equation is written in standard form and can be graphed using its intercepts. The line is solid because the equal sign is included in the inequality. Since (0, 0) works in the inequality, shade the half plane below the line.
The second equation can be rewritten in slope-intercept form to graph.
\begin{align*}&\qquad \qquad \qquad \qquad \qquad \qquad \Rightarrow \\ &-4y \le -x+12 && y \ge \frac{x}{4}-3\end{align*}
The division by –4 causes the inequality to reverse. The line is solid again because the equal sign is included in the inequality. Shade the half plane above the boundary line because (0, 0) works in the original inequality.
When we combine the graphs, we see that the blue and red shaded regions overlap. This overlap is where both inequalities work. Thus the purple region denotes the solution of the system, the feasible region.
The kind of solution displayed in this example is called unbounded, because it continues forever in at least one direction (in this case, forever upward and to the left).
Bounded regions occur when more than two inequalities are graphed on the same coordinate plane, as you will see in Example C.
Writing Systems of Linear Inequalities
In some cases, you are given the feasible region and asked to write the system of inequalities. To do this, you work in reverse order of graphing.
• Write the equation for the boundary line.
• Determine whether the sign should include “or equal to.”
• Determine which half plane is shaded.
• Repeat for each boundary line in the feasible region.
Example B
Write the system of inequalities shown below.
Solution:
There are two boundary lines, so there are two inequalities. Write each one in slope-intercept form.
\begin{align*}y & \le \frac{1}{4} x + 7\\ y & \ge -\frac{5}{2} x - 5\end{align*}
Linear Programming – Real-World Systems of Linear Inequalities
Entire careers are devoted to using systems of inequalities to ensure a company is making the most profit by producing the right combination of items or is spending the least amount of money to make certain items. Linear programming is the mathematical process of analyzing a system of inequalities to make the best decisions given the constraints of the situation.
Constraints are the particular restrictions of a situation due to time, money, or materials.
The goal is to locate the feasible region of the system and use it to answer a profitability, or optimization, question.
Theorem: The maximum or minimum values of an optimization equation occur at the vertices of the feasible region – at the points where the boundary lines intersect.
This theorem provides an important piece of information. While the individual colors of the inequalities will overlap, providing an infinite number of possible combinations, only the vertices will provide the maximum (or minimum) solutions to the optimization equation.
Let’s look at the following situation.
Example C
James is trying to expand his pastry business to include cupcakes and personal cakes. He has 40 hours available to decorate the new items and can use no more than 22 pounds of cake mix. Each personal cake requires 2 pounds of cake mix and 2 hours to decorate. Each cupcake order requires one pound of cake mix and 4 hours to decorate. If he can sell each personal cake for $14.99 and each cupcake order for$16.99, how many personal cakes and cupcake orders should James make to make the most revenue?
There are four inequalities in this situation. First, state the variables. Let \begin{align*}p=\end{align*} the number of personal cakes and \begin{align*}c=\end{align*} the number of cupcake orders.
Translate this into a system of inequalities.
\begin{align*}2p+1c \le 22\end{align*} – This is the amount of available cake mix.
\begin{align*}2p+4c \le 40\end{align*} – This is the available time to decorate.
\begin{align*}p \ge 0\end{align*} – You cannot make negative personal cakes.
\begin{align*}c \ge 0\end{align*} – You cannot make negative cupcake orders.
Now graph each inequality and determine the feasible region.
The feasible region has four vertices: {(0, 0),(0, 10),(11, 0),(8, 6)}. To find each of these points, select the inequalities in pairs and solve them as a system of equations.
According to our theorem, the optimization answer will only occur at one of these vertices.
Write the optimization equation. How much of each type of order should James make to bring in the most revenue?
\begin{align*}14.99p+16.99c=maximum \ revenue\end{align*}
Substitute each ordered pair to determine which makes the most money.
\begin{align*}(0,0) & \rightarrow \0.00\\ (0,10) & \rightarrow 14.99(0)+16.99(10)=\169.90\\ (11,0) &\rightarrow 14.99(11)+16.99(0)=\164.89\\ (8,6) & \rightarrow 14.99(8)+16.99(6)=\221.86\end{align*}
To make the most revenue, James should make 8 personal cakes and 6 cupcake orders.
Example D
Two oil refineries produce three grades of gasoline: A, B, and C. At each refinery, the various grades of gasoline are produced in a single operation so that they are in fixed proportions. Assume that one operation at Refinery 1 produces 1 unit of A, 3 units of B, and 1 unit of C. One operation at Refinery 2 produces 1 unit of A, 4 units of B, and 5 units of C. Refinery 1 charges $3000 for what is produced in one operation, and Refinery 2 charges$5000 for the production of one operation. A consumer needs 100 units of A, 340 units of B, and 150 units of C. How should the orders be placed if the consumer is to meet his needs most economically?
There are five inequalities in this situation. First, state the variables. Let x = the number of operations ordered at Refinery 1 and y = the number of operations ordered at Refinery 2.
Translate this into a system of inequalities.
\begin{align*}1x+1y \ge 100\end{align*} – This is the amount of units of gasoline A produced.
\begin{align*}3x+4y \ge 340\end{align*} – This is the amount of units of gasoline B produced.
\begin{align*}1x+5y \ge 150\end{align*} – This is the amount of units of gasoline C produced.
\begin{align*}x \ge 0\end{align*} – You cannot make negative orders at Refinery 1.
\begin{align*}y \ge 0\end{align*} – You cannot make negative orders at Refinery 2.
Now graph each inequality and determine the feasible region.
Looking at the feasible region we see it has four vertices. To find each of these points, we select the inequalities whose lines cross at each point and find their intersection. Starting from left to right we have \begin{align*}x=0\\ x+y=100\end{align*}, \begin{align*}x+y=100\\ 3x+4y=340\end{align*}, \begin{align*}3x+4y=340\\ x+5y=150\end{align*}, and \begin{align*}y=0\\ x+5y=150\end{align*}. Solving them by substitution or elimination the 4 vertices are at \begin{align*}\left (0, 100 \right )\end{align*}, \begin{align*}\left (60, 40 \right )\end{align*}, \begin{align*}\left (100, 10 \right )\end{align*}, and \begin{align*}\left (150, 0 \right )\end{align*}.
According to our theorem, the optimization answer will only occur at one of these vertices.
Write the optimization equation. How many orders at each refinery should be made to complete the production most economically?
\begin{align*}3000x + 5000y &= cost \ of \ production\end{align*}
Substitute each ordered pair to determine which costs the least money.
\begin{align*}(0,100) & \rightarrow 3000(0)+5000(100)=\500,000\\ (60,40) & \rightarrow 3000(60)+5000(40)=\380,000\\ (100,10) &\rightarrow 3000(100)+5000(10)=\350,000\\ (150,0) & \rightarrow 3000(150)+5000(0)=\450,000\\\end{align*}
To be the most economical, 100 orders should be made at Refinery 1 and 10 at Refinery 2.
Guided Practice
1. Find the solution set to the following system.
\begin{align*}y & > 3x-4\\ y & < -\frac{9}{4}x+2\\ x & \ge 0\\ y & \ge 0\end{align*}
2. The Reliable Appliance store wishes to stock at most 100 refrigerators and stoves. They are sure they cannot sell more than 50 refrigerators. Each refrigerator weighs 200 pounds and each stove 100 pounds and they are limited to a total weight of 12,000 pounds. They make $35 on each refrigerator and$20 on each stove. How many of each should they stock in order to maximize profits?
Solution:
1. Graph each line and shade appropriately.
\begin{align*}y > 3x-4\end{align*}
\begin{align*}y < - \frac{9}{4}x+2\end{align*}
Finally, we graph \begin{align*}x \ge 0\end{align*} and \begin{align*}y \ge 0\end{align*}, and the intersecting region is shown in the following figure.
2. There are five inequalities in this situation. First, state the variables. Let R = the number of refrigerators and S = the number of stoves.
Translate this into a system of inequalities.
\begin{align*}1R+1S \le 100\end{align*} – Stock at most 100.
\begin{align*}R \le 50\end{align*} – Can sell no more than 50 refrigerators.
\begin{align*}200R+100S \le 12000\end{align*} – Total weight limit.
\begin{align*}R \ge 0\end{align*} – You cannot stock negative refrigerators.
\begin{align*}S \ge 0\end{align*} – You cannot stock negative stoves.
Now graph each inequality and determine the feasible region.
Looking at the feasible region we see it has five vertices. To find each of these points, we select the inequalities whose lines cross at each point and find their intersection. We have \begin{align*}R=0\\ S=0\end{align*}, \begin{align*}R=0\\ R+S=100\end{align*}, \begin{align*}R+S=100\\ 200R+100S=12000\end{align*}, \begin{align*}200R+100S=12000\\ R=50\end{align*}, and \begin{align*}S=0\\ R=50\end{align*}. Solving them by substitution or elimination the 5 vertices are at \begin{align*}\left (0, 0 \right )\end{align*}, \begin{align*}\left (0, 100 \right )\end{align*}, \begin{align*}\left (20, 80 \right )\end{align*}, \begin{align*}\left (50, 20 \right )\end{align*}, and \begin{align*}\left (50, 0 \right )\end{align*}.
According to our theorem, the optimization answer will only occur at one of these vertices.
Write the optimization equation. How many of each should they stock in order to maximize profits?
\begin{align*}35R + 20S &= profit\end{align*}
Substitute each ordered pair to determine which maximizes profit.
\begin{align*}(0,0) & \rightarrow 35(0)+20(0)=\0\\ (0,100) & \rightarrow 35(0)+20(100)=\2000\\ (20,80) & \rightarrow 35(20)+20(80)=\2300\\ (50,20) & \rightarrow 35(50)+20(20)=\2150\\ (50,0) & \rightarrow 35(50)+20(0)=\1750\\\end{align*}
To make the most profit, they should stock 20 refrigerators and 80 stoves.
Notes/Highlights Having trouble? Report an issue.
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Cos double angle identity in terms of Tan
Formula
$\cos{(2\theta)} \,=\, \dfrac{1-\tan^2{\theta}}{1+\tan^2{\theta}}$
The cos of double angle equals to the quotient of the subtraction of square of tan function from one by the sum of one and square of tan function is called the cos of double angle identity in terms of tan functions.
Introduction
Let theta represents an angle of a right triangle. The cosine of double angle is written as $\cos{2\theta}$ and the square of tangent of angle theta is written as $\tan^2{\theta}$ in mathematical form.
The $\cos{2\theta}$ is equal to the quotient of subtraction of $\tan^2{\theta}$ from $1$ by the addition of $1$ and .$\tan^2{\theta}$.
$\cos{(2\theta)} \,=\, \dfrac{1-\tan^2{\theta}}{1+\tan^2{\theta}}$
It is called the cos double angle identity in terms of tan function and also used as a formula in trigonometry.
Usage
The cosine of double angle rule is used in two cases in mathematics.
1. To expand a cos double angle function as the quotient of subtraction of square of tan function from one by the addition of one and square of tan function.
2. To simplify the quotient of subtraction of square of tan function from one by the sum of one and square of tan function as the cos double angle function.
Popular forms
The cos of double angle rule is written in terms of tan functions popularly in the following two different forms.
$(1) \,\,\,$ $\cos{(2x)} \,=\, \dfrac{1-\tan^2{x}}{1+\tan^2{x}}$
$(2) \,\,\,$ $\cos{(2A)} \,=\, \dfrac{1-\tan^2{A}}{1+\tan^2{A}}$
Proof
Learn how to derive the cos double angle formula in terms of tan functions in trigonometric mathematics.
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# Linear Equations in Two Variables Word Problems
Inverse variation refers to a nonlinear relationship in which the product of two variables is constant. Linear equations are used in many real life fields like engineering, accounting, banking etc. make their jobs easier. Linear equations word problems often intimidate students. Even to those students who are very good at Math find solving word problems difficult. This is usually because word problems require converting the problem into a mathematical equation which can be solved to arrive at an outcome. Even students who are good at Math may not always be good reading a problem and arriving at an equation. Linear equations word problems relate mathematical concepts to real world situations.
## Word Problems
Algebra word problems make use of many an algebraic models to answer questions rising in real life situations. If students solve Math word problems every day, their confidence grows rapidly as they feel more confident about being able to solve them. Usually Math word problems describe real life situations most of the time, so, learning how to solve these problems helps students immensely.
Problem 1:
Aarthi went to US with her Mom for vacations. While returning back to India she gave her savings to her mom to convert into Indian currency. Her mom found that, Aarthi has collected 150 coins in his Piggy Bank, all consisting of dimes and quarters. If the total worth of the coins is 30 dollars, how many dimes and quarters does Aarthi have?
Solution: Let x and y be correspondingly the number of dimes and quarters in the Piggy Bank.
We can form two equations, one on the total number of coins and another on the value of the coins.
x + y = 150 ....(1)
0.10x + 0.25y = 30 ....(2)
Because 1 dime value = 0.10 dollars and 1 quarter value = 0.25 dollars
The system can be solved using substitution as follows:
Solve equation (1) for y
y = 150 - x
Put the value of y in equation (2), to get the value of x
0.10x + 0.25(150 - x) = 30
0.10x + 0.25 $\times$ 150 - 0.25x = 30
0.10x - 0.25x = 30 - 37.5
-0.15x = -7.5
x = 7.5/0.15 = 50
Again from equation (1)
50 + y = 150
y = 150 - 50 = 100
Number of Dimes = 50
Number of Quarters = 100
Problem 2: The linear model P(d) = 62.5d + 2117 is used to find the pressure (lb/ft^2) at d feet below the surface of the water.
(a) What does the constant 2117 represent?
(b) What information do you get from the number 62.5?
(c) What is the pressure 200 ft below water surface?
Solution:
The model is given in slope intercept form y = mx + b , where m is the slope and b the y intercept.
a)The constant 2117, which can be viewed as the y intercept is the function value when d = 0.
This means the pressure on the surface of water = 2117 lb/ft^2.
b)The number 62.5 can be related to 'm' the slope in a linear equation. It is the rate at which the pressure is increasing for every ft below the water surface.
c)
P(d) = 62.5d + 2117
P(200) = 62.5(200) + 2117 = 12,500 + 2,117 = 14,617 lb/ft^2.
Problem 3: The sum of two number is 50 and their difference is 10. Find the numbers.
Solution: Convert the statement into simpler equations:
Consider two numbers, say x and y
Sum of two numbers = x + y = 50
Difference of two numbers = x - y = 10
Add both the equations as coefficient of variable y are same and opposite in signs.
x + y = 50
x - y = 10
_________
2x + 0y = 60
2x =60
x = 30
Again, put value of x in x - y = 10
30 - y = 10
y = 30 - 10 = 20
Practice Problem 1: If 1 is added to the numerator and denominator of any fraction, it becomes $\frac{1}{2}$. If 2 is subtracted from the numerator and denominator fraction become $\frac{3}{2}$. Find the fractions. |
## Number Yard
In this lesson, children will create their own number yard books by drawing the number of objects that correspond with the numbers on the pages of the book.
### Lesson for:
Toddlers/Preschoolers
(See Step 5: Adapt lesson for toddlers or preschoolers.)
### Content Area:
Algebra
Numbers and Operations
### Learning Goals:
This lesson will help toddlers and preschoolers meet the following educational standards:
• Understand numbers, ways of representing numbers, relationships among numbers and number systems
• Understand patterns, relations and functions
### Learning Targets:
After this lesson, toddlers and preschoolers should be more proficient at:
• Developing understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connection
• Sorting, classifying and ordering objects by size, number and other properties
• Connecting number words and numerals to the quantities that they represent, using various physical models and representations
• Counting with understanding and recognizing “how many” in sets of objects
## Number Yard
### Lesson plan for toddlers/preschoolers
#### Step 1: Gather materials.
• The book, Grandpa Gazillion’s Number Yard by Laurie Keller
• A pre-made book for each child with the numbers 1-10 on each page (Write the number in words and numerals.)
• Markers or crayons
Note: Small parts pose a choking hazard and are not appropriate for children age five or under. Be sure to choose lesson materials that meet safety requirements.
#### Step 2: Introduce activity.
1. Explain that you are going to read a story about your crazy grandpa, who has a number yard instead of a junk yard. At this number yard, he uses numbers to help people. Ask the children if they can think of ways that numbers might help people.
2. Explain that they are also going to create their own number junkyard books. Invite the children to follow along as you read the story.
#### Step 3: Engage children in lesson activities.
1. Begin reading the story. After you read a page with a number on it, pause and have the children draw an object that corresponds with the number in their number books. For example, after you read the following statement from the book: “You are hungry and stuck on a tall mountain peak? A lollipop NINE can last you all week!” Ask the children to draw nine of something that they would want if they were stuck on a tall mountain peak. Continue this throughout the book until their number books are finished.
2. After the number books are completed, the children might want to dictate and have you write down what they want to say about each page.
• You can also create a number book with large numbers 1-20 on each page. Instead of representing the numbers with pictorial objects, the children can create other objects from the already existing numbers and incorporate the numbers into their drawings. For example, the number 8 can become glasses on the face of person.
#### Step 4: Vocabulary.
• Count: To identify the amount of something by number (e.g.,”Count the number of objects needed for the number nine.”)
• Represent: A quantity of something whose characteristics represent the entire batch (e.g.,”Draw the correct number of objects that represent the number nine.”)
#### Step 5: Adapt lesson for toddlers or preschoolers.
###### Toddlers may:
• not know number names or understand the count sequence
###### Child care providers may:
• Demonstrate what a group of nine looks like using manipulatives, count the manipulatives with the children and then support the children as they draw nine objects on the 9 page
• Read the book slowly (read a page or two a day and reinforce each number that is being worked on throughout the day)
###### Preschoolers may:
• Be able to successfully represent each number with pictorial representation
###### Child care providers may:
• Create a number book with large numbers 1-20 on each page.
• Instead of representing the number with pictorial objects, the children can create other objects from the already existing numbers and incorporate the numbers into their drawings (e.g., the number 8 can become glasses on the face of person)
### Suggested Books
• Grandpa Gazillion’s Number Yard by Laurie Keller (New York: Henry Holt and Company, 2005)
### Outdoor Connections
Get some exercise! Call out a number and an exercise and have the children do the corresponding exercises to that number while counting along: 10 jumping jacks, run in place for 15 seconds—counting as you complete the exercises or the time. |
# Vectors Class 12 Notes Maths Chapter 10 - CBSE
## What are Vectors ?
The dot mark field are mandatory, So please fill them in carefully
Vectors come in handy when we study those physical objects that have direction as well as magnitude. Let’s look at some of the terms associated with vectors :
## Scalar
It is a quantity which has magnitude but no direction. Examples : Mass, length, distance, speed etc.
## Vector
It is a quantity which has a magnitude as well as a specific direction. Examples : Weight, displacement, velocity etc. Denoted as
$$\overrightarrow{AB}$$
## Magnitude
The distance between initial and terminal points of a vector is called the magnitude of the vector.
$$\text{Denoted as}\space|\overrightarrow{AB}| \space \text{which is read as modulus of}\space \overrightarrow{AB}.$$
## Direction Cosines
Direction cosines of a vector are the cosines of the angle between the vector and the three coordinate axes.
• As you see in the figure the angle α, β,γ made by the vector A with the positive directions of x,y and z-axis respectively, are called its direction angles. The cosine values of these angles ,i.e., cos α, cos β and cos γ are called direction cosines.
• Denoted by l,m and n.
## Direction Ratios
Numbers that are proportional to the direction cosines of the line are called direction ratios of the line.
• Ax = A cos α, Ay = A cos β and Az = A cos γ
• Denoted as a, b and c.
• Let (l, m, n) be the direction cosines of a line and direction ratios of the line be (a, b, c). Then
$$\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=\frac{\sqrt{l^2+m^2+n^2}}{\sqrt{a^2+b^2+c^2}}=\pm \space \frac{1}{\sqrt{a^2+b^2+c^2}}\qquad\\l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}}(\because l^2+m^2+n^2=1)\\m=\pm\frac{b}{\sqrt{a^2+b^2+c^2}}\space\space\space\space n=\pm\frac{c}{\sqrt{a^2+b^2+c^2}}$$
## Vectors
Equal Vectors
Two vectors are said to be equal vectors if they have same magnitude and direction.
$$\overrightarrow{AB}=\overrightarrow{CD}$$
## Unit Vectors
If the magnitude of the vector is one , then it is called unit vector.
$$•\space\hat {a} \text{ is a unit vector in the direction of}\space \vec{a} \space\text{and}\space \hat a =\frac{\vec{a}}{|\vec{a}|}\\ \text{• Unit vectors in the directions of the positive x-axis, y-axis and z-axis are i, j, k.}$$
## Zero Vectors
If the magnitude of the vector is zero , then it is called null vector or zero vector. It can have any arbitrary direction.
## Parallel Vectors
Vectors which have same direction or exactly the opposite direction are called parallel vectors . The angle between them is 0° or 180°. Parallel vectors
are of two types :
## Like Parallel Vectors
If the angle between the vectors is 0°, then they are called like parallel vectors.
## Unlike Parallel Vectors
If the angle between the vectors is 180°, then they are called unlike parallel vectors.
## Collinear Vectors
Two or more vectors are said to be collinear vectors if they are parallel in same line irrespective of their magnitudes.
## Position Vector Of A Point
The vector OA is said to be a position vector of A with respect to Origin O, if A be any terminal point and O is the origin which is fixed.
## Negative Of A Vector
Two vectors are called negative vectors of each other if they have same magnitude but opposite direction. Denoted as
$$-\vec{a}$$
## Components Of A Vector
$$\overrightarrow{\text{OA}},\overrightarrow{\text{OB}}\space\text{AND}\space \overrightarrow{\text{OC}}\text{are the unit vectors along the axes OX, OY, and OZ respectively,}\text{and denoted by}\space \hat{i}, \hat{j}, \hat{k}.$$
$$\text{Now, consider the position vector}\space\overrightarrow{\text{OP}}\text{of a point P (x,y,z) as in fig}.\\\text{Let}\space \text{P}_1\space \text{be the foot of the perpendicular from P on the plane XOY. We, thus, See that}\\\text{P}_1 \text{P is parallel to z-}\hat{axis}.\hat{}\space \hat{\text{As}}\text{i, j, k are the unit vectors along the x,y and z-axes respectively, and by the definition of the coordinates of P,}\text{we have}\overrightarrow{P_1P}=\overrightarrow{OR}=Z\hat{k}. \space\text{Similarly,}\\\overrightarrow{QP}_1=\overrightarrow{OS}=y\hat{j}\space\text{and}\space\overrightarrow{OQ}=x\hat{i}.$$
$$\text{Therefore, it follows that}\space\overrightarrow{\text{OP}_1}={\overrightarrow{\text{OQ}}}+{\overrightarrow{\text{OP}}_1}=x\hat{i}+y\hat{j}\\\text{And}\overrightarrow{\text\space{OP}}=\overrightarrow{\text{OP}}_1+\overrightarrow{\text{P}_1P}=x\hat{i}+y\hat{j}+z\hat{k}\\\text{Hence , the position vector of P with reference to O is given by}\\\overrightarrow{\text{OP}}=r=x\hat{i}+y\hat{j}+z\hat{k}\\\text{This form of any vector is called its component form. Here x,y,z are called as the scalar components of }\text{are called the vector components of r along the respective axes.}$$
Triangle Law
$$\overrightarrow{\text{AC}}=\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}\implies \vec{c}=\vec{a}+\vec{b}\\•\space \vec{\text{c}}\space \text{is called the resultant of\space }\vec{\text{a}}\space\text{and}\space\vec{\text{b}}$$
The addition of vectors is achieved by ‘tail to nose’ placing of the directed line segments in a triangle.
Parallelogram Law
The result of adding two co-initial vectors is the vector represented by the diagonal of the parallelogram formed with the component vectors as adjacent sides. eg: in the fig. below,
## Multiplication Of A Vector By A Scalar
$$\text{\text{The product of}}\space \text{by a scalar ‘m’ is denoted as m.}\vec{a}\space \text{such that :}\\\text{• The support of ma is same or parallel to that of }\vec{a},\\ • |\text{m}\vec{\text{a}}|=|\text{m}||\vec{\text{a}}|\\ \text{• m}\vec{\text{a}}\space \text{has the direction same or opposite to that of a, according as m is +ve or -ve.}$$
## Position Vector Of A Point Dividing A Line Segment In A Given Ratio
For Internal Division
$$\text{\text{Let}}\space\vec{\text{a}}\space \text{and}\space \vec{\text{b}}\space \text{be the position vectors of two points A and B respectively and let C be a point with position vector}\space\vec{\text{c}}\space \text{dividing AB internally in the ratio m : n. Then the position vector of C is}\\\vec{\text{c}}=\overrightarrow{\text{OC}}=\frac{\text{m}\vec{\text{b}}+\text{n}\vec{\text{a}}}{\text{m+n}}$$
For External Division
$$\text{\text{Let}}\space\vec{\text{a}}\space \text{and}\space \vec{\text{b}}\space \text{be the position vectors of two points A and B respectively and let C be the point with position vector}\space \vec{\text{c}} \text{dividing AB externally in the ratio m : n. Then the position vector of C is}\\ \vec{\text{c}}=\overrightarrow{\text{OC}}=\frac{\text{m}\vec{\text{b}}-\text{n}\vec{\text{a}}}{\text{m+n}}$$
## Properties And Application Of Scalar(Dot) Product Of Vectors
$$\text{(i)}\space \vec{a}.\vec{a}=|\vec{a}|^2=a^2\\\qquad\text{(ii)}\space \vec{a}.\vec{b}=\vec{b}.\vec{a} \space \text{(commutativity)}\\\qquad\text{(iii)} \space \vec{a}.\vec{0}=0\\\qquad\text{(iv)}\space \vec{a}.(\vec{b}+\vec{c})=\vec{a}.\vec{b}+\vec{a}.\vec{c}\space\text{(Distributivity)}\\\qquad\text{(v)} \space\vec{a}.(\vec{b}-\vec{c})=\vec{a}.\vec{b}-\vec{a}.\vec{c}\\\qquad\text{(vi)}\space \vec{a}.\vec{b}=|\vec{a}|.|\vec{b}|.\text{cos} \space \theta\\\qquad \text{(vii) If x is any scalar, then}\space (x\vec{a}).\vec{b}=x(\vec{a}.\vec{b})=\vec{a}.(x\vec{b})\\\qquad\text{(viii)}\space(x\vec{a}).(y\vec{b})=\text{xy}(\vec{a}.\vec{b})\text{where x, y are scalars.}\\\qquad\text{(ix)}\space (\vec{a}+\vec{b}).(\vec{a}-\vec{b})=|\vec{a}|^2-|\vec{b}|^2\\\qquad\text{(x)}\space \text{(a + b)}^2=|\vec{a}|^2+|\vec{b}|^2+2\vec{a}.\vec{b}\\\qquad\text{(xi)}\space (\vec{\text{a}}-\vec{\text{b}})^2=|\vec{a}|^2+|\vec{b}|^2-2\vec{a}.\vec{b}\\\qquad\text{(xii) If}\space \hat i, \hat j, \hat k\space\text{are three-unit vectors along three mutually perpendicular lines, then}\\ \qquad •\space\space \hat i .\hat i=\hat j.\hat j=\hat k.\hat k=1\\•\space \hat{i} .\hat{j} = \hat{j} . \hat{k} = \hat{k} . \hat{i} = \hat{j} . \hat{i} = \hat{k} . \hat{j} = \hat{i} . \hat{k} = 0.\\\qquad\text{(xiii) If OX, OY, OZ are three mutually perpendicular axes and}\space\hat{i},\hat{j},\hat{k}\space \text{are unit vectors along these axes. If co-ordinates of any point P, are (x, y, z) then its position vector is}\\\text{x}\hat{i}+\text{y}\hat{j}+\text{z}\hat{k}.$$
## Properties Of Vector (Cross) Product Of Vectors
$$\text{(i)}\space \vec{a}×\vec{b}=-(\vec{b}×\vec{a})\\\qquad\text{(ii)}\space\vec{a}×\vec{a}=\vec{0}\\\qquad\text{(iii)}\space x(\vec{a}×\vec{b})=(x\vec{a})×\vec{b}=\vec{a}×(x\vec{b})\\\qquad\text{(iv)}\space(x\vec{a})×(y\vec{b})=\text{xy}(\vec{a}×\vec{b}).\\\qquad\text{(v)}\space|\vec{a}×\vec{b}|^2=|\vec{a}|^2|\vec{b}|^2-(\vec{a}.\vec{b})^2.\\\qquad\text{(vi)}\space \hat{i}×\hat{i}=\hat{j}×\hat{j}=\hat{k}×\hat{k}=0.\\\qquad\text{(vii)}\space \hat{i}×\hat{j}=\hat{k},\hat{j}×\hat{k}=\hat{i},\hat{k}×\hat{i}=\hat{j}\\\qquad\text{(viii) Two non-zero vectors are collinear if their cross product is equal to zero vector.}\\\qquad\text{(ix)} \space\text{If} \space \vec{a}_1\hat{i}+\vec{a_2}\hat{j}+\vec{a_3}\hat{k}\\\text{And} \space\vec{b}=\vec{b}_1\hat{i}+\vec{b_2}\hat{j}+\vec{b_3}\hat{k}\\\text{Then}\space\vec{a}×\vec{b}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3 \end{vmatrix}=(a_2b_3-a_3b_2)\hat{i}-(a_1b_3-a_3b_1)\hat{j}+(a_1b_2-a_2b_1)\hat{k}\\\qquad\text{(x) If a, b, c are three vectors, then}\\\vec{a}×(\vec{b}+\vec{c})=\vec{a}×\vec{b}+\vec{a}×\vec{c}\\(\vec{b}+\vec{c})×\vec{a}=\vec{b}×\vec{a}+\vec{c}×\vec{a}$$ |
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
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# Mixed Numbers as Improper Fractions
## 2 and 3/4 = [(4 x 2)+3]/4=11/4
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Mixed Numbers as Improper Fractions
Have you ever ordered multiple pizzas and then had to keep track of how many slices were eaten and how many weren't? Well, the sixth graders are having the same sort of dilemma.
Casey ordered eight pizzas for the drama club to enjoy. Each pizza was cut into ten slices. At the end of the pizza party, there were two whole pizzas and two slices left.
How many slices weren't eaten?
Can you express this as an improper fraction of pizza slices? Do you know what an improper fraction is?
This Concept is all about mixed numbers and improper fractions. At the end of it, you will know how to figure all of this out.
### Guidance
In the last Concept on measuring to a fraction of an inch, sometimes we ended up with measurements that included whole inches and parts of a whole inch or a fraction of an inch. When we have wholes and parts together, we have a new type of number. It is called a mixed number .
A mixed number is a number that has both wholes and parts in it.
$5 \frac{1}{4}$
Here we have a mixed number. We have five whole items and one-fourth of a whole.
Now you know how to identify a mixed number. The opposite of a mixed number is an improper fraction.
What is an improper fraction?
An improper fraction is a fraction that has a larger numerator and a smaller denominator.
Huh? What does this mean?
$\frac{12}{5}$
Let’s think about what this means. If the denominator tells us how many parts the whole has been divided into, then this whole has been divided into 5 parts. The numerator tells us how many parts of the whole we have in this case, we have twelve parts. What? If we have twelve out of five parts, then we have MORE than one whole. One whole would be five out of five parts, but we have 12 out of 5 parts. This is where mixed numbers come in.
How do we write a mixed number as an improper fraction?
To write a mixed number as an improper fraction, we want to write a fraction in terms of parts instead of in terms of wholes and parts.
Change $2 \frac{1}{3}$ to an improper fraction.
To do this, we multiply the whole number times the denominator and add the numerator. Then we put this over the original denominator.
2 $\times$ 3 + 1 $=$ 7
Our original denominator is 3.
Our answer is $2 \frac{1}{3} = \frac{7}{3}$ .
Notice that the mixed number and the improper fraction are also equivalent.
Change the following mixed numbers to improper fractions.
#### Example A
$3 \frac{1}{3}$
Solution: $\frac{10}{3}$
#### Example B
$5 \frac{2}{3}$
Solution: $\frac{17}{3}$
#### Example C
$6 \frac{1}{8}$
Solution: $\frac{49}{8}$
Now back to the pizza dilemma. Have you figured out the answer? Here is the original problem once again.
Casey ordered eight pizzas for the drama club to enjoy. Each pizza was cut into ten slices. At the end of the pizza party, there were two whole pizzas and two slices left.
How many slices weren't eaten?
Can you express this as an improper fraction of pizza slices? Do you know what an improper fraction is?
First, let's think about how many slices weren't eaten.
Two whole pizzas and two slices = $2 \frac{2}{10}$
We can convert that to an improper fraction.
Our answer is $\frac{22}{10}$ slices of pizza weren't eaten.
### Vocabulary
Mixed Number
a number made up of a whole number and a fraction
Improper Fractions
a fraction where the numerator is greater than the denominator
Equivalent
means equal
### Guided Practice
Here is one for you to try on your own.
Express $4 \frac{7}{8}$ as an improper fraction.
To do this, we multiply the denominator of the fraction with the whole number and add the numerator. This will give us our new numerator which is put over the original denominator.
$\frac{39}{8}$
### Practice
Directions: Write each mixed number as an improper fraction.
1. $2 \frac{1}{2}$
2. $3 \frac{1}{4}$
3. $5 \frac{1}{3}$
4. $4 \frac{2}{3}$
5. $6 \frac{1}{4}$
6. $6 \frac{2}{5}$
7. $7 \frac{1}{3}$
8. $8 \frac{2}{5}$
9. $7 \frac{4}{5}$
10. $8 \frac{2}{7}$
11. $8 \frac{3}{4}$
12. $9 \frac{5}{6}$
13. $6 \frac{5}{8}$
14. $9 \frac{2}{3}$
15. $5 \frac{1}{2}$
16. $16 \frac{1}{4}$
### Vocabulary Language: English
Equivalent
Equivalent
Equivalent means equal in value or meaning.
improper fraction
improper fraction
An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
Mixed Number
Mixed Number
A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$. |
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# How to find Mean, Median, Mode and Range ?
## Finding Mean, Median, Mode and Range
This post helps you to understand few very important and most commonly asked Statistics terms like Mean, Median, Mode and Range. After reading this complete post you are able to solve any questions based on these terms. Previously we have all ready posted Statistics notes in PDF .
These terms are explained in crystal clear manner with illustrative examples. If you still have any problem please comment below.
### MEAN, MEDIAN, MODE AND RANGE
#### MEAN
Mean – Mean is average of a group of numbers.
To find the mean add all the numbers together then divide that sum by the amount of numbers.
Another name for mean is average.
Key Point- “Mean is find by evening out the numbers
Example:
Q. Find Mean of 3,6,3,2,6
Ans- Mean= (3+6+3+2+6)/5 = 4
Illustrative Example:
Q. Find Mean of 9,11,13,19,23,27
Ans- Mean= (9+11+13+19+23+27)/6 = 17
Q. Find Mean of 3,10,15,28
Ans- Mean= (3+10+15+28)/4 = 14
Q. Find Mean of 2,3,8,12,20
Ans- Mean= (2+3+8+12+20)/5 = 9
Q. Find Mean of 27,34,42,53
Ans- Mean= (27+34+42+53)/4 = 39
#### MEDIAN
Median – The middle number in a set of ordered numbers is called median.
To find the median list all the numbers in correct order, being sure to include any doubles. Then find the middle number.
If there is an odd amount of numbers - the median is the middle number.
If there is an even amount of numbers - the median is found by adding the middle two numbers together and dividing the sum by 2 . The resulting number is the median.
Steps:
Step-1. - Arrange the given set of number in an order of least to greatest.
Step-2. - If there is an odd amount of numbers - the median is the middle number.
Step-3. - If there are two middle numbers, find the mean of these two numbers.
Key Point- “Median is in the Middle
Example:
Q. Find median of given set of numbers- 2,4,8,11,14
Solution:
First arrange in an order of least to greatest – 2,4,8,11,14
Now find middle of this set- 8
So median = 8 (Ans)
Q. Find median of given set of numbers- 22,19,25,20,28
Solution:
First arrange in an order of least to greatest – 19,20,22,25,28
Now find middle of this set- 22
So median = 22 (Ans)
Q. Find median of given set of numbers- 19,20,22,26,28,29
Solution:
First arrange in an order of least to greatest – 19,20,22,26,28,29
Now find middle of this set- 22 and 26
Here two number in middle So median= (22+26)/2= 48/2= 24
So median = 24 (Ans)
Q. Find median of given set of numbers- 17,11,8
Solution:
First arrange in an order of least to greatest – 8,11,17
Now find middle of this set- 11
So median = 11 (Ans)
Q. Find median of given set of numbers- 30,9,5,12,20
Solution:
First arrange in an order of least to greatest –5,9,12,20,30
Now find middle of this set- 12
So median = 12 (Ans)
Q. Find median of given set of numbers- 32,8,3,13,15,20
Solution:
First arrange in an order of least to greatest – 3,8,13,15,20,32
Now find middle of this set- 13 and 15
Here two number in middle So median= (13+15)/2= 28/2= 14
So median = 14 (Ans)
Q. Find median of given set of numbers- 54,6,82,68,26,79
Solution:
First arrange in an order of least to greatest – 6,26,54,68,79,82
Now find middle of this set- 54 and 68
Here two number in middle So median= (54+68)/2= 112/2= 61
So median = 61 (Ans)
#### MODE
Mode - The mode is that number which appears the most frequently in a set of given numbers.
There can be more than one mode provided they are listed the same amount of times.
Steps:
Step 1 – Arrange the numbers in order of least to greatest.
Step 2 – Find the number that is repeated the most.
Key Point- “Mode is the Most Popular”
Example:
Q. Find Mode of given set of numbers – 2,2,4,8,11,14
Solution:
First arrange these numbers in an order of least to greatest – 2,2,4,8,11,14
Now find number that is repeated the most- 2
So, Mode= 2 (Ans)
Q. Find Mode of given set of numbers – 22,19,25,20,19
Solution:
First arrange these numbers in an order of least to greatest – 19,19,20,22,25
Now find number that is repeated the most- 19
So, Mode= 19 (Ans)
Q. Find Mode of given set of numbers – 30,9,5,9,20
Solution:
First arrange these numbers in an order of least to greatest – 5,9,9,20,30
Now find number that is repeated the most- 9
So, Mode= 9 (Ans)
Q. Find Mode of given set of numbers – 2,3,3,10,10,5,10,11
Solution:
First arrange these numbers in an order of least to greatest – 2,3,3,5,10,10,10,11
Now find number that is repeated the most- 10
So, Mode= 10 (Ans)
Q. Find Mode of given set of numbers – 23,23,28,32,22,33
Solution:
First arrange these numbers in an order of least to greatest – 22,22,23,28,32,33
Now find number that is repeated the most- 22
So, Mode= 22 (Ans)
#### RANGE
Range - is the difference between the greatest and the least value in a set of numbers.
It is found by subtracting the lowest from the highest.
Steps:
Step 1 – Arrange the numbers in order from least to greatest.
Step 2 – Find the lowest and highest numbers.
Step 3 – Find the difference between these 2 numbers.
Key Point- “ Range is the distance between the lowest and highest number.”
Example:
Q. What is the Range?
2,2,4,8,11,14
Solution:
First arrange these numbers in an order of least to greatest – 2,2,4,8,11,14
Now find the lowest and the highest numbers – 2 and 14
So, Range= 14-2 = 12 (Ans)
Q. What is the Range?
22,19,25,20,28
Solution:
First arrange these numbers in an order of least to greatest – 19,20,22,25,28
Now find the lowest and the highest numbers – 19 and 28
So, Range= 28-19 = 9 (Ans)
Q. What is the Range?
30,9,5,9,20
Solution:
First arrange these numbers in an order of least to greatest – 5,9,9,20.30
Now find the lowest and the highest numbers – 5 and 30
So, Range= 30-5 = 25 (Ans)
Q. What is the Range?
23,22,28,32,22,33
Solution:
First arrange these numbers in an order of least to greatest – 22,22,23,28,32,33
Now find the lowest and the highest numbers – 22 and 33
So, Range= 33-22 = 11 (Ans)
Q. What is the Range?
32,9,4,12,20
Solution:
First arrange these numbers in an order of least to greatest – 4,9,12,20,32
Now find the lowest and the highest numbers – 4 and 32
So, Range= 32-4 = 28 (Ans) |
# Step Functions Also known as Discontinuous Functions
The graph below is an example of a step function. As you examine the graph, determine why you think it might be called a step function.
Do you see what looks like a set of steps? This is one reason why it is called a step function. It is better known as a discontinuous function.
Why do you think it is called a discontinuous function? Yes, it is not a continuous line, it stops and starts repeatedly.
So, the question may be, is it a function? Does it pass the vertical line test? Let's see!
It looks like the vertical lines may touch two points on the graph at the same time. However, take a look at the points. One is a closed circle and one is an open circle.
If you review our inequalities lesson, you will remember that a closed circle means that the point includes that particular point. But... an open circle does NOT include that point.
So, in this case, where it looks like the vertical line is touching two points, it is really only touching one point, because the open circle does not include that point. So, to answer our question, yes this is considered a function. It's not linear, and it's not quadratic. We call it a step function or a discontinuous function.
## How Do We Read and Interpret a Discontinuous Graph?
Let's take a look at our postage graph again.
This graph describes how much it will cost to send a letter depending on the weight of the letter. I've labeled the steps so that you better understand the explanation below.
Step 1: If the weight of the letter is over 0 oz and up to 1 oz (including 1 oz, since the circle is closed), it will cost 39 cents.
Step 2: If the weight of the letter is more than 1 oz (not 1 oz exactly because the circle is open) and up to 2 oz (including 2 oz since the circle is closed), then the price is 41 cents.
Step 3: If the weight of the letter is more than 2 oz (not 2 oz exactly because the circle is open) and up to 3 oz (including 3 oz since the circle is closed), then the price is 43 cents.
Steps 4-6 follow the same pattern as steps 1-3 described above.
As you can see, this graph tells you exactly how much your letter will cost depending on the weight. A discontinuous graph must be used because the price stays the same between ounces, but then changes to the next price as you reach a whole ounce.
Let's take a look at a few other discontinuous graphs and determine whether or not they are functions. These graphs may not look like "steps", but they are considered discontinuous.
This graph is not a function because when utilizing the vertical line test, it touches in two points.
Both points at x = 1 are solid, therefore the graph is discontinuous, but not a function.
This graph is a function because it passes the vertical line test.
Each vertical line only touches the graph at one point. (Although it looks like it touches at two points at x = -3, since one circle is "open" we do not include that as a point.)
Therefore, it is considered a discontinuous function. It is discontinuous at x = -3.
Let's practice creating and interpreting a graph for a discontinuous function.
## Example 1: Step Functions
A wholesale t-shirt manufacturer charges the following prices for t-shirt orders:
• \$20 per shirt for shirt orders up to 20 shirts.
• \$15 per shirt for shirt between 21 and 40 shirts.
• \$10 per shirt for shirt orders between 41 and 80 shirts.
• \$5 per shirt for shirt orders over 80 shirts.
• Sketch a graph of this discontinuous function.
• You've ordered 40 shirts and must pay shipping fees of \$10. How much is your total order?
## Solution
If I ordered 40 shirts and must pay \$10 in shipping fees, then my total order will cost \$610. (40 * \$15) +10 = 610.
## Example 2
This would be a great problem for you to try on your own:
In many states a "sales tax" is added to most goods that you buy. The tax rate varies from state to state. Let's suppose that your particular state issues a sales tax on any goods purchased.
You are selling candy bars. The taxable amounts and tax imposed up to \$1 are shown below.
For amounts between \$0.01 and \$0.20, the tax is \$.01.
For amounts greater than \$0.20 and less than or equal to \$0.40, the tax is \$0.02.
For amounts greater than \$0.40 and less than or equal to \$0.60, the tax is \$0.03.
For amounts greater than \$0.60 and less than or equal to \$0.80, the tax is \$0.04
For amounts greater than \$0.80 and less than or equal to \$1.00, the tax is \$0.05.
Complete the graph to show the tax that imposed on candy bars.
Use the graph to answer the following questions:
1. A candy bar costs \$0.55. What is the total cost with tax?
2. Your aunt purchased three candy bars at \$0.55 a piece. What is the total cost with tax?
3. Someone purchased 4 candy bars at \$0.55 a piece. They gave you \$2 and a quarter. Is this enough money to cover the candy bars and the tax? Explain your answer.
## Solution
The following is the completed graph for the discontinuous function.
The red circles indicate open circles.
1. If a candy bar costs \$0.55, then the total cost with tax is \$0.58. (0.55 +0.03)
2. The total cost of three candy bars is \$1.74.
3. If someone gave me \$2.25 for 4 candy bars, they would not have given me enough money. The total cost would be \$2.31
Great Job! You have successfully completed the Step functions lesson. |
# 2a Alge Tiles presentationNov20071
```Alge-Tiles
Making the Connection between the
Concrete ↔ Symbolic
(Alge-tiles)
↔
(Algebraic)
What are Alge-Tiles?
Alge-Tiles are rectangular and square
shapes (tiles) used to represent integers and
polynomials.
Examples: 1→
1x →
1x2 →
Objectives for this lesson
Using Alge-Tiles for the following:
- Combining like terms
- Multiplying polynomials
- Factoring
- Solving equations
Allow students to work in small groups when
doing this lesson.
Construction of Alge-Tiles
1 (let the side = one unit)
For one unit tile:
(it is a square tile)
1
Area = (1)(1) = 1
x
For a 1x tile
(it is a rectangular tile)
(unknown length therefore let it = x)
1
Side of unit tile = side of x tile
Area = (1)(x) = 1x
x
Side of x2 tile = side of x tile
For x2 tile:
(It is a square tile)
Other side of x2 tile = side of x tile
Area = (x)(x) = x2
x
Part I: Combining Like Terms
Prerequisites: prior to this lesson students would have been taught the
Zero Property
Outcomes: Grade 7 - B11, B12, B13
Use the Alge Tiles to represent the following:
3x
3
2x2
Part I: Combining Like Terms
For negative numbers use the other side of each
tile (the white side)
Use the Alge Tiles to represent the following:
-2x →
-4 →
-3x - 4 →
Part I: Combining Like Terms
Represent “2x” with tiles
Represent “3” with tiles
Can 2x tiles be combined with the tiles for 3 to make
one of our three shapes? Why or why not?
Therefore: simplify 2x + 3 =
2x + 3 can’t be simplified any further (can’t touch this)
Part I: Combining Like Terms
Combine like terms (use the tiles):
+
2x + 2x →
1 +1x +2 →
+
-2x + 3x +1→
Using the zero property
= 4x
= 1x+3 (ctt)
+
+
+
= 1x +1(ctt)
Part I: Combining Like Terms
After mastering several questions where students
were combing terms you could then pose the
question to the class working in groups:
“Is there a pattern or some kind of rule you
can come up with that you can use in all
situations when combining polynomials.”
In conclusion, when combining like terms you can
only combine terms that have the same tile shape
(concrete) → Algebraic: Can combine like terms if
they have the same variable and exponent.
Part II: Multiplying Polynomials
Prerequisites: Students were taught the distributive
property and finding the area of a rectangle.
Area(rectangle) = length x width
When multiplying polynomials the terms in each
bracket represents the width or length of a rectangle.
Find the area of a rectangle with sides 2 and 3. Two
can be the width and 3 would be the length.
The area of the rectangle would = (2)•(3) = 6
Part II: Multiplying Polynomials
We will use tiles to find the answer. The same premise will be
used as finding the area of a rectangle.
Make the length = 3 tiles
The width = 2 tiles
The tiles form a rectangle, use other tiles to fill in the rectangle
Once the rectangle is filled in remove the sides and what is left
is your answer in this case it is 6 or 6 unit tiles
Part II: Multiplying Polynomials
Try: (2x)(3x)→
Side: 3x
Side: 2x
Remove the sides
Therefore: (2x)(3x) = 6x2
Part II: Multiplying Polynomials
Try (1x + 2)(3)
Side: 1x + 2)
Side: 3
Therefore: (1x + 2)(3) = 3x + 6 (ctt)
Make rectangle, fill rectangle
Remove sides
Part II: Multiplying Polynomials
Try (1x +2)(1x -1)
Side: 1x - 1
Side: 1x + 2
Tiles remaining:
Simplify to get:
x2 + 2x – 1x – 2
x2 + 1x – 2 (ctt)
Part II: Multiplying Polynomials
Pattern: After mastering several questions where
students were combing terms you could then pose
the question to the class working in groups:
“Is there a pattern or some kind of rule you
can come up with that you can use in all
situations when multiplying polynomials.”
This can lead to a larger discussion where students
can put forth their ideas.
Part III: Factoring
Outcomes: Grade 9 – B9, B10, Grade 10 – B1, B3, C16
Take an expression like 2x + 4 and use the rectangle to factor.
You will go in reverse when being compared to multiplying
polynomials. (make the rectangle to help find the sides)
The factors will be the sides of the rectangle
i. Construct a rectangle using 2 ‘x’ tiles and 4 unit one tiles. This
can be tricky until you get the hang of it.
Part III: Factoring
Now make the sides; width and length of the
rectangle using the alge-tiles.
Side 1 : (1x + 2)
Side 2 : (2)
2x + 4 = (2)(1x +2)
Remove the rectangle and what is left are the factors of 2x +4
Part III: Factoring
Try factoring 3x + 6 with your tiles.
1x + 2
First make a rectangle
Make the sides
Remove the rectangle
3
The sides are the factors
Factors → (1x + 2)(3)
3x + 6 = (3)(1x + 2)
Part III: Factoring
Try factoring x2 + 5x + 6 (make rectangle)
(1x + 3)
**Hint: when the expression has x2,
Next, place the 6 unit tiles at the
bottom right hand corner of the x2 tile.
You will make a small rectangle(1x
with+ 2)
the unit tiles.
Then add the x tiles where needed to
complete the rectangle
3
2
When the rectangle is finished examine
to see
if+the
tiles
combine
to give
you
2it +
x
5x
6
=
(1x
+
3)
(1x
+
2
the original expression → x + 5x + 6
Next make the sides for the rectangle
Remove the rectangle and you have the factors. (1x + 3) (1x + 2)
2)
Part III: Factoring
What if someone tried the following:
Factor: x2 + 5x + 6 (make rectangle)
with the 6 unit tiles.
Now complete the rectangle using the x
tiles.
1
When the rectangle is finished examine it to
see if the tiles combine to give you the
original expression → x2 + 5x + 6
6
When the tiles are combined, the result is
x2 + 7x + 6, where is the mistake?
The unit tiles must be arranged in a rectangle so when the x tiles are used
to complete the rectangle they will combine to equal the middle term, in
this case 5x.
Factoring
Have students try to factor more trinomials
(refer to Alge-tile binder – Factoring section: F – 3b for additional
questions)
After mastering several questions where students were
factoring trinomials you could then pose the question to the
class :
“Is there a pattern or some kind of rule
you can come up with that you can use when
factoring trinomials?”
Part III: Factoring (negatives)
Try factoring:
x2 - 1x – 6
this case -6 which is 6 white unit tiles.
1x - 3
Remember to make a rectangle at the
bottom corner of the x2 tiles where the sides
have to add to equal the coefficient of the
1x + 2
middle term, -1.
-3
Next fill in the x tiles to make the
rectangle.
2
Now the rectangle is complete check
to see if the tiles combine to equal
x2 - 1x – 6.
Therefore x2 - 1x – 6 = (x – 3) (x + 2)
Fill in the sides and remove the rectangle
to give you the factors.
Part IV: Solving for X
Solve 2x + 1 = 5 using alge-tiles
Set up 2x + 1= 5 using tiles
=
1x = 2
Using the zero property to remove the 1 tile you add a -1 tile to both sides
On the left side -1 tile and +1 tile give us zero and you are left with 2 ‘x’ tiles
On the right side adding -1 tile gives you +4 tiles
Now 2 ‘x’ tiles = 4 unit tiles, (how many groups of 2 are in 4)
Therefore 1 ‘x’ tile = 2 unit tiles
Part IV: Solving for X
Solve 3x + 1 = 7
=
1x = 2
Add a -1 tile to both sides
Zero Property takes place
What’s left? 3 ‘x’ tiles = 6 unit tiles (how many groups of 3 are in 6)
Therefore 1x tile = 2 unit tiles
Part IV: Solving for X
Solve for x:
2x – 1 = 1x + 3
=
1x = 4
Now add +1 tile to both sides… zero property
You are left with 2x = 1x + 4
Add -1x tile to each side… zero property
Leaving 1x = 4
Alge-Tile Conclusion
Assessment: While students are working on
question sheet handout, go around to each group
and ask students to do some questions for you to
demonstrate what they have learned.
For practice refer to handout of questions for all four
sections:
Part I: Combining Like Terms
Part II: Multiplying Polynomials
Part III: Factoring
Part IV: Solving for an unknown
(P.S. the answers are at the end)
``` |
# 13.6 Facts about the correlation coefficient for linear regression (Page 3/3)
Page 3 / 3
• ${H}_{o}$ : $\rho$ = 0
• ${H}_{a}$ : $\rho$ ≠ 0
• $\alpha$ = 0.05
• The p-value is 0.026 (from LinRegTTest on your calculator or from computer software)
• The p-value, 0.026, is less than the significance level of $\alpha$ = 0.05
• Decision: Reject the Null Hypothesis ${H}_{o}$
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from 0.
• Because $r$ is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
## Method 2: using a table of critical values to make a decision
The 95% Critical Values of the Sample Correlation Coefficient Table at the end of this chapter (before the Summary ) may be used to give you a good idea of whether the computed value of $r$ is significant or not . Compare $r$ to the appropriate critical value in the table. If $r$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $r$ is significant, then you may want to use the line for prediction.
Suppose you computed $r=0.801$ using $n=10$ data points. $\text{df}=n-2=10-2=8$ . The critical values associated with $\text{df}=8$ are -0.632 and + 0.632. If $r$ $()\text{negative critical value}$ or $r>\text{positive critical value}$ , then $r$ is significant. Since $r=0.801$ and $0.801>0.632$ , $r$ is significant and the line may be used for prediction. If you view this example on a number line, it will help you.
Suppose you computed $r=-0.624$ with 14 data points. $\text{df}=14-2=12$ . The critical values are -0.532 and 0.532. Since $-0.624$ $()$ $-0.532$ , $r$ is significant and the line may be used for prediction
Suppose you computed $r=0.776$ and $n=6$ . $\text{df}=6-2=4$ . The critical values are -0.811 and 0.811. Since $-0.811$ $()$ $0.776$ $()$ $0.811$ , $r$ is not significant and the line should not be used for prediction.
## Third exam vs final exam example: critical value method
• Consider the third exam/final exam example .
• The line of best fit is: $\stackrel{^}{y}=-173.51+\text{4.83x}$ with $r=0.6631$ and there are $\mathrm{n = 11}$ data points.
• Can the regression line be used for prediction? Given a third exam score ( $x$ value), can we use the line to predict the final exam score (predicted $y$ value)?
• ${H}_{o}$ : $\rho$ = 0
• ${H}_{a}$ : $\rho$ ≠ 0
• $\alpha$ = 0.05
• Use the "95% Critical Value" table for $r$ with $\text{df}=\text{n}-2=11-2=9$
• The critical values are -0.602 and +0.602
• Since $0.6631>0.602$ , $r$ is significant.
• Decision: Reject ${H}_{o}$ :
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from 0.
• Because $r$ is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.
## Additional practice examples using critical values
Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if $r$ is significant and the line of best fit associated with each $r$ can be used to predict a $y$ value. If it helps, draw a number line.
1. $r=-0.567$ and the sample size, $n$ , is 19. The $\text{df}=n-2=17$ . The critical value is -0.456. $-0.567$ $()$ $-0.456$ so $r$ is significant.
2. $r=0.708$ and the sample size, $n$ , is 9. The $\text{df}=n-2=7$ . The critical value is 0.666. $0.708>0.666$ so $r$ is significant.
3. $r=0.134$ and the sample size, $n$ , is 14. The $\text{df}=\mathrm{14}-2=12$ . The critical value is 0.532. 0.134 is between -0.532 and 0.532 so $r$ is not significant.
4. $r=0$ and the sample size, $n$ , is 5. No matter what the dfs are, $r=0$ is between the two critical values so $r$ is not significant.
## Assumptions in testing the significance of the correlation coefficient
Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between $x$ and $y$ in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between $x$ and $y$ in the population.
The regression line equation that we calculate from the sample data gives the best fit line for our particular sample. We want to use this best fit line for the sample as an estimate of the best fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.
## The assumptions underlying the test of significance are:
• There is a linear relationship in the population that models the average value of $y$ for varying values of $x$ . In other words, the expected value of $y$ for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The $y$ values for any particular $x$ value are normally distributed about the line. This implies that there are more $y$ values scattered closer to the line than are scattered farther away. Assumption (1) above implies that these normal distributions are centered on the line: the means of these normal distributions of $y$ values lie on the line.
• The standard deviations of the population $y$ values about the line are equal for each value of $x$ . In other words, each of these normal distributions of $y$ values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
**With contributions from Roberta Bloom
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
A B C DHomework
Solutions for Session 4, Homework
See solutions for Problems: H1 | H2 | H3
Problem H1
Chris's method will produce tea that has too much mix. This method would work only if the original recipe called for the same amount of tea and water, which it doesn't. In terms of fractions, 22/30 is not the same proportion as 12/20.
• Pat's method is correct: 6/10 is the same proportion as 12/20, so the new mixture will have the right proportion of mix. • Sandy's method is also correct: 18/30 is the same proportion as 12/20. • Lee's method is identical to Chris's method. If you keep the difference between tea and water the same, there will be too much tea mix added at 30 cups. • Chris and Lee's suggestions are incorrect because they are absolute comparisons, where a relative comparison is needed to keep the proportion of tea mix the same.
Problem H2
d. The triangle with side lengths 6, 8, and 10 is similar looking. A good comparison is the measures of the three angles of the triangle.
Problem H3
a. The twins are Adam and David. We know they start at the starting line, so they must be the ones without any constants in their equations. b. The oldest must be Eric, since he is the one whose equation includes the instruction "-5", which means he begins 5 meters behind the starting line. c. Adam started at the starting line, and ran at 6 meters per second. Brett started 7 meters ahead, and ran at 4 meters per second. Caleb started 4 meters ahead, and ran at 5 meters per second. David started at the starting line, and ran at 5 meters per second. Eric started 5 meters behind the starting line, and ran at 7 meters per second.
d. The points of intersection on the graph represent when one brother passes another during the race. Their times (on the horizontal axis) and their distance from the start (on the vertical axis) are the same. e. Use t = 2 in all five equations, or refer to the graph. The order is: Brett (15 m), Caleb (14), Adam (12), David (10), and Eric (9). f. Caleb and David, who run at the same speed. We know this because their graphs form parallel lines, which always stay the same distance apart. g. Find the finishing times for the five by solving the equations for d = 30, or referring to the graph (draw a horizontal line at distance d = 30). Solving the equations shows that Adam and Eric tie for first (5 seconds), then Caleb (5.2), Brett (5.75), and David (6 seconds). h. Adam and David. Their graphs pass through the origin (0, 0) and their equations are in the form y = kx.
Session 4: Index | Notes | Solutions | Video |
# Basic set operations
In mathematics, a set is a collection of well-defined and distinct objects, where an object is something that is, or can be, formally defined. Numbers, integers, permutations, combinations, functions, points, lines, and segments are just a few examples of mathematical objects.
Set theory can be used in deductive reasoning and mathematical proofs, and as such, can be seen as a foundation through which most math can be derived.
There are four basic operations in set theory: unions, intersections, complements, and Cartesian products.
## Unions
Unions are a result of "adding" two sets together. Unions are denoted by the symbol "∪."
The union of two sets, A and B, written A ∪ B, includes all objects that are members of A, B, or both.
Example:
Given the following sets:
A = {4, 5, 6} B = {6, 7, 8} A ∪ B = {4,5,6} ∪ {6,7,8} = {4,5,6,7,8}
Below is a depiction of a union of two sets:
The shaded grey area represents the union of A and B.
## Intersections
Intersections are the result of determining the common members between two sets. Intersections are denoted as "∩."
The intersection of two sets, A and B, written A ∩ B, includes all objects that are in both A and B.
Example:
Given the following sets:
A = {4, 5, 6} B = {6, 7, 8} A ∩ B = {4,5,6} ∩ {6,7,8} = {6}
Below is a depiction of the intersection of two sets:
The shaded grey area represents the intersection of A and B.
## Complements
The complement of a set, A, refers to the elements that are not in A. Among other notations, the complement of A can be denoted as Ac. For example, in a case where all integers are being considered, if A were the set of all even integers, Ac would be the set of all odd integers.
Sets can be "subtracted." The difference between two sets, A and B, can be denoted as A \ B. This difference can be referred to as the relative complement of B in A and represents the set of all elements in A that are not in B. This difference can be depicted as follows:
In the context of complements, a universal set, U, can be said to contain all the subsets being discussed. In such a case, U \ A would be the complement of A. In other words, Ac = U \ A.
Below is a depiction of Ac:
The grey area represents the complement of A.
## Cartesian products
Cartesian products are the set of all ordered pairs, (a, b), where a and b are members of the sets A and B respectively. Cartesian products make it possible to construct a new set by associating each element in one set with each element in another set. Cartesian products can be denoted as A x B.
Example:
Given the following sets:
A = {4, 5, 6}
B = {6, 7, 8}
A×B = {4,5,6}×{6,7,8} = {(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,6),(6,7),(6,8)} |
# Let g(x) = f(x) + f(2+x), where $$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1 - \left| x \right|,\;\;\left| x \right| \le 1}\\ {0,\;\;\left| x \right| > 1} \end{array}} \right.$$The number of points where the function g is not differentiable is
1. 4
2. 5
3. 6
4. 7
Option 2 : 5
## Detailed Solution
Concept :
⇒ Let say the above graph depicts a function h(x).
⇒ Now according to the graph h(x) is continuous from A to C and differentiable at all points except B since it has a sharp turn.
⇒ h(x) is not continuous at C, hence also not differentiable at C.
⇒ h(x) is continuous from D and also differentiable from D except at E since the graph took a sharp turn at E.
Explanation :
⇒ From the above concept, we can find the points where a function is differentiable or not with a graph.
Given function g(x) = f(x) + f(2+x).
⇒ $$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1 - \left| x \right|,\;\;\left| x \right| \le 1}\\ {0,\;\;\left| x \right| > 1} \end{array}} \right.$$.
⇒ |x| is always ≥ 0. Thud f(x) can be rewritten as :
⇒ $$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0 ,x <{-1}}\\ {1+x, x \in [-1,0)}\\ {1-x, x \in [0,1]}\\ {0, x > 1} \end{array}} \right.$$. Thus the graph of f(x) would be as follows :
⇒
⇒ With this f(x), g(x) can be shorten to $$g\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0 ,x < -3}\\ {3+x, x \in [-3,-2)}\\ {-1-x, x \in [-2,-1)}\\ {1+x, x \in [-1,0)}\\ {1-x, x \in [0,1]}\\ {0, x >1} \end{array}} \right.$$
⇒ The graph of g(x) would be as follows :
⇒
⇒ Thus, according to the concept explained, g(x) is not differentiable at 5 points. |
Find the just common denominator $$\frac{4}{6}$$ + $$\frac{2}{3}$$
$$\frac{4}{6}$$ + $$\frac{2}{3}$$
First, simplify $$\frac{4}{6}$$ to $$\frac{2}{3}$$ / Your problem should look like: $$\frac{2}{3}$$ + $$\frac{2}{3}$$
Second, join the denominators. / Your problem should look like: $$\frac{2+2}{3}$$
Third, simplify. / Your problem should look like: $$\frac{4}{3}$$
Fourth, convert to mixed fraction. / Your problem should look like: $$1\frac{1}{3}$$
Mixed number form: $$1\frac{1}{3}$$
Exact form: $$\frac{4}{3}$$
$$\frac{4}{6} + \frac{2}{3}$$ =?
You need to find a common denominator to add them which is 6
6×1=6 and 3×2=6
What you do on the denominator, you must do in the numerator
$$\frac{4}{6} + \frac{4}{5}$$ = $$\frac{8}{6}$$
To reduce, find a common factor which in this case is 2
8÷2=4 and 6÷2=3 ⇒ $$\frac{4}{3}$$
To change to a mix fraction, divide the numerator by the denominator
$$\frac{4}{3}$$ ⇒ $$1 \frac{1}{3}$$
Answer: $$\frac{4}{3}$$ as an improper fraction or $$1 \frac{1}{3}$$ as a mixed fraction
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# Ex. 6.6 Q7 Triangles Solution - NCERT Maths Class 10
Go back to 'Ex.6.6'
## Question
In Fig. below, two chords $$AB$$ and $$CD$$ intersect each other at the point $$P.$$
Prove that:
(i) $$\Delta APC\,\text{~}\Delta {\text{ }}DPB$$
(ii) $$AP. PB = CP. DP$$
Video Solution
Triangles
Ex 6.6 | Question 7
## Text Solution
Reasoning:
As we know that, two triangles, are similar if:
(i) Their corresponding angles are equal.
(ii) Their corresponding sides are in the same ratio.
As we know that angles in the same segment of a circle are equal.
Steps:
(i) In, $$\Delta APC \;\text{and}\; \Delta DPB$$
$$\angle APC = \angle DPB$$
(Vertically opposite angles)
$$\angle PAC = \angle PDB$$
(Angles in the same segment)
$$\Rightarrow \Delta APC\text{~}\Delta DPB$$
(A.A criterion)
(ii) In, $$\Delta APC \;\text{and}\; \Delta DPB$$
\begin{align} \frac{{AP}}{{PD}}& = \frac{{PC}}{{PB}} = \frac{{AC}}{{DB}} \\ & \quad \left[ {\Delta APC\text{~}\Delta DPB} \right]\\ \frac{{AP}}{{PD}} &= \frac{{PC}}{{PB}} \end{align}
$$\Rightarrow \;\;AP.PB = PC.PD$$
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# Arithmetic & Geometric sequences, recursive formulae
Page 1 / 2
## Introduction
In this chapter we extend the arithmetic and quadratic sequences studied in earlier grades, to geometric sequences. We also look at series, which is the summing of the terms in a sequence.
## Arithmetic sequences
The simplest type of numerical sequence is an arithmetic sequence .
Arithmetic Sequence
An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term
For example, $1,2,3,4,5,6,...$ is an arithmetic sequence because you add 1 to the current term to get the next term:
first term: 1 second term: 2=1+1 third term: 3=2+1 $⋮$ ${n}^{\mathrm{th}}$ term: $n=\left(n-1\right)+1$
## Common difference :
Find the constant value that is added to get the following sequences and write out the next 5 terms.
1. $2,6,10,14,18,22,...$
2. $-5,-3,-1,1,3,...$
3. $1,4,7,10,13,16,...$
4. $-1,10,21,32,43,54,...$
5. $3,0,-3,-6,-9,-12,...$
## General equation for the ${n}^{th}$ -term of an arithmetic sequence
More formally, the number we start out with is called ${a}_{1}$ (the first term), and the difference between each successive term is denoted d , called the common difference .
The general arithmetic sequence looks like:
$\begin{array}{ccc}\hfill {a}_{1}& =& {a}_{1}\hfill \\ \hfill {a}_{2}& =& {a}_{1}+d\hfill \\ \hfill {a}_{3}& =& {a}_{2}+d=\left({a}_{1}+d\right)+d={a}_{1}+2d\hfill \\ \hfill {a}_{4}& =& {a}_{3}+d=\left({a}_{1}+2d\right)+d={a}_{1}+3d\hfill \\ \hfill ...\\ \hfill {a}_{n}& =& {a}_{1}+d·\left(n-1\right)\hfill \end{array}$
Thus, the equation for the ${n}^{th}$ -term will be:
${a}_{n}={a}_{1}+d·\left(n-1\right)$
Given ${a}_{1}$ and the common difference, $d$ , the entire set of numbers belonging to an arithmetic sequence can be generated.
Arithmetic Sequence
An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term:
$\begin{array}{c}\hfill {a}_{n}={a}_{n-1}+d\end{array}$
where
• ${a}_{n}$ represents the new term, the ${n}^{th}$ -term, that is calculated;
• ${a}_{n-1}$ represents the previous term, the ${\left(n-1\right)}^{th}$ -term;
• $d$ represents some constant.
Test for Arithmetic Sequences
A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant:
${a}_{2}-{a}_{1}={a}_{3}-{a}_{2}={a}_{n}-{a}_{n-1}=d$
This is quite an important equation, and is the definitive test for an arithmetic sequence. If this condition does not hold, the sequence is not an arithmetic sequence.
## Plotting a graph of terms in an arithmetic sequence
Plotting a graph of the terms of sequence sometimes helps in determining the type of sequence involved.For an arithmetic sequence, plotting ${a}_{n}$ vs. $n$ results in:
## Geometric sequences
Geometric Sequences
A geometric sequence is a sequence in which every number in the sequence is equal to the previous number in the sequence, multiplied by a constant number.
This means that the ratio between consecutive numbers in the geometric sequence is a constant. We will explain what we mean by ratio after looking at the following example.
## What is influenza?
Influenza (commonly called “the flu”) is caused by the influenza virus, which infects the respiratory tract (nose, throat, lungs). It can cause mild to severeillness that most of us get during winter time. The main way that the influenza virus is spread is from person to person in respiratory droplets of coughs and sneezes. (This is called “dropletspread”.) This can happen when droplets from a cough or sneeze of an infected person are propelled (generally, up to a metre) through the air and deposited on the mouth or nose of people nearby. Itis good practise to cover your mouth when you cough or sneeze so as not to infect others around you when you have the flu.
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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How do you simplify (x^2)^4*3x^5 and write it using only positive exponents?
$3 {x}^{13}$
Explanation:
We have
${\left({x}^{2}\right)}^{4} \times 3 {x}^{5}$
We can rewrite the first term using the rule ${\left({x}^{a}\right)}^{b} = {x}^{a b}$:
${x}^{8} \times 3 {x}^{5}$
Before we go further combining x terms, let's work with the 3. I'll do that by rewriting the equation this way:
${x}^{8} \times 3 {x}^{5} = {x}^{8} \times 3 \times {x}^{5}$
We can now combine the x terms using the rule ${x}^{a} \times {x}^{b} = {x}^{a + b}$
${x}^{13} \times 3$
and this just simplifies down to
$3 {x}^{13}$ |
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Level Three > Geometry and Measurement
# Fred's Rent-a-Fence
Student Activity:
Click on the image to enlarge it. Click again to close. Download PDF (1027 KB)
Purpose:
This is a level 3 number and measurement activity from the Figure It Out series.
Specific Learning Outcomes:
use multiplication to solve perimeter and area problems.
Description of mathematics:
Use this activity to encourage transition from advanced additive strategies (stage 6) to advanced multiplicative strategies (stage 7).
Required Resource Materials:
FIO, Levels 3-4, Multiplicative Thinking, Fred's Rent-a-Fence, page 19
Activity:
#### Activity
This activity is based on the perimeter and area of rectangles. As a general introduction, have your students look at this rectangle where the side lengths are given as l and w.
The area can be found by: area = l x w.
The perimeter can be found by: perimeter = 2 x l + 2 x w.
Problems that involve maximising or minimising one measurement while either holding the other constant or minimising it are common in the real world. Fred’s fence is typical of constrained maximisation or minimisation problems.
Students exploring question 1 are likely to try different side lengths that will result in an area of 80 square metres. The problem requires a systematic approach, so encourage your students to organise their results in a table or organised list:
Side A Side B Area Perimeter 1 80 80 162 2 40 80 84 4 20 80 48 5 16 80 42 8 10 80 36
In this way, the students can find all the solutions with whole-number side measurements and calculate the perimeters at the same time. They may notice that the closer the side measurements become to each other, the smaller the perimeter becomes.
Encourage your students to explore the minimum perimeters for rectangles with the areas 16, 36, and 64 (square numbers). They will find that the perimeter is minimised when the rectangle is a square. In this situation, the length of each side is the square root of the area. They can then go back to question 1 with the knowledge that the solution is the closest whole number to √80 = 8.944 (to 4 significant figures). Students are likely to argue that the question asked for a rectangle and that a rectangle is not a square. It is worth stopping to discuss this reasonable view. In everyday use, a rectangle and a square are different shapes, but in mathematics, a square is just a special case of a rectangle.
Provide the students with a set of rectangles and squares and ask them to describe the attributes of these shapes. Encourage them to come up with minimal definitions, listing just the attributes that are absolutely necessary to define the shape. Students will typically say that a rectangle has:
• 4 sides
• 4 right-angled corners
• 2 pairs of parallel sides.
If you ask them to draw a 4-sided polygon that has right-angled corners but does not have 2 pairs of parallel sides, they will find that this is impossible. So it is not necessary to state that opposite sides must be parallel. This gives us the minimal definition for a rectangle. The minimal definition of a square is “a 4-sided polygon with right-angled corners and equal sides”. Squares are therefore a subclass of rectangles.
In the Investigation, students try to find rectangles that have the same number for the measurement of their perimeter as they do for the measurement of their area.
One solution is a square with sides of 4 metres. Its perimeter is 16 metres, and its area is 16 square metres. If they are systematic, students should be able to establish the existence of two other whole-number solutions.
They could begin by setting the length (at, say, 2 metres) and exploring what widths might work. They will discover that no whole-number solution will work for a side length of 2. But if they then try 3, they will find that a 3 x 6 rectangle has an area of 18 square metres and a perimeter of 18 metres. 6 x 3 is a third solution, but this is not a genuinely different rectangle.
Having got this far, your students may guess that there are other rectangles that meet the requirement but that they do not have whole-number sides. There are in fact an infinite number of such rectangles. In the table below, there are six rectangles that happen to have a whole-number measurement for one of their two dimensions. You could give your students the length of side b and challenge them to find the length of side a (in bold in the table), using a trial-and-improvement strategy.
Side a Side b Area Perimeter 7 2.8 19.6 19.6 10 2.5 25 25 12 2.4 28.8 28.8 18 2.25 40.5 40.5 22 2.2 48.4 48.4 27 2.16 58.32 58.32
There is an algebraic relationship between the pairs of values of a and b that satisfy the requirement that the number of perimeter units must be equal to the number of units of area. The relationship can be expressed in this way:
(To find the length of the second side, double the length of the first and divide by its length less 2.) Students who are developing an understanding of symbolic notation may like to try using this formula to find other pairs for a and b with the help of a calculator or spreadsheet program such as that shown.
Numeracy Project materials (see http://www.nzmaths.co.nz/numeracy-projects)
Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics Investigating Area, page 11
Figure It Out
Number: Book Three, Years 7–8, Level 4 Orchard Antics, page 23
Number Sense and Algebraic Thinking: Book One, Levels 3–4 Tile the Town, Tiny!, pages 20–21
1. a. 5 different rectangular shapes are: 1 m by 80 m, 2 m by 40 m, 4 m by 20 m, 5 m by 16 m, and 8 m by 10 m. Only the last two shapes would suit the dodgems (the other three would be too narrow).
b. The 8 m by 10 m rectangle would use 36 panels and cost \$108, which is cheaper than the other options. It is one of the shapes that would suit the dodgems.
2. a. There are 12 different-sized rectangles that could be made.
Width 1 2 3 4 5 6 7 8 9 10 11 12 Length 23 22 21 20 19 18 17 16 15 14 13 12
b. The largest option is 12 m wide and 12 m long, which gives an area of 144 m2. The shape of this area is a square.
3. 50 m. The length must be 15 m because 10 x 15 = 150. 2 x (10 + 15) = 2 x 25 = 50 gives the perimeter.
Investigation
Answers may vary. There are three whole-number solutions: 4 x 4, 3 x 6, and 6 x 3 (which is the same as 3 x 6). There is an infinite number of solutions if rectangles with only one whole-number side or no whole-number sides are included.
AttachmentSize
FredsRent-a-Fence.pdf1 MB
## Square Skills
This is a level 3 number and measurement activity from the Figure It Out series.
## Rubber Band Rectangles
This is a level 3 number and measurement activity from the Figure It Out series.
## Your Half or My Half?
This is a level 3 measurement activity from the Figure It Out theme series.
## Measure Up
This is a level 3 measurement and transformation activity from the Figure It Out theme series.
## Is there Room?
This is a level 3 measurement activity from the Figure It Out theme series. |
# ML Aggarwal Knowing Our Numbers Exe-1.5 Class 6 ICSE Maths Solutions
ML Aggarwal Knowing Our Numbers Exe-1.5 Class 6 ICSE Maths Solutions. We Provide Step by Step Answer of Exe-1.5 Questions for Knowing Our Numbers as council prescribe guideline for upcoming board exam. Visit official Website CISCE for detail information about ICSE Board Class-6.
## ML Aggarwal Knowing Our Numbers Exe-1.5 Class 6 ICSE Maths Solutions
Board ICSE Publications Avichal Publishig Company (APC) Subject Maths Class 6th Chapter-1 Knowing Our Numbers Writer ML Aggarwal Book Name Understanding Topics Solution of Exe-1.5 Questions Edition 2023-2024
### Knowing Our Numbers Exe-1.5
ML Aggarwal Class 6 ICSE Maths Solutions
Page-20
#### Question 1. Round off each of the following numbers to their nearest tens:
(i) 77
(ii) 903
(iii) 1205
(iv) 999
(i) 77–nearest tens= 80
(ii) 903 nearest tens= 900
(iii) 1205 nearest tens= 1210
(iv) 999 nearest tens= 1000
#### Question- 2. Estimate each of the following numbers to their nearest hundreds:
(i) 1246
(ii) 32057
(iii) 53961
(iv) 555555
(i) 1246 — nearest hundreds 1200
(ii) 32057– nearest hundreds 32100
(iii) 53961 nearest hundreds 5400
(iv) 555555 nearest hundreds 555600
#### Question 3. Estimate each of the following numbers to their nearest thousands:
(i) 5706
(ii) 378
(iii) 47,599
(iv) 1,09,736
(i) 5706– nearest thousands:6000
(ii) 378– nearest thousands:0
(iii) 47,599– nearest thousands:48000
(iv) 1,09,736– nearest thousands:1,10,000
#### Question 4. Give a rough estimate (by rounding off to nearest hundreds) and also a closer estimate (by rounding off to nearest tens):
(i) 439 + 334 + 4317
(ii) 8325 – 491
(iii) 1,08,734-47,599
(iv) 4,89,348 – 48,365
(i) 439 + 334 + 4317
rounding off to nearest hundreds
400+300+4300
=5000
a closer estimate (by rounding off to nearest tens):
440+330+4320
=5090
(ii) 8325 – 491
rounding off to nearest hundreds
8300-500
=7800
a closer estimate (by rounding off to nearest tens):
8330-490
=7840
(iii) 1,08,734-47,599
rounding off to nearest hundreds
108700 – 47600 = 61100
a closer estimate (by rounding off to nearest tens):
108730 – 47600
= 61130
(iv) 4,89,348 – 48,365
rounding off to nearest hundreds
489300 – 48400 = 440900
a closer estimate (by rounding off to nearest tens):
489350 – 48370
= 440980
#### Question 5. Estimate each of the following by rounding off each number nearest to its greatest place:
(i) 730 + 998
(ii) 5,290 + 17,986
(iii) 796-314
(iv) 28,292 – 21,496
(i) 730 + 998= 700+1000= 1700
(ii) 5,290 + 17,986= 5000+18000=23000
(iii) 796-314=800-300=500
(iv) 28,292 – 21,496=28000-21000=7000
#### Question 6. Estimate the following products by rounding off each of its factors nearest to its greatest place:
(i) 578 × 161
(ii) 9650 × 27
(i) 578 × 161=600 x 200 = 120000
(ii) 9650 × 27= 10000 x 30 = 300000
#### Question -7. Estimate the following products by rounding off each of its factors nearest to its hundreds place:
(i) 5281 × 3491
(ii) 1387 × 888
(i) 5281 × 3491= 5300 x 3500 = 1,85,50,000
(ii) 1387 × 888= 1400 x 900 = 12,60,000
#### Question -8. Estimate the following quotients by rounding off each number to its nearest ens:
(i) 423 ÷ 29
(ii) 777 ÷ 27
(i) 423 ÷ 29 = 430 ÷ 30 = 14
(ii) 777 ÷ 27 = 780 ÷ 30 = 26
### Knowing Our Numbers Exe-1.5
ML Aggarwal Class 6 ICSE Maths Solutions
Page-21
#### Question -9. Estimate the following quotients by rounding off each number to its nearest hundreds :
(i) 2472 ÷ 493
(ii) 7459 ÷ 286
(i) 2472 ÷ 493 = 2500 ÷ 500 = 5
(ii) 7459 ÷ 286 = 7500 ÷ 300 = 25
— : End of ML Aggarwal Knowing Our Numbers Exe-1.5 Class 6 ICSE Maths Solutions :–
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# MP Board Class 6th Maths Solutions Chapter 14 Practical Geometry Ex 14.5
## MP Board Class 6th Maths Solutions Chapter 14 Practical Geometry Ex 14.5
Question 1.
Draw $$\overline{A B}$$ of length 7.3 cm and find its axis of symmetry.
Solution:
Axis of symmetry of line segment $$\overline{A B}$$ will be the perpendicular bisector of $$\overline{A B}$$. So, draw the perpendicular bisector of AB.
Steps of construction:
(i) Draw $$\overline{A B}$$ = 7.3 cm.
(ii) With A as centre and radius more than half of AB, draw two arcs, one on each side of AB.
(iii) With B as a centre and the same radius as in step (ii), draw arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the axis of symmetry of the line segment AB.
Question 2.
Draw a line segment of length 9.5 cm and construct its perpendicular bisector.
Solution:
Steps of construction :
(i) Draw $$\overline{A B}$$ = 9.5 cm.
(ii) With A as centre and radius more than half of AB, draw two arcs one on each side of AB.
(iii) With B as a centre and the same radius as in step (ii), draw arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the perpendicular bisector of $$\overline{A B}$$.
Question 3.
Draw the perpendicular bisector of $$\overline{X Y}$$ whose length is 10.3 cm.
(a) Take any point P on the bisector drawn. Examine whether PX = PY.
(b) If M is the mid point of $$\overline{X Y}$$, what can you say about the lengths MX and XY?
Solution:
Steps of construction :
(i) Draw $$\overline{X Y}$$ = 10.3 cm.
(ii) With X as centre and radius more than half of XY, draw two arcs one on each side of XY.
(iii) With Y as centre and the same radius as in step (ii), draw two arcs cutting the arcs drawn in the previous step at C and D.
(iv) Join CD. Then CD is the required perpendicular bisector of $$\overline{X Y}$$.
Now
(a) Take any point P on the bisector drawn. With the help of divider we can check that $$\overline{P X}=\overline{P Y}$$ if P is the point of intersection of XY and CD.
(b) If M is the mid-point of $$\overline{X Y}$$, then $$\overline{M X}=\frac{1}{2} \overline{X Y}$$
Question 4.
Draw a line segment of length 12.8 cm. Using compasses, divide it into four equal parts. Verify by actual measurement.
Solution:
Steps of construction:
(i) Draw AB = 12.8 cm.
(ii) Draw the perpendicular bisector of $$\overline{A B}$$ which cuts it at C. Thus, C is the mid-point of $$\overline{A B}$$.
(iii) Draw the perpendicular bisector of $$\overline{A C}$$ which cuts it at D. Thus D is the mid-point of AC.
(iv) Again, draw the perpendicular bisector of $$\overline{C B}$$ which cuts it at E. Thus, E is the mid-point of $$\overline{C B}$$.
(v) Now, point C, D and E divide $$\overline{A B}$$ in four equal parts.
(vi) By actual measurement, we find that $$\overline{A D}=\overline{D C}=\overline{C E}=\overline{E B}$$ = 3.2 cm
Question 5.
With $$\overline{P Q}$$ of length 6.1 cm as diameter, draw a circle.
Solution:
Steps of construction :
(i) Draw a line segment $$\overline{P Q}$$ = 6.1 cm.
(ii) Draw the perpendicular bisector of PQ which cuts, it at O. Thus O is the mid-point of $$\overline{P Q}$$.
(iii) Taking O as centre and OP or OQ as radius draw a circle where $$\overline{P Q}$$ is the diameter.
Question 6.
Draw a circle with centre C and radius 3.4 cm. Draw any chord $$\overline{A B}$$. Construct the perpendicular bisector of $$\overline{A B}$$ and examine if it passes through C.
Solution:
Steps of construction:
(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw any chord $$\overline{A B}$$.
(iii) With A as center and radius more than half of $$\overline{A B}$$, draw two arcs one on each side of AB.
(iv) With B as a centre and the radius same as in step (iii), draw two arcs cutting the arcs drawn in the previous step at P and Q.
(v) Join PQ. Then PQ is the perpendicular bisector of $$\overline{A B}$$.
(vi) This perpendicular bisector of $$\overline{A B}$$ passes through the centre C of the circle.
Question 7.
Repeat Question 6, if $$\overline{A B}$$ happens to be a diameter.
Solution:
Steps of construction :
(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw its diameter AB.
(iii) With A as center and radius more than half of AB, draw two arcs one on each side of AB.
(iv) With B as a centre and the radius same as in step (iii), draw two arcs cutting the arcs drawn in the previous step at P and Q.
(v) Join PQ. Then PQ is the perpendicular bisector of $$\overline{A B}$$.
(vi) We observe that this perpendicular bisector of $$\overline{A B}$$ intersect it at the centre C of the circle.
Question 8.
Draw a circle of radius 4 cm. Draw any two of its chords. Construct the perpendicular bisectors of these chords. Where do they meet?
Solution:
Steps of construction :
(i) Draw the circle with centre O and radius 4 cm.
(ii) Draw any two chords AB and CD in this, circle.
(iii) With A as center and radius more than half AB, draw two arcs one on each side of AB.
(iv) With B as centre and radius same as in step (ii), draw two arcs cutting the arcs drawn in previous step at E and F.
(v) Join EF. Thus EF is the perpendicular bisector of chord AB.
(vi) Similarly, draw GH the perpendicular bisector of chord CD.
(vii) These two perpendicular bisectors meet at O, the centre of the circle.
Question 9.
Draw any angle with vertex O. Take a point A on one’ of its arms and B on another such that OA = OB. Draw the perpendicular bisectors of $$\overline{O A}$$ and $$\overline{O B}$$. Let them meet at P. Is PA = PB?
Solution:
Steps of construction :
(i) Draw any angle with vertex O.
(ii) Take a point A on one of its arms and B on another such that OA = OB.
(iii) Draw perpendicular bisector of OA and OB.
(iv) Let they meet at P. Join PA and PB.
(v) With the help of divider, we check that PA = PB. |
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Kindergarten Counting and Comparing Numbers CCSS: K.CC.B.5
How many squares?
1. 5
2. 10
3. 15
4. 20
Kindergarten Counting and Comparing Numbers CCSS: K.CC.A.3
Grade 1 Counting and Comparing Numbers CCSS: 1.NBT.A.1
How many squares?
1. 12
2. 24
3. 48
4. 60
Kindergarten Counting and Comparing Numbers CCSS: K.CC.A.1
Which problem best explains the following model?
1. I planted 3 rows of corn in my garden. There were 5 plants in each row.
2. I planted 15 rows of corn in my garden. There were 3 plants in each row.
3. I planted 5 rows of corn in my garden. There were 15 plants in each row.
4. I planted 4 rows of corn in my garden. There were 3 plants in each row.
Which problem best explains the following model?
1. There are 2 rows of chairs with 4 chairs in each row.
2. There are 4 rows of chairs with 5 chairs in each row.
3. There are 5 rows of chairs with 6 chairs in each row.
4. There are 2 rows of chairs with 5 chairs in each row.
Kindergarten Counting and Comparing Numbers CCSS: K.MD.B.3
Kindergarten Counting and Comparing Numbers CCSS: K.CC.C.6
Kindergarten Counting and Comparing Numbers CCSS: K.CC.B.5
How many squares?
1. 4
2. 6
3. 11
4. 14
Kindergarten Counting and Comparing Numbers CCSS: K.CC.C.6
Grade 6 Number Properties CCSS: 6.EE.A.3
Which equation represents the model?
$=$
1. $(x + 1) = 2x$
2. $2(x + 1) = 2x + 2$
3. $(x + 1)(x + 1) = 2 (x + 1)$
4. $2(x + 1)(x + 1) = 4x + 2$
Kindergarten Counting and Comparing Numbers CCSS: K.CC.B.5
How many squares?
1. 4
2. 8
3. 12
4. 16
Kindergarten Counting and Comparing Numbers CCSS: K.CC.A.2
Kindergarten Counting and Comparing Numbers CCSS: K.CC.A.1
How many squares?
1. 10
2. 20
3. 30
4. 40
Kindergarten Counting and Comparing Numbers CCSS: K.CC.A.2
1. $3 + 4 = 7$
2. $3 + 3 + 3 = 9$
3. $4 + 4 + 4 = 12$ |
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Quadratic Inequalities ( Read ) | Analysis | CK-12 Foundation
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Progress
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How would you express the following as a function?
You are supposed to mow your square-shaped lawn for your parents, but the mower only has part of a tank of gas. If you can mow 2500 sf per gallon, and the mower has approximately 2.5 gallons in it, what is the maximum length of one side of the lawn you can mow? If your lawn is 75 feet long, will you need more gas?
### Guidance
Quadratic inequalities are inequalities that have one of the following forms
$ax^{2}+bx+c>0$
and
$ax^{2}+bx+c<0$
We can solve these inequalities by using the techniques that we have learned about solving quadratic equations. For example, consider the graph of the equation:
$y=f(x)=x^{2}+x-6$
Notice that the curve intersects the $x-$ axis at -3 and 2. From graph, we notice the followings
• If x<3$x< -3$ then f(x)>0$f(x)>0$
• If 3<x<2$-3 < x < 2$ , then f(x)<0$f(x)<0$
• If x>2$x>2$ , then f(x)>0$f(x)>0$
Therefore, $x^{2}+x-6>0$ whenever $x<-3$ or $x>2$ , and $x^{2}+x-6<0$ when $-3< x < 2$ .
#### Example A
What is the solution set of the inequality $2x^{2}+7x-4<0$ ?
Solution:
It is best to graph the function $f(x)=2x^{2}+7x-4$ and look for the values of $x$ such that the inequality $f(x)<0$ is true.
Thus from graph, $2x^{2}+7x-4<0$ only if
$-4 < x < \frac{1}{2}$
So the solution set is $x\in\left ( -4,\frac{1}{2} \right )$ or in set builder notation, $\left \{x | -4 < x < \frac{1}{2} \right \}$ .
Although the method of graphing to find the solution set of an inequality is easy to follow, another algebraic method can be used. The algebraic method involves finding the $x-$ intercepts of the graph and then dividing the $x-$ axis into intervals separated by the $x-$ intercepts. The examples below illustrate the method.
#### Example B
Find the solution set of the quadratic inequality $x^{2}+2x-8>0$ without graphing.
Solution:
To find the solution set without graphing, first factor:
$x^2 + 2x - 8 = 0$
$(x + 4)(x - 2) = 0$
Recalling the zero product rule , we can see that the two solutions to this quadratic equation are $x = -4$ and $x = 2$ , thus, the $x-$ intercepts of the function $f(x) = x^{2} + 2x - 8$ are -4 and 2.
These points divide the $x-$ axis into three intervals: $(-\infty,-4) | (-4,2) | (2,\infty)$ . We can choose a test point from each interval, substitute it into $f(x)$ and see if the function is negative or positive with that value as x . This procedure can be simplified by making a table as shown below:
Interval Test Point Is $x^{2}+2x-8$ positive or negative? Part of Solution set?
$(-\infty,-4)$ $-5$ $+$ $yes$
$(-4, 2)$ $1$ $-$ $no$
$(2,+\infty)$ $3$ $+$ $yes$
From the table, we conclude that since $x^2 +2x -8 > 0$ if and only if $x < -4$ and $x > 2$ . The solution set can also be written as:
$x\in(-\infty,-4)\cup(2,+\infty)$
Some problems in science involve quadratic inequalities. The example below illustrates one such application.
#### Example C
A rectangle has a length 10 meters more than twice the width. Find all of the possible widths that result in the area of the rectangle not exceeding 100 squared meters.
Solution:
Let $w$ be the width of the rectangle and $l$ its length. Given the information in the question, we can say:
$l = 10 + 2w$
Then we can use the formula for the area of a rectangle:
$area = l \times w$
Substituting $10 + 2w$ in for $l$ gives:
$area = w \times (10 + 2w)$
$=2w^2 + 10w$
The area cannot exceed $100 \ m^{2}$ so
$10w+2w^{2}<100$
or
$2w^{2}+10w-100<0$
Simplify by dividing both sides by 2:
$w^{2}+5w-50<0$
Factor the trinomial:
$w^{2}+5w-50=(w+10)(w-5)$
So the partition points are 5 and -10, which means we have three intervals. Since width cannot be negative, we can safely ignore -10. That means the maximum area is $100 \ m^{2}$
$\therefore w < 5$ .
The width must be less than 5 meters.
Concept question wrap-up:
Were you able to solve the question about mowing a lawn that was discussed at the beginning of the lesson?
'If you can mow 2500sf of grass per gallon of gas, and the mower has 2.5 gallons in it, what is the maximum length of one side of the lawn you can mow?
If your lawn is 75ft long, will you need more gas?'
By applying the process from Ex#3, we know that the function $S^{2} < 6250$ describes the possible side lengths of square shapes you could mow before running out of gas.
Solving for S gives:
$\sqrt{s^2} = \sqrt{6250}$
$s = 79$
With 2.5gal of gas, you could mow a square up to apx 79ft on each side.
You should not need more gas if the lawn is only 75ft long on each side.
### Vocabulary
Quadratic Inequality: A term describing a squared function that is specified to be smaller or larger than a given value.
Roots: The roots of a quadratic function are the values of x that make y equal to zero.
### Guided Practice
1) Find the solution set of the inequality $x^2 \leq 16$
2) Find the solution set: $x^2 - 7x > 30$
3) Find the solution set: $x^2 - 13x > -36$
4) Graph the solution set: $(x - 3)(x + 4) \geq 0$
1) Set the function equal to zero:
$x^2 - 16 = 0$
Factor to find the critical values (points where the graph crosses the x axis, thereby changing signs):
$(x - 4)(x + 4) = 0$
By the zero product rule: $x = 4$ or $x = -4$
That gives us three sections on the graph:
$x \leq -4$
$-4 \leq x \leq 4$
$4 \leq x$
Test one sample value from each division to identify possible solution sets.
Set $x \leq -4$ $-4 \leq x \leq 4$ $4 \leq x$
Test value $-5$ $0$ $5$
$f(x)$ true with value? $4\cdot(-5)^2 = 100 \nleq 16$ $\therefore No$ $4\cdot 0^2 = 0 \leq 16$ $\therefore Yes$ $4\cdot5^2 = 100 \nleq 16$ $\therefore No$
Therefore the solution set is $- 4 \leq x \leq 4$
2) Follow the same process as #1:
Set the function equal to zero: $x^2 - 7x - 30 = 0$
Factor: $(x - 10)(x + 3) = 0$
Identify critical values: $x = { -3, 10 }$
The three sections are: $x < -3$ and $-3 < x < 10$ and $10 < x$
Test one sample value from each division to identify possible solution sets.
Set $x < -3$ $-3 < x < 10$ $10 < x$
Test value $-10$ $0$ $20$
$f(x)$ true with value? $(-10)^2 -7(-10) = 170$ $170 > 30$ $\therefore Yes$ $0^2 -7(0) = 0$ $0 < 30$ $\therefore No$ $20^2 -7(20) = 260$ $260 > 30$ $\therefore Yes$
The solution set is: $x < -3$ and $10 < x$
3) Use the same process again:
Set the function equal to zero: $x^2 - 13x + 36 = 0$
Factor: $(x - 9)(x - 4) = 0$
Identify critical values: $x = {4, 9}$
The three sections are: $x < 4$ and $4 < x < 9$ and $9 < x$
Test one sample value from each division to identify possible solution sets.
Set $x < 4$ $4 < x < 9$ $9 < x$
Test value $0$ $6$ $12$
$f(x)$ true with value? $(0)^2 -13(0) = 0$ $0 > -36$ $\therefore Yes$ $6^2 -13(6) = -42$ $-42 < -36$ $\therefore No$ $12^2 -13(12) = -12$ $-12 > -36$ $\therefore Yes$
The solution set is: $x < 4$ and $9 < x$
4) The solutions to $(x - 3)(x + 4) \geq 0$ can be identified with the rules for multiplying negative numbers:
Recall from Pre-Algebra that an even number of negatives yields a positive answer, and an odd number of negatives yields a negative answer.
Since $(x - 3)(x + 4) \geq 0$ we know we need a positive answer or zero.
Therefore either:
Case #1: $(x - 3) \geq 0 \to x \geq 3$ and $(x + 4) \geq 0 \to x \geq -4$
or
Case #2: $(x - 3) \leq 0 \to x \leq 3$ and $(x + 4) \leq 0 \to x \leq -4$
Since any number greater than 3 is already greater than -4, from Case #1 we get: $x \geq 3$
Since any number less than -4 is already less than 3, from Case #2 we get $x \leq -4$
Therefore our answer is $x \leq -4$ or $x \geq 3$
In set notation: $x\in(-\infty,-4]\cup[3,+\infty)$
To graph this information, we draw a line graph, and mark the values that x can be , with solid dots on the end numbers to indicate that those values are included.
Visually that is:
### Explore More
Graph the solutions sets below on a number line:
1. $x < 3$ or $x > 4$
2. $x \geq -5$ and $x \geq 3$
3. $x < 6$ and $x \geq -2$
4. $x > 7$ or $x \geq -4$
5. $x \leq -8$ and $x > 3$
Identify critical points, solve, and graph:
1. $x^2 + 9x > -14$
2. $x^2 -5x \leq 50$
3. $x^2 + 2x \leq 48$
4. $x - \frac{20}{x} - 8 < 0$ (hint: multiply both sides by x first)
5. $x + 10 \geq -\frac{21}{x}$
6. $(x + 6)(x - 3) > 0$
7. $(x - 8)(x + 1) > 0$
8. $x^2 - x \geq 90$
9. $3x^2 - 23x \leq 8$
10. $x^2 + x - 6 > 0$ |
CBSE Class 10 Maths Notes Chapter 2 Polynomials Pdf free download is part of Class 10 Maths Notes for Quick Revision. Here we have given NCERT Class 10 Maths Notes Chapter 2 Polynomials. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks. https://www.cbselabs.com/polynomials-cbse-notes-class-10-maths/
## CBSE Class 10 Maths Notes Chapter 2 Polynomials
• “Polynomial” comes from the word ‘Poly’ (Meaning Many) and ‘nomial’ (in this case meaning Term)-so it means many terms.
• A polynomial is made up of terms that are only added, subtracted or multiplied.
• A quadratic polynomial in x with real coefficients is of the form ax² + bx + c, where a, b, c are real numbers with a ≠ 0.
• Degree – The highest exponent of the variable in the polynomial is called the degree of polynomial. Example: 3x3 + 4, here degree = 3.
• Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomial respectively.
• A polynomial can have terms which have Constants like 3, -20, etc., Variables like x and y and Exponents like 2 in y².
• These can be combined using addition, subtraction and multiplication but NOT DIVISION.
• The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.
If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then
$$sum\quad of\quad zeros,\alpha +\beta =\frac { -b }{ a } =\frac { -coefficient\quad of\quad x }{ coefficient\quad of\quad { x }^{ 2 } }$$
$$product\quad of\quad zeros,\alpha \beta =\frac { c }{ a } =\frac { constant\quad term }{ coefficient\quad of\quad { x }^{ 2 } }$$
If α, β, γ are the zeroes of the cubic polynomial ax3 + bx2 + cx + d = 0, then
$$\alpha +\beta +\gamma =\frac { -b }{ a } =\frac { -coefficient\quad of\quad { x }^{ 2 } }{ coefficient\quad of\quad { x }^{ 3 } }$$
$$\alpha \beta +\beta \gamma +\gamma \alpha =\frac { c }{ a } =\frac { coefficient\quad of\quad { x } }{ coefficient\quad of\quad { x }^{ 3 } }$$
$$\alpha \beta \gamma =\frac { -d }{ a } =\frac { -constant\quad term }{ coefficient\quad of\quad { x }^{ 3 } }$$
Zeroes (α, β, γ) follow the rules of algebraic identities, i.e.,
(α + β)² = α² + β² + 2αβ
∴(α² + β²) = (α + β)² – 2αβ
Degree of a Polynomial Calculator is a free online tool that helps students to calculate the polynomial expression degree value in no time with show work.
Polynomials Class 10 Notes Chapter 2
DIVISION ALGORITHM:
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then
p(x) = g(x) × q(x) + r(x)
Dividend = Divisor x Quotient + Remainder
Remember this!
• If r (x) = 0, then g (x) is a factor of p (x).
• If r (x) ≠ 0, then we can subtract r (x) from p (x) and then the new polynomial formed is a factor of g(x) and q(x). |
# 2013 AIME I Problems/Problem 7
## Problem 7
A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.
## Solution 1
Let the height of the box be $x$.
After using the Pythagorean Theorem three times, we can quickly see that the sides of the triangle are 10, $\sqrt{\left(\frac{x}{2}\right)^2 + 64}$, and $\sqrt{\left(\frac{x}{2}\right)^2 + 36}$. Since the area of the triangle is $30$, the altitude of the triangle from the base with length $10$ is $6$.
Considering the two triangles created by the altitude, we use the Pythagorean theorem twice to find the lengths of the two line segments that make up the base of $10$.
We find: $$10 = \sqrt{\left(28+x^2/4\right)}+x/2$$
Solving for $x$ gives us $x=\frac{36}{5}$. Since this fraction is simplified: $$m+n=\boxed{041}$$
## Solution 2
We may use vectors. Let the height of the box be $2h$. Without loss of generality, let the front bottom left corner of the box be $(0,0,0)$. Let the center point of the bottom face be $P_1$, the center of the left face be $P_2$ and the center of the front face be $P_3$.
We are given that the area of the triangle $\triangle P_1 P_2 P_3$ is $30$. Thus, by a well known formula, we note that $\frac{1}{2}|\vec{P_1P_2} \text{x} \vec{P_1P_3}|=30$ We quickly attain that $\vec{P_1P_2}=<-6,0,h>$ and $\vec{P_1P_3}=<0,-8,h>$ (We can arbitrarily assign the long and short ends due to symmetry)
Computing the cross product, we find: $$\vec{P_1P_2} x \vec{P_1P_3}=-<6h,8h,48>$$
Thus: $$\sqrt{(6h)^2+(8h)^2+48^2}=2*30=60$$ $$h=3.6$$ $$2h=7.2$$
$$2h=36/5$$
$$m+n=\boxed{041}$$
## Solution 3
Let the height of the box be $x$.
After using the Pythagorean Theorem three times, we can quickly see that the sides of the triangle are 10, $\sqrt{(x/2)^2 + 64}$, and $\sqrt{(x/2)^2 + 36}$. Therefore, we can use Heron's formula to set up an equation for the area of the triangle.
The semiperimeter is $\left(10 + \sqrt{(x/2)^2 + 64} + \sqrt{(x/2)^2 + 36}\right)/2$. Therefore, when we square Heron's formula, we find
\begin{align*}900 &= \frac{1}{2}\left(\left(10 + \sqrt{(x/2)^2 + 64} + \sqrt{(x/2)^2 + 36}\right)/2\right)\times\left(\left(10 + \sqrt{(x/2)^2 + 64} + \sqrt{(x/2)^2 + 36}\right)/2 - 10\right)\\&\qquad\times\left(\left(10 + \sqrt{(x/2)^2 + 64} + \sqrt{(x/2)^2 + 36}\right)/2 - \sqrt{(x/2)^2 + 64}\right)\\&\qquad\times\left(\left(10 + \sqrt{(x/2)^2 + 64} + \sqrt{(x/2)^2 + 36}\right)/2 - \sqrt{(x/2)^2 + 36}\right).\end{align*}
Solving, we get $\boxed{041}$.
## Solution 4
It isn't hard to see that the triangle connecting the centers of the faces of the rectangular prism is congruent to the triangle connecting the midpoints of three edges that concur. So we can now apply de Guas theorem to see that:
$30^2=24^2+(3x)^2+(4x)^2$
Where $x$ is half the desired length of the height.
Solving yields $2x=\frac{36}{5}$
And thus $36+5=\boxed{041}$
---Solution 4 contributed by Siddharth Namachivayam |
# Lesson 2
Match Representations of Tens
## Warm-up: Estimation Exploration: Cubes and Towers (10 minutes)
### Narrative
The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. When students notice that they can make a more accurate estimate more readily when the connecting cubes are grouped in towers of 10 they make use of base-ten structure (MP7).
This is the first time students participate in this routine in grade 1. The teacher can begin by saying, “Today we are doing a new warm-up called Estimation Exploration. In this activity we are going to see a collection of objects. Your job is to look at the image and think about how many objects there are.” Let students know that they should estimate, or find a value that is close to the correct value, without counting each one. As needed, clarify that the number they come up with is called an estimate.
### Launch
• Groups of 2
• Display the first image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time
### Activity
• 1 minute: partner discussion
• Record responses.
• “Let’s look at another image of the same collection.”
• Display the second image.
• “Based on the second image, do you want to revise or change your estimates?”
### Student Facing
Round 1: How many cubes?
Record an estimate that is:
too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$
Round 2: How many cubes?
Record an estimate that is:
too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$
### Student Response
For access, consult one of our IM Certified Partners.
### Activity Synthesis
• “Let’s look at our revised estimates. Why were these estimates more accurate the second time?” (Some of the cubes are organized. We could see there were more than 2 tens.)
• “There are 30 cubes.”
## Activity 1: Representations of Tens (20 minutes)
### Narrative
The purpose of this activity is for students to make connections between representations of multiples of 10. Students match cards that show multiples of 10 with pictures, numbers, and words. Each student is given a card and then moves around the room to find the other two students whose cards have the same value.
During the synthesis, the teacher and students create a chart to display the matches. Students discuss the direct connection between the number, the written number of tens, and the representation of towers of 10. When students notice and describe that the first digit in each number matches the written number of tens and the number of towers of 10, they look for and make sense of the base-ten structure (MP7).
Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from starting with more accessible values. For example, cards with a value of 10 or 20.
Supports accessibility for: Social-Emotional Functioning, Visual-Spatial Processing
### Required Materials
Materials to Copy
• Representations of Tens
### Required Preparation
• Create a set of cards from the blackline master for the class. Each student will only need one card.
• Create a large chart titled “Representations of Tens” at the top. Draw three columns. Label columns “Number,” “Number of Tens,” and “Drawing.”
### Launch
• Groups of 2
• Give each student a card.
• “Today you will play a game in which you try to find cards that match.”
• “To play this game, each of you will get one card. Find the other two students who have cards with the same value. Each time you compare cards with another student, discuss whether or not your cards have the same value and how you know.”
### Activity
• 5 minutes: partner work time
• If time: Shuffle the cards and have students repeat the activity with new groups.
• As students work, consider asking
• “How do you know these cards match?”
• “How many tens does this card have?”
### Student Response
For access, consult one of our IM Certified Partners.
### Activity Synthesis
• “Let’s look at the matches we saw during the game today.”
• Write 10 in the first column of the chart.
• “Who has 10? How do you know?”
• Repeat for all multiples of 10, discussing how students know their cards match for each multiple of 10.
## Activity 2: It's a Match! (15 minutes)
### Narrative
The purpose of this activity is for students to connect written multiples of 10 to base-ten representations. Students notice a connection between how the numbers are said, written, and represented with tens. For example, “70” has the word “seven” as it is said, the digit “7” when it is written, and is represented with seven tens. Each number name is written under the number in this activity. Students are not expected to spell number names.
MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, “How did you match the towers of 10 to the written numbers?” This gives both students an opportunity to produce language.
### Required Materials
Materials to Gather
Materials to Copy
• It's a Match (10-90) words, numbers, pictures
### Launch
• Groups of 2
• Give each student a blackline master and access to connecting cubes.
### Activity
• 5 minutes: independent work time
• 3 minutes: partner discussion
• Monitor for students who count the number of towers, match correctly, and can explain how they knew.
### Student Response
For access, consult one of our IM Certified Partners.
### Activity Synthesis
• Invite previously identified students to share.
• “How did you know which representations matched?” (I counted by ten and found the number. I said 10, 20, 30, 40, 50, and found 50.)
## Lesson Synthesis
### Lesson Synthesis
“Today we thought about how to count things that are arranged in groups of 10 and looked at the numbers that represent them.”
Display one row of the chart made during the activity 1 synthesis; for example: 30, 3 towers of 10, 3 tens
“What do you notice about the towers of 10 and the number that represents them?” (30 has a 3 in it, there are 3 towers of 10 or 3 tens.)
“The 3 in the number 30 tells us how many tens there are.”
Display another row on the chart.
“How many towers are there?” (7)
“How many tens are there?” (7 tens because each tower is 10.)
“How many cubes are there?” (70)
## Cool-down: How Many Are There? (5 minutes)
### Cool-Down
For access, consult one of our IM Certified Partners. |
Class 6 - Maths - Practical Geometry
Exercise 14.1
Question 1:
Draw a circle of radius 3.2 cm.
Steps of construction:
(a) Open the compass for the required radius of 3.2 cm.
(b) Make a point with a sharp pencil where we want the centre of circle to be.
(c) Name it O.
(d) Place the pointer of compasses on O.
(e) Turn the compasses slowly to draw the circle.
Hence, it is the required circle.
Question 2:
With the same centre O, draw two circles of radii 4 cm and 2.5 cm.
Steps of construction:
(a) Marks a point O with a sharp pencil where we want the centre of the circle.
(b) Open the compasses 4 cm.
(c) Place the pointer of the compasses on O.
(d) Turn the compasses slowly to draw the circle.
(e) Again open the compasses 2.5 cm and place the pointer of the compasses on D.
(f) Turn the compasses slowly to draw the second circle.
Hence, it is the required figure.
Question 3:
Draw a circle and any two of its diameters. If you join the ends of these diameters, what
is the figure obtained if the diameters are perpendicular to each other? How do you check
(i) By joining the ends of two diameters, we get a rectangle. By measuring, we find AB = CD = 3
cm, BC = AD = 2 cm, i.e., pairs of opposite sides are equal and also
angle A = angle B = angle C = angle D = 90 degree
i.e. each angle is of 90 degree. Hence, it is a rectangle.
(ii) If the diameters are perpendicular to each other, then by joining the ends of two
diameters, we get a square.
By measuring, we find that AB = BC = CD = DA = 2.5 cm, i.e., all four sides are equal
Also, angle A = angle B = angle C = angle D = 90 degree
i.e. each angle is of 90 degree. Hence, it is a square.
Question 4:
Draw any circle and mark points A, B and C such that:
(a) A is on the circle.
(b) B is in the interior of the circle.
(c) C is in the exterior of the circle.
(i) Mark a point O with sharp pencil where we want centre of the circle.
(ii) Place the pointer of the compasses at O. Then move the compasses slowly to draw a circle.
(a) Point A is on the circle.
(b) Point B is in interior of the circle.
(c) Point C is in the exterior of the circle
Question 5:
Let A, B be the centers of two circles of equal radii; draw them so that each one of them passes through the centre of the other.
Let them intersect at C and D. Examine whether AB and CD are at right angles.
Draw two circles of equal radii taking A and B as their centre such that one of them passes
through the centre of the other. They intersect at C and D. Join AB and CD.
Yes, AB and CD intersect at right angle as angle COB is 90 degree.
Exercise 14.2
Question 1:
Draw a line segment of length 7.3 cm, using a ruler.
(i) Place the zero mark of the ruler at a point A.
(ii) Mark a point B at a distance of 7.3 cm from A.
(iii) Join AB.
Hence, AB is the required line segment of length 7.3 cm.
Question 2:
Construct a line segment of length 5.6 cm using ruler and compasses.
(i) Draw a line l. Mark a point A on this line.
(ii) Place the compasses pointer on zero mark of the ruler. Open it to place the pencil point up
to 5.6 cm mark.
(iii) Without changing the opening of the compasses. Place the pointer on A and cut an arc
l at B.
Now, AB is the required line segment of length 5.6 cm.
Question 3:
Construct AB of length 7.8 cm. From this cut off AC of length 4.7 cm. Measure BC.
(i) Place the zero mark of the ruler at A.
(ii) Mark a point B at a distance 7.8 cm from A.
(iii) Again, mark a point C at a distance 4.7 from A.
Hence, by measuring BC, we find that BC = 3.1 cm
Question 4:
Given AB of length 3.9 cm, construct PQ such that the length PQ is twice that of AB.
Verify by measurement.
(Hint: Construct PX such that length of PX = length of AB; then cut off XQ such that XQ also has the length of AB.
(i) Draw a line l.
(ii) Construct PX such that length of PX = length of AB
(iii) Then cut of XQ such that XQ also has the length of AB.
(iv) Thus, the length of PX and the length of XQ added together make twice the length of AB.
Verification:
Hence, by measurement we find that PQ = 7.8 cm
= 3.9 cm + 3.9 cm
= AB + AB
= 2 * AB
Question 5:
Given AB of length 7.3 cm and CD of length 3.4 cm, construct a line segment XY such that the length of XY is equal to the difference between the lengths of AB and CD.
Verify by measurement.
Steps of construction:
(i) Draw a line l and take a point X on it.
(ii) Construct XZ such that length XZ = length of AB = 7.3 cm
(iii) Then cut off ZY = length of CD = 3.4 cm
(iv) Thus, the length of XY = length of AB - length of CD
Verification:
Hence, by measurement we find that length of XY = 3.9 cm
= 73. Cm - 3.4 cm
= AB - CD
Exercise 14.3
Question 1:
Draw any line segment PQ. Without measuring PQ, construct a copy of PQ.
(i) Given PQ whose length is not known.
(ii) Fix the compasses pointer on P and the pencil end on Q. The opening of the instrument
now gives the length of PQ .
(iii) Draw any line l. Choose a point A on l Without changing the compasses setting, place the
pointer on A.
(iv) Draw an arc that cuts l at a point, say B.
Hence, AB is the copy of PQ.
Question 2:
Given some line segment AB, whose length you do not know, construct PQ such that the length of PQ is twice that of AB.
(i) Given AB whose length is not known.
(ii) Fix the compasses pointer on A and the pencil end on B. The opening of the instrument
now gives the length of AB.
(iii) Draw any line l. Choose a point P on l. Without changing the compasses setting, place the
pointer on Q.
(iv) Draw an arc that cuts l at a point R.
(v) Now place the pointer on R and without changing the compasses setting, draw another arc
that cuts l at a point Q.
Hence, PQ is the required line segment whose length is twice that of AB.
Exercise 14.4
Question 1:
Draw any line segment AB. Mark any point M on it. Through M, draw a perpendicular to
1. (Use ruler and compasses
Steps of construction:
(i) With M as centre and a convenient radius, draw an arc intersecting the line AB at two points
C and B.
(ii) With C and D as centres and a radius greater than MC, draw two arcs, which cut each other
at P.
(iii) Join PM. Then PM is perpendicular to AB through the point M.
Question 2:
Draw any line segment PQ. Take any point R not on it. Through R, draw a perpendicular
To PQ. (Use ruler and set-square)
Steps of construction:
(i) Place a set-square on PQ such that one arm of its right angle aligns along PQ .
(ii) Place a ruler along the edge opposite to the right angle of the set-square.
(iii) Hold the ruler fixed. Slide the set square along the ruler till the point R touches the other
arm of the set square.
(iv) Join RM along the edge through R meeting PQ at M.
Then RM Ʇ PQ.
Question 3:
Draw a line l and a point X on it. Through X, draw a line segment XY perpendicular to l.
Now draw a perpendicular to XY to Y. (use ruler and compasses)
(i) Draw a line l and take point X on it.
(ii) With X as centre and a convenient radius,
draw an arc intersecting the line l at two points A and B.
(iii) With A and B as centres and a radius greater than XA,
draw two arcs, which cut each other at C.
(iv) Join AC and produce it to Y. Then XY is perpendicular to l.
(v) With D as centre and a convenient radius, draw an arc intersecting XY at two points C and
D.
(vi) With C and D as centres and radius greater than YD, draw two arcs which cut each other at
F.
(vii) Join YF, then YF is perpendicular to XY at Y.
Exercise 14.5
Question 1:
Draw AB of length 7.3 cm and find its axis of symmetry.
Axis of symmetry of line segment AB will be the perpendicular bisector of AB.
So, draw the perpendicular bisector of AB.
Steps of construction:
(i) Draw a line segment AB = 7.3 cm
(ii) Taking A and B as centres and radius more than half of AB, draw two arcs which intersect each other at C and D.
(iii) Join CD. Then CD is the axis of symmetry of the line segment AB.
Question 2:
Draw a line segment of length 9.5 cm and construct its perpendicular bisector.
Steps of construction:
(i) Draw a line segment AB = 9.5 cm
(ii) Taking A and B as centres and radius more than half of AB,
draw two arcs which intersect each other at C and D.
(iii) Join CD. Then CD is the perpendicular bisector of AB
Question 3:
Draw the perpendicular bisector of XY whose length is 10.3 cm.
(a) Take any point P on the bisector drawn. Examine whether PX = PY.
(b)If M is the mid-point of XY, what can you say about the lengths MX and XY?
(i) Draw a line segment XY = 10.3 cm
(ii) Taking X and Y as centres and radius more than half of AB, draw two arcs which intersect each other at C and D.
(iii) Join CD. Then CD is the required perpendicular bisector of XY.
Now:
(a) Take any point P on the bisector drawn. With the help of divider we can check that
PX = PY.
(b) If M is the mid-point of XY, then MX = XY/2
Question 4:
Draw a line segment of length 12.8 cm. Using compasses, divide it into four equal parts.
Verify by actual measurement.
(i) Draw a line segment AB = 12.8 cm
(ii) Draw the perpendicular bisector of AB which cuts it at C. Thus, C is the midpoint of AB.
(iii) Draw the perpendicular bisector of AC which cuts it at D. Thus D is the midpoint of AC.
(iv) Again draw the perpendicular bisector of CB which cuts it at E.
Thus, E is the mid-point of CB.
(v) Now, point C, D and E divide the line segment AB in the four equal parts.
(vi) By actual measurement, we find that
AD = DC = CE = EB = 3.2 cm
Question 5:
With PQ of length 6.1 cm as diameter, draw a circle.
Steps of construction:
(i) Draw a line segment PQ = 6.1 cm.
(ii) Draw the perpendicular bisector of PQ which cuts, it at O.
Thus O is the mid-point of PQ.
(iii) Taking O as centre and OP or OQ as radius draw a circle where diameter is the line segment PQ.
Question 6:
Draw a circle with centre C and radius 3.4 cm. Draw any chord AB. Construct the perpendicular bisector AB and examine if it passes through C.
Steps of construction:
(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw any chord AB.
(iii) Taking A and B as centres and radius more than half of AB,
draw two arcs which cut each other at P and Q.
(iv) Join PQ. Then PQ is the perpendicular bisector of AB.
(v) This perpendicular bisector of AB passes through the centre C of the circle.
Question 7:
Repeat Question 6, if AB happens to be a diameter.
Steps of construction:
(i) Draw a circle with centre C and radius 3.4 cm.
(ii) Draw its diameter AB.
(iii) Taking A and B as centres and radius more than half of it,
draw two arcs which intersect each other at P and Q.
(iv) Join PQ. Then PQ is the perpendicular bisector of AB.
(v) We observe that this perpendicular bisector of AB passes through the centre C of the circle.
Question 8:
Draw a circle of radius 4 cm. Draw any two of its chords. Construct the perpendicular bisectors of these chords. Where do they meet?
(i) Draw the circle with O and radius 4 cm.
(ii) Draw any two chords AB and CD in this circle.
(iii) Taking A and B as centres and radius more than half AB,
draw two arcs which intersect each other at E and F.
(iv) Join EF. Thus EF is the perpendicular bisector of chord CD.
(v) Similarly draw GH the perpendicular bisector of chord CD.
(vi) These two perpendicular bisectors meet at O, the centre of the circle.
Question 9:
Draw any angle with vertex O. Take a point A on one of its arms and B on another such that OA = OB. Draw the perpendicular bisectors of OA and OB.
Let them meet at P. Is PA = PB?
Steps of construction:
(i) Draw any angle with vertex O.
(ii) Take a point A on one of its arms and B on another
such that OA = OB.
(iii) Draw perpendicular bisector of OA and OB.
(iv) Let them meet at P. Join PA and PB.
(v) With the help of divider, we check that PA = PB.
Exercise 14.6
Question 1:
Draw angle POQ of measure 750 and find its line of symmetry.
Steps of construction:
(a) Draw a line l and mark a point O on it.
(b) Place the pointer of the compasses at O and draw an arc
of any radius which intersects the line l at A.
(c) Taking same radius, with centre A, cut the previous arc at B.
(d)Join OB, then angle BOA = 600.
(e) Taking same radius, with centre B, cut the previous arc at C.
(f) Draw bisector of angle BOC. The angle is of 900. Mark it at D. Thus, angle DOA = 900
(g) Draw OP as bisector of angle DOB.
Thus, angle POA = 750
Question 2:
Draw an angle of measure 1470 and construct its bisector.
Steps of construction:
(a) Draw a ray OA.
(b) With the help of protractor, construct angle AOB = 1470.
(c) Taking centre O and any convenient radius, draw an arc
which intersects the arms OA and OB at P and Q respectively.
(d) Taking P as centre and radius more than half of PQ, draw an arc.
(e) Taking Q as centre and with the same radius, draw another arc which intersects the
previous at R.
(f) Join OR and produce it.
(g) Thus, OR is the required bisector of angle AOB.
Question 3:
Draw a right angle and construct its bisector.
Steps of construction:
(a) Draw a line PQ and take a point O on it.
(b) Taking O as centre and convenient radius, draw an arc which intersects PQ at A and B.
(c) Taking A and B as centres and radius more than half of AB, draw two arcs which intersect
each other at C.
(d) Join OC. Thus, angle COQ is the required right angle.
(e) Taking B and E as centre and radius more than half of BE, draw two arcs which intersect
each other at the point D.
(f) Join OD. Thus, OD is the required bisector of angle COQ.
Question 4:
Draw an angle of measure 1530 and divide it into four equal parts.
Steps of construction:
(a) Draw a ray OA.
(b) At O, with the help of a protractor, construct angle AOB = 153.
(c) Draw OC as the bisector of angle AOB.
(d) Again, draw OD as bisector of angle AOC.
(e) Again, draw OE as bisector of angle BOC.
(f) Thus, OC, OD and OE divide angle AOB in four equal arts.
Question 5:
Construct with ruler and compasses, angles of following measures:
(a) 600 (b) 300 (c) 900 (d) 1200 (e) 450 (f) 1350
(a) 600
(i) Draw a ray OA.
(ii) Taking O as centre and convenient radius, mark an arc,
which intersects OA at P.
(iii) Taking P as centre and same radius, cut previous arc at Q.
(iv) Join OQ.
Thus, angle BOA is required angle of 600.
(b) 300
(i) Draw a ray OA.
(ii) Taking O as centre and convenient radius,
mark an arc, which intersects OA at P.
(iii) Taking P as centre and same radius, cut previous arc at Q.
(iv) Join OQ. Thus, angle BOA is required angle of 600.
(v) Put the pointer on P and mark an arc.
(vi) Put the pointer on Q and with same radius, cut the previous arc at C.
Thus, angle COA is required angle of 300.
(c) 900
(i) Draw a ray OA.
(ii) Taking O as centre and convenient radius,
mark an arc, which intersects OA at X.
(iii) Taking X as centre and same radius cut previous arc at Y.
(iv) Taking Y as centre and same radius, draw another arc intersecting the same arc at Z.
(v) Taking Y and Z as centres and same radius, draw two arcs intersecting each other at S.
(vi) Join OS and produce it to form a ray OB.
Thus, angle BOA is required angle of 900.
(d) 1200
(i) Draw a ray OA.
(ii) Taking O as centre and convenient radius,
mark an arc, which intersects OA at P.
(iii) Taking P as centre and same radius, cut previous arc at Q.
(iv) Taking Q as centre and same radius cut the arc at S.
(v) Join OS.
Thus, AOD is required angle of 1200.
(e) 450
(i) Draw a ray OA.
(ii) Taking O as centre and convenient radius,
mark an arc, which intersects OA at X.
(iii) Taking X as centre and same radius,
cut previous arc at Y.
(iv) Taking Y as centre and same radius, draw another arc intersecting the same arc at Z.
(v) Taking Y and Z as centres and same radius, draw two arcs intersecting each other at S.
(vi) Join OS and produce it to form a ray OB. Thus, angle BOA is required angle of 90.
(vii) Draw the bisector of angle BOA.
Thus, angle MOA is required angle of 450.
(f) 1350
(i) Draw a line PQ and take a point O on it.
(ii) Taking O as centre and convenient radius,
mark an arc, which intersects PQ at A and B.
(iii) Taking A and B as centres and radius more than half of AB,
draw two arcs intersecting each other at R.
(iv) Join OR. Thus, angle QOR = angle POQ = 900.
(v) Draw OD the bisector of angle POR.
Thus, angle QOD is required angle of 135.
Question 6:
Draw an angle of measure 450 and bisect it.
Steps of construction:
(a) Draw a line PQ and take a point O on it.
(b) Taking O as centre and a convenient radius,
draw an arc which intersects PQ at two points A and B.
(c) Taking A and B as centres and radius more than half of AB,
draw two arcs which intersect each other at C.
(d) Join OC. Then angle COQ is an angle of 900
(e) Draw OE as the bisector of angle COE. Thus, angle QOE = 450
(f) Again draw OG as the bisector of angle QOE.
Thus, angle QOG = angle EOG = 22½
Question 7:
Draw an angle of measure 1350 and bisect it.
Steps of construction:
(a) Draw a line PQ and take a point O on it.
(b) Taking O as centre and convenient radius, mark an arc, which intersects PQ at A and B.
(c) Taking A and B as centres and radius more than half of AB, draw two arcs intersecting each
other at R.
(d) Join OR. Thus, angle QOR = angle POQ = 90.
(e) Draw OD the bisector of POR. Thus, angle QOD is required angle of 135.
(f) Now, draw OE as the bisector of angle QOD.
Thus, angle QOE = angle DOE = 67½
Question 8:
Draw an angle of 700. Make a copy of it using only a straight edge and compasses.
(a) Draw an angle 70 with protractor, i.e., angle POQ = 700
(b) Draw a ray AB.
(c) Place the compasses at O and draw an arc to cut the rays of angle POQ at L and M.
(d) Use the same compasses, setting to draw an arc with A as centre, cutting AB at X.
(e) Set your compasses setting to the length LM with the same radius.
(f) Place the compasses pointer at X and draw the arc to cut the arc drawn earlier at Y.
(g) Join AY.
Thus, angle YAX = 700
Question 9:
Draw an angle of 400. Copy its supplementary angle.
(a) Draw an angle of 400 with the help of protractor, naming angle AOB.
(b) Draw a line PQ.
(c) Take any point M on PQ.
(d) Place the compasses at O and draw an arc to cut the rays of angle AOB at L and N.
(e) Use the same compasses setting to draw an arc O as centre, cutting MQ at X.
(f) Set your compasses to length LN with the same radius.
(g) Place the compasses at X and draw the arc to cut the arc drawn earlier Y.
(h)Join MY.
Thus, angle QMY = 400 and angle PMY is supplementary of it. |
# How To Subtract Mixed Numbers
Enlarge
## How To Subtract Mixed Numbers
Having trouble subtracting mixed numbers? This video shows you how to ace your math test by converting mixed numbers to improper fractions and finding common denominators.
Let's say you want to subtract from three and one fifths, one and one third. The first step here is to get our mixed numbers back into the form of improper fractions. We know that three here means that we have three wholes, and we also have one fifth.
The equivalent way of writing this is to write sixteen divided by five, or sixteen fifths. Because three multiplied by five is fifteen, and we have one here. Sixteen.
To get this number in its improper fractional form, we do three multiplied by one is three, and we add the one here, to get four divided by three. We now need to get this subtraction in a form where we can subtract the numbers from each other. We notice that these two numbers do not have the same denominator, which means that we need to multiply this numerator and this denominator by three, and this numerator and this denominator by five.
So let's do that. Sixteen multiplied by three is forty-eight. Five multiplied by three is fifteen.
Four multiplied by five is twenty. And three multiplied by five is fifteen. Add a subtraction sign, and we have one more step.
The final step is to subtract only the numerators to give us twenty-eight fifteenths. Let's just recap. Our first step is to get these mixed numbers back in the form of improper fractions.
We did this by multiplying the denominator by the number in front and adding the existing numerator, in this case, three times five plus one is sixteen. We do this for both numbers and realize now that we have to get these numbers with the same denominator. The way we do that is we cross-multiply three to this numerator and denominator, and we cross-multiply five to this numerator and denominator, giving us this subtraction.
We finally subtract this numerator from this numerator, and leave the denominators unchanged, to give us twenty-eight divided by fifteen, or twenty-eight fifteenths. And this is how to subtract mixed numbers. |
You are here: Home Vectors and scalars
## Introduction to vectors and scalars
We come into contact with many physical quantities in the natural world on a daily basis. For example, things like time, mass, weight, force, and electric charge, are physical quantities with which we are all familiar. We know that time passes and physical objects have mass. Things have weight due to gravity. We exert forces when we open doors, walk along the street and kick balls. We experience electric charge directly through static shocks in winter and through using anything which runs on electricity.
There are many physical quantities in nature, and we can divide them up into two broad groups called vectors and scalars.
Chapter introduction
## Scalars and vectors
Scalars are physical quantities which have only a number value or a size (magnitude). A scalar tells you how much of something there is.
Definition 1: Scalar
A scalar is a physical quantity that has only a magnitude (size).
For example, a person buys a tub of margarine which is labelled with a mass of 500 g. The mass of the tub of margarine is a scalar quantity. It only needs one number to describe it, in this case, 500 g.
Vectors are different because they are physical quantities which have a size and a direction. A vector tells you how much of something there is andwhich direction it is in.
Definition 2: Vector
A vector is a physical quantity that has both a magnitude and a direction.
For example, a car is travelling east along a freeway at 100 km·h−1. What we have here is a vector called the velocity. The car is moving at 100 km·h−1 (this is the magnitude) and we know where it is going – east (this is the direction). These two quantities, the speed and direction of the car, (a magnitude and a direction) together form a vector we call velocity.
Examples of scalar quantities:
• mass has only a value, no direction
• electric charge has only a value, no direction
Examples of vector quantities:
• force has a value and a direction. You push or pull something with some strength (magnitude) in a particular direction
• weight has a value and a direction. Your weight is proportional to your mass (magnitude) and is always in the direction towards the centre of the earth.
### Exercise 1: Vectors and scalars
Classify the following as vectors or scalars
1. length
2. force
3. direction
4. height
5. time
6. speed
7. temperature
1. scalar
2. vector
3. scalar
4. scalar
5. scalar
6. vector
7. scalar
## Vector notation
Vectors are different to scalars and must have their own notation. There are many ways of writing the symbol for a vector. In this book vectors will be shown by symbols with an arrow pointing to the right above it. For example, $F →$, $W →$ and $v →$ represent the vectors of force, weight and velocity, meaning they have both a magnitude and a direction.
Sometimes just the magnitude of a vector is needed. In this case, the arrow is omitted. For the case of the force vector:
• $F →$ represents the force vector
• F represents the magnitude of the force vector |
Quadratic Equations Questions and Solutions
Today we will discuss about the quadratic equation $\boxed{ax^2+bx+c=0}$.
The way to derive the Quadratic Formula is to use completing the square.
First, we divide throughout by a, where a is assumed to be non-zero (otherwise it isn’t a quadratic equation)
$\displaystyle x^2+\frac{b}{a}x+\frac{c}{a}=0$
The next step is the key step: add and subtract the square of b/2a to complete the square.
$\displaystyle x^2+\frac{b}{a}x\mathbf{+(\frac{b}{2a})^2-(\frac{b}{2a})^2}+\frac{c}{a}=0$
$\displaystyle (x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0$
Move all the constant terms to the right:
$\displaystyle (x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}$
Taking square root on both sides,
$\displaystyle x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}$
Hence, $\displaystyle\boxed{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$
Using the above information, we can practice and solve some quadratic equations! (Answers provided) More than ample questions are listed here, you can just pick a few of them! If you arrived on this page from a search engine, press Ctrl-F to search for your quadratic equation!
Q1) Solve 4x^2+49x+12=0
Ans: x=-0.25 or x=-12
Q2) Solve 14x^2+13x+3=0
Ans: x=-0.43 or x=-0.5
Q3) Solve x^2+2x+1=0
Ans: x=-1 or x=-1
Q4) Solve 36x^2+31x+3=0
Ans: x=-0.11 or x=-0.75
Q5) Solve 6x^2+11x+4=0
Ans: x=-0.5 or x=-1.33
Q6) Solve 32x^2+44x+9=0
Ans: x=-0.25 or x=-1.12
Q7) Solve 9x^2+30x+25=0
Ans: x=-1.67 or x=-1.67
Q8) Solve 48x^2+19x+1=0
Ans: x=-0.06 or x=-0.33
Q9) Solve 12x^2+40x+25=0
Ans: x=-0.83 or x=-2.5
Q10) Solve 48x^2+50x+7=0
Ans: x=-0.17 or x=-0.87
Q11) Solve 13x^2+40x+3=0
Ans: x=-0.08 or x=-3
Q12) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q13) Solve 4x^2+24x+20=0
Ans: x=-1 or x=-5
Q14) Solve 10x^2+28x+16=0
Ans: x=-0.8 or x=-2
Q15) Solve 42x^2+43x+11=0
Ans: x=-0.5 or x=-0.52
Q16) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q17) Solve x^2+27x+50=0
Ans: x=-2 or x=-25
Q18) Solve 4x^2+18x+20=0
Ans: x=-2 or x=-2.5
Q19) Solve 16x^2+38x+12=0
Ans: x=-0.37 or x=-2
Q20) Solve 5x^2+23x+18=0
Ans: x=-1 or x=-3.6
Q21) Solve 14x^2+45x+9=0
Ans: x=-0.21 or x=-3
Q22) Solve 8x^2+29x+15=0
Ans: x=-0.62 or x=-3
Q23) Solve 7x^2+8x+1=0
Ans: x=-0.14 or x=-1
Q24) Solve 38x^2+27x+4=0
Ans: x=-0.21 or x=-0.5
Q25) Solve 15x^2+26x+11=0
Ans: x=-0.73 or x=-1
Q26) Solve 26x^2+29x+8=0
Ans: x=-0.5 or x=-0.62
Q27) Solve 6x^2+35x+44=0
Ans: x=-1.83 or x=-4
Q28) Solve 3x^2+25x+22=0
Ans: x=-1 or x=-7.33
Q29) Solve 30x^2+38x+8=0
Ans: x=-0.27 or x=-1
Q30) Solve 4x^2+20x+16=0
Ans: x=-1 or x=-4
Q31) Solve 30x^2+47x+17=0
Ans: x=-0.57 or x=-1
Q32) Solve 27x^2+48x+21=0
Ans: x=-0.78 or x=-1
Q33) Solve 5x^2+46x+48=0
Ans: x=-1.2 or x=-8
Q34) Solve 2x^2+44x+42=0
Ans: x=-1 or x=-21
Q35) Solve 18x^2+46x+28=0
Ans: x=-1 or x=-1.56
Q36) Solve 42x^2+39x+9=0
Ans: x=-0.43 or x=-0.5
Q37) Solve 14x^2+37x+24=0
Ans: x=-1.14 or x=-1.5
Q38) Solve 14x^2+43x+3=0
Ans: x=-0.07 or x=-3
Q39) Solve 14x^2+34x+20=0
Ans: x=-1 or x=-1.43
Q40) Solve 12x^2+50x+42=0
Ans: x=-1.17 or x=-3
Q41) Solve 24x^2+46x+15=0
Ans: x=-0.42 or x=-1.5
Q42) Solve 10x^2+34x+24=0
Ans: x=-1 or x=-2.4
Q43) Solve 15x^2+28x+5=0
Ans: x=-0.2 or x=-1.67
Q44) Solve 15x^2+40x+20=0
Ans: x=-0.67 or x=-2
Q45) Solve 26x^2+27x+7=0
Ans: x=-0.5 or x=-0.54
Q46) Solve 25x^2+28x+3=0
Ans: x=-0.12 or x=-1
Q47) Solve 27x^2+44x+17=0
Ans: x=-0.63 or x=-1
Q48) Solve 12x^2+11x+2=0
Ans: x=-0.25 or x=-0.67
Q49) Solve 14x^2+47x+30=0
Ans: x=-0.86 or x=-2.5
Q50) Solve 9x^2+30x+24=0
Ans: x=-1.33 or x=-2
Q51) Solve 36x^2+37x+7=0
Ans: x=-0.25 or x=-0.78
Q52) Solve 27x^2+39x+14=0
Ans: x=-0.67 or x=-0.78
Q53) Solve 48x^2+48x+12=0
Ans: x=-0.5 or x=-0.5
Q54) Solve 18x^2+39x+15=0
Ans: x=-0.5 or x=-1.67
Q55) Solve 32x^2+44x+12=0
Ans: x=-0.37 or x=-1
Q56) Solve 3x^2+20x+25=0
Ans: x=-1.67 or x=-5
Q57) Solve 16x^2+38x+21=0
Ans: x=-0.87 or x=-1.5
Q58) Solve 14x^2+36x+22=0
Ans: x=-1 or x=-1.57
Q59) Solve 45x^2+33x+2=0
Ans: x=-0.07 or x=-0.67
Q60) Solve 21x^2+13x+2=0
Ans: x=-0.29 or x=-0.33
Q61) Solve 45x^2+38x+8=0
Ans: x=-0.4 or x=-0.44
Q62) Solve 8x^2+41x+50=0
Ans: x=-2 or x=-3.12
Q63) Solve 5x^2+32x+12=0
Ans: x=-0.4 or x=-6
Q64) Solve 8x^2+17x+2=0
Ans: x=-0.12 or x=-2
Q65) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q66) Solve 11x^2+24x+13=0
Ans: x=-1 or x=-1.18
Q67) Solve 27x^2+33x+8=0
Ans: x=-0.33 or x=-0.89
Q68) Solve 12x^2+35x+25=0
Ans: x=-1.25 or x=-1.67
Q69) Solve 22x^2+48x+26=0
Ans: x=-1 or x=-1.18
Q70) Solve 24x^2+31x+10=0
Ans: x=-0.62 or x=-0.67
Q71) Solve 2x^2+9x+4=0
Ans: x=-0.5 or x=-4
Q72) Solve 15x^2+45x+30=0
Ans: x=-1 or x=-2
Q73) Solve 18x^2+50x+32=0
Ans: x=-1 or x=-1.78
Q74) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q75) Solve 42x^2+37x+5=0
Ans: x=-0.17 or x=-0.71
Q76) Solve 5x^2+29x+24=0
Ans: x=-1 or x=-4.8
Q77) Solve 14x^2+19x+6=0
Ans: x=-0.5 or x=-0.86
Q78) Solve 21x^2+38x+5=0
Ans: x=-0.14 or x=-1.67
Q79) Solve 34x^2+19x+1=0
Ans: x=-0.06 or x=-0.5
Q80) Solve 32x^2+44x+12=0
Ans: x=-0.37 or x=-1
Q81) Solve 5x^2+48x+43=0
Ans: x=-1 or x=-8.6
Q82) Solve 48x^2+38x+5=0
Ans: x=-0.17 or x=-0.62
Q83) Solve 6x^2+17x+12=0
Ans: x=-1.33 or x=-1.5
Q84) Solve 13x^2+44x+31=0
Ans: x=-1 or x=-2.38
Q85) Solve 2x^2+21x+49=0
Ans: x=-3.5 or x=-7
Q86) Solve 10x^2+20x+10=0
Ans: x=-1 or x=-1
Q87) Solve 6x^2+11x+5=0
Ans: x=-0.83 or x=-1
Q88) Solve 3x^2+21x+18=0
Ans: x=-1 or x=-6
Q89) Solve 4x^2+5x+1=0
Ans: x=-0.25 or x=-1
Q90) Solve 18x^2+27x+9=0
Ans: x=-0.5 or x=-1
Q91) Solve 16x^2+30x+9=0
Ans: x=-0.37 or x=-1.5
Q92) Solve 12x^2+41x+24=0
Ans: x=-0.75 or x=-2.67
Q93) Solve 28x^2+33x+9=0
Ans: x=-0.43 or x=-0.75
Q94) Solve 18x^2+18x+4=0
Ans: x=-0.33 or x=-0.67
Q95) Solve 2x^2+9x+9=0
Ans: x=-1.5 or x=-3
Q96) Solve x^2+21x+38=0
Ans: x=-2 or x=-19
Q97) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q98) Solve 20x^2+48x+16=0
Ans: x=-0.4 or x=-2
Q99) Solve 32x^2+36x+7=0
Ans: x=-0.25 or x=-0.87
Q100) Solve 16x^2+40x+9=0
Ans: x=-0.25 or x=-2.25
Q101) Solve 9x^2+30x+16=0
Ans: x=-0.67 or x=-2.67
Q102) Solve 9x^2+34x+25=0
Ans: x=-1 or x=-2.78
Q103) Solve 50x^2+20x+2=0
Ans: x=-0.2 or x=-0.2
Q104) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q105) Solve 10x^2+33x+23=0
Ans: x=-1 or x=-2.3
Q106) Solve 12x^2+49x+49=0
Ans: x=-1.75 or x=-2.33
Q107) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q108) Solve 10x^2+31x+3=0
Ans: x=-0.1 or x=-3
Q109) Solve 18x^2+37x+19=0
Ans: x=-1 or x=-1.06
Q110) Solve 15x^2+22x+8=0
Ans: x=-0.67 or x=-0.8
Q111) Solve x^2+6x+5=0
Ans: x=-1 or x=-5
Q112) Solve 45x^2+46x+1=0
Ans: x=-0.02 or x=-1
Q113) Solve 8x^2+10x+3=0
Ans: x=-0.5 or x=-0.75
Q114) Solve x^2+31x+30=0
Ans: x=-1 or x=-30
Q115) Solve 20x^2+44x+17=0
Ans: x=-0.5 or x=-1.7
Q116) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q117) Solve 14x^2+40x+26=0
Ans: x=-1 or x=-1.86
Q118) Solve 11x^2+27x+10=0
Ans: x=-0.45 or x=-2
Q119) Solve 28x^2+48x+20=0
Ans: x=-0.71 or x=-1
Q120) Solve 42x^2+46x+4=0
Ans: x=-0.1 or x=-1
Q121) Solve 21x^2+37x+10=0
Ans: x=-0.33 or x=-1.43
Q122) Solve 2x^2+34x+32=0
Ans: x=-1 or x=-16
Q123) Solve 45x^2+33x+2=0
Ans: x=-0.07 or x=-0.67
Q124) Solve 40x^2+42x+8=0
Ans: x=-0.25 or x=-0.8
Q125) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q126) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q127) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q128) Solve 30x^2+42x+12=0
Ans: x=-0.4 or x=-1
Q129) Solve 4x^2+25x+34=0
Ans: x=-2 or x=-4.25
Q130) Solve 29x^2+35x+6=0
Ans: x=-0.21 or x=-1
Q131) Solve 34x^2+31x+7=0
Ans: x=-0.41 or x=-0.5
Q132) Solve 6x^2+17x+12=0
Ans: x=-1.33 or x=-1.5
Q133) Solve 17x^2+35x+2=0
Ans: x=-0.06 or x=-2
Q134) Solve 14x^2+41x+30=0
Ans: x=-1.43 or x=-1.5
Q135) Solve 14x^2+21x+7=0
Ans: x=-0.5 or x=-1
Q136) Solve 25x^2+26x+1=0
Ans: x=-0.04 or x=-1
Q137) Solve 25x^2+25x+4=0
Ans: x=-0.2 or x=-0.8
Q138) Solve 16x^2+47x+30=0
Ans: x=-0.94 or x=-2
Q139) Solve 14x^2+37x+18=0
Ans: x=-0.64 or x=-2
Q140) Solve x^2+26x+25=0
Ans: x=-1 or x=-25
Q141) Solve 8x^2+16x+6=0
Ans: x=-0.5 or x=-1.5
Q142) Solve 23x^2+31x+8=0
Ans: x=-0.35 or x=-1
Q143) Solve 2x^2+38x+36=0
Ans: x=-1 or x=-18
Q144) Solve x^2+44x+43=0
Ans: x=-1 or x=-43
Q145) Solve 24x^2+27x+3=0
Ans: x=-0.12 or x=-1
Q146) Solve 3x^2+19x+16=0
Ans: x=-1 or x=-5.33
Q147) Solve 6x^2+24x+24=0
Ans: x=-2 or x=-2
Q148) Solve 12x^2+10x+2=0
Ans: x=-0.33 or x=-0.5
Q149) Solve 35x^2+34x+8=0
Ans: x=-0.4 or x=-0.57
Q150) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q151) Solve 9x^2+24x+12=0
Ans: x=-0.67 or x=-2
Q152) Solve 26x^2+29x+8=0
Ans: x=-0.5 or x=-0.62
Q153) Solve 22x^2+26x+4=0
Ans: x=-0.18 or x=-1
Q154) Solve 34x^2+39x+11=0
Ans: x=-0.5 or x=-0.65
Q155) Solve 26x^2+37x+11=0
Ans: x=-0.42 or x=-1
Q156) Solve 34x^2+41x+7=0
Ans: x=-0.21 or x=-1
Q157) Solve 17x^2+42x+16=0
Ans: x=-0.47 or x=-2
Q158) Solve 7x^2+15x+8=0
Ans: x=-1 or x=-1.14
Q159) Solve 20x^2+41x+2=0
Ans: x=-0.05 or x=-2
Q160) Solve 35x^2+33x+4=0
Ans: x=-0.14 or x=-0.8
Q161) Solve 9x^2+6x+1=0
Ans: x=-0.33 or x=-0.33
Q162) Solve 3x^2+22x+7=0
Ans: x=-0.33 or x=-7
Q163) Solve 7x^2+25x+22=0
Ans: x=-1.57 or x=-2
Q164) Solve 22x^2+25x+7=0
Ans: x=-0.5 or x=-0.64
Q165) Solve 18x^2+23x+5=0
Ans: x=-0.28 or x=-1
Q166) Solve 2x^2+18x+40=0
Ans: x=-4 or x=-5
Q167) Solve 9x^2+33x+28=0
Ans: x=-1.33 or x=-2.33
Q168) Solve 15x^2+47x+32=0
Ans: x=-1 or x=-2.13
Q169) Solve 9x^2+38x+33=0
Ans: x=-1.22 or x=-3
Q170) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q171) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q172) Solve 8x^2+30x+7=0
Ans: x=-0.25 or x=-3.5
Q173) Solve 21x^2+37x+12=0
Ans: x=-0.43 or x=-1.33
Q174) Solve 7x^2+24x+9=0
Ans: x=-0.43 or x=-3
Q175) Solve 9x^2+38x+29=0
Ans: x=-1 or x=-3.22
Q176) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q177) Solve 10x^2+23x+13=0
Ans: x=-1 or x=-1.3
Q178) Solve x^2+6x+9=0
Ans: x=-3 or x=-3
Q179) Solve 16x^2+32x+15=0
Ans: x=-0.75 or x=-1.25
Q180) Solve 11x^2+32x+20=0
Ans: x=-0.91 or x=-2
Q181) Solve 50x^2+40x+6=0
Ans: x=-0.2 or x=-0.6
Q182) Solve 7x^2+23x+6=0
Ans: x=-0.29 or x=-3
Q183) Solve x^2+44x+43=0
Ans: x=-1 or x=-43
Q184) Solve 28x^2+31x+6=0
Ans: x=-0.25 or x=-0.86
Q185) Solve 2x^2+31x+42=0
Ans: x=-1.5 or x=-14
Q186) Solve 21x^2+47x+22=0
Ans: x=-0.67 or x=-1.57
Q187) Solve 18x^2+27x+7=0
Ans: x=-0.33 or x=-1.17
Q188) Solve 24x^2+22x+3=0
Ans: x=-0.17 or x=-0.75
Q189) Solve 47x^2+48x+1=0
Ans: x=-0.02 or x=-1
Q190) Solve 11x^2+36x+28=0
Ans: x=-1.27 or x=-2
Q191) Solve 48x^2+49x+11=0
Ans: x=-0.33 or x=-0.69
Q192) Solve 18x^2+50x+28=0
Ans: x=-0.78 or x=-2
Q193) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q194) Solve 12x^2+38x+30=0
Ans: x=-1.5 or x=-1.67
Q195) Solve 20x^2+34x+12=0
Ans: x=-0.5 or x=-1.2
Q196) Solve 17x^2+38x+21=0
Ans: x=-1 or x=-1.24
Q197) Solve 10x^2+42x+8=0
Ans: x=-0.2 or x=-4
Q198) Solve 8x^2+42x+10=0
Ans: x=-0.25 or x=-5
Q199) Solve 10x^2+25x+10=0
Ans: x=-0.5 or x=-2
Q200) Solve 12x^2+43x+21=0
Ans: x=-0.58 or x=-3
Q201) Solve 10x^2+49x+18=0
Ans: x=-0.4 or x=-4.5
Q202) Solve 9x^2+42x+40=0
Ans: x=-1.33 or x=-3.33
Q203) Solve 30x^2+47x+16=0
Ans: x=-0.5 or x=-1.07
Q204) Solve 35x^2+45x+10=0
Ans: x=-0.29 or x=-1
Q205) Solve 9x^2+31x+22=0
Ans: x=-1 or x=-2.44
Q206) Solve 14x^2+45x+19=0
Ans: x=-0.5 or x=-2.71
Q207) Solve 40x^2+23x+3=0
Ans: x=-0.2 or x=-0.37
Q208) Solve x^2+9x+8=0
Ans: x=-1 or x=-8
Q209) Solve 4x^2+33x+50=0
Ans: x=-2 or x=-6.25
Q210) Solve 30x^2+49x+19=0
Ans: x=-0.63 or x=-1
Q211) Solve 15x^2+47x+6=0
Ans: x=-0.13 or x=-3
Q212) Solve 12x^2+22x+6=0
Ans: x=-0.33 or x=-1.5
Q213) Solve 5x^2+30x+25=0
Ans: x=-1 or x=-5
Q214) Solve 44x^2+31x+5=0
Ans: x=-0.25 or x=-0.45
Q215) Solve 8x^2+37x+29=0
Ans: x=-1 or x=-3.62
Q216) Solve 6x^2+21x+9=0
Ans: x=-0.5 or x=-3
Q217) Solve 26x^2+25x+6=0
Ans: x=-0.46 or x=-0.5
Q218) Solve 2x^2+15x+13=0
Ans: x=-1 or x=-6.5
Q219) Solve 12x^2+27x+15=0
Ans: x=-1 or x=-1.25
Q220) Solve 10x^2+18x+8=0
Ans: x=-0.8 or x=-1
Q221) Solve 18x^2+50x+28=0
Ans: x=-0.78 or x=-2
Q222) Solve 5x^2+20x+15=0
Ans: x=-1 or x=-3
Q223) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q224) Solve 5x^2+32x+27=0
Ans: x=-1 or x=-5.4
Q225) Solve 35x^2+38x+8=0
Ans: x=-0.29 or x=-0.8
Q226) Solve 25x^2+30x+5=0
Ans: x=-0.2 or x=-1
Q227) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q228) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q229) Solve 8x^2+39x+45=0
Ans: x=-1.87 or x=-3
Q230) Solve 9x^2+20x+11=0
Ans: x=-1 or x=-1.22
Q231) Solve 11x^2+20x+9=0
Ans: x=-0.82 or x=-1
Q232) Solve 3x^2+19x+20=0
Ans: x=-1.33 or x=-5
Q233) Solve 22x^2+49x+27=0
Ans: x=-1 or x=-1.23
Q234) Solve 4x^2+20x+16=0
Ans: x=-1 or x=-4
Q235) Solve 15x^2+17x+4=0
Ans: x=-0.33 or x=-0.8
Q236) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q237) Solve 3x^2+31x+50=0
Ans: x=-2 or x=-8.33
Q238) Solve 34x^2+29x+6=0
Ans: x=-0.35 or x=-0.5
Q239) Solve 35x^2+31x+6=0
Ans: x=-0.29 or x=-0.6
Q240) Solve 32x^2+44x+12=0
Ans: x=-0.37 or x=-1
Q241) Solve 2x^2+35x+17=0
Ans: x=-0.5 or x=-17
Q242) Solve 15x^2+22x+7=0
Ans: x=-0.47 or x=-1
Q243) Solve 15x^2+29x+12=0
Ans: x=-0.6 or x=-1.33
Q244) Solve x^2+12x+11=0
Ans: x=-1 or x=-11
Q245) Solve 20x^2+29x+6=0
Ans: x=-0.25 or x=-1.2
Q246) Solve 4x^2+50x+46=0
Ans: x=-1 or x=-11.5
Q247) Solve 41x^2+42x+1=0
Ans: x=-0.02 or x=-1
Q248) Solve 28x^2+32x+9=0
Ans: x=-0.5 or x=-0.64
Q249) Solve 32x^2+50x+3=0
Ans: x=-0.06 or x=-1.5
Q250) Solve 40x^2+30x+5=0
Ans: x=-0.25 or x=-0.5
Q251) Solve 2x^2+49x+47=0
Ans: x=-1 or x=-23.5
Q252) Solve 24x^2+10x+1=0
Ans: x=-0.17 or x=-0.25
Q253) Solve 2x^2+21x+45=0
Ans: x=-3 or x=-7.5
Q254) Solve 21x^2+33x+12=0
Ans: x=-0.57 or x=-1
Q255) Solve 37x^2+44x+7=0
Ans: x=-0.19 or x=-1
Q256) Solve 9x^2+28x+19=0
Ans: x=-1 or x=-2.11
Q257) Solve 17x^2+49x+30=0
Ans: x=-0.88 or x=-2
Q258) Solve 18x^2+38x+20=0
Ans: x=-1 or x=-1.11
Q259) Solve 32x^2+50x+17=0
Ans: x=-0.5 or x=-1.06
Q260) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q261) Solve 28x^2+44x+3=0
Ans: x=-0.07 or x=-1.5
Q262) Solve 24x^2+40x+6=0
Ans: x=-0.17 or x=-1.5
Q263) Solve 13x^2+49x+30=0
Ans: x=-0.77 or x=-3
Q264) Solve 8x^2+34x+33=0
Ans: x=-1.5 or x=-2.75
Q265) Solve x^2+14x+49=0
Ans: x=-7 or x=-7
Q266) Solve 3x^2+22x+7=0
Ans: x=-0.33 or x=-7
Q267) Solve 22x^2+50x+12=0
Ans: x=-0.27 or x=-2
Q268) Solve 40x^2+21x+2=0
Ans: x=-0.12 or x=-0.4
Q269) Solve 12x^2+48x+48=0
Ans: x=-2 or x=-2
Q270) Solve 9x^2+23x+14=0
Ans: x=-1 or x=-1.56
Q271) Solve 2x^2+47x+45=0
Ans: x=-1 or x=-22.5
Q272) Solve 20x^2+28x+8=0
Ans: x=-0.4 or x=-1
Q273) Solve 28x^2+33x+5=0
Ans: x=-0.18 or x=-1
Q274) Solve 24x^2+43x+5=0
Ans: x=-0.12 or x=-1.67
Q275) Solve 16x^2+39x+14=0
Ans: x=-0.44 or x=-2
Q276) Solve 34x^2+27x+5=0
Ans: x=-0.29 or x=-0.5
Q277) Solve 7x^2+26x+24=0
Ans: x=-1.71 or x=-2
Q278) Solve 45x^2+46x+1=0
Ans: x=-0.02 or x=-1
Q279) Solve 29x^2+49x+20=0
Ans: x=-0.69 or x=-1
Q280) Solve x^2+10x+25=0
Ans: x=-5 or x=-5
Q281) Solve 30x^2+40x+10=0
Ans: x=-0.33 or x=-1
Q282) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q283) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q284) Solve 6x^2+39x+18=0
Ans: x=-0.5 or x=-6
Q285) Solve 9x^2+42x+33=0
Ans: x=-1 or x=-3.67
Q286) Solve 42x^2+43x+6=0
Ans: x=-0.17 or x=-0.86
Q287) Solve 2x^2+20x+48=0
Ans: x=-4 or x=-6
Q288) Solve 26x^2+30x+4=0
Ans: x=-0.15 or x=-1
Q289) Solve 10x^2+23x+6=0
Ans: x=-0.3 or x=-2
Q290) Solve 7x^2+23x+6=0
Ans: x=-0.29 or x=-3
Q291) Solve 28x^2+36x+11=0
Ans: x=-0.5 or x=-0.79
Q292) Solve 7x^2+38x+40=0
Ans: x=-1.43 or x=-4
Q293) Solve 11x^2+41x+38=0
Ans: x=-1.73 or x=-2
Q294) Solve 2x^2+19x+44=0
Ans: x=-4 or x=-5.5
Q295) Solve 12x^2+46x+34=0
Ans: x=-1 or x=-2.83
Q296) Solve 24x^2+43x+19=0
Ans: x=-0.79 or x=-1
Q297) Solve 5x^2+16x+11=0
Ans: x=-1 or x=-2.2
Q298) Solve 20x^2+45x+25=0
Ans: x=-1 or x=-1.25
Q299) Solve 33x^2+35x+8=0
Ans: x=-0.33 or x=-0.73
Q300) Solve 36x^2+38x+2=0
Ans: x=-0.06 or x=-1
Q301) Solve 25x^2+10x+1=0
Ans: x=-0.2 or x=-0.2
Q302) Solve 25x^2+49x+24=0
Ans: x=-0.96 or x=-1
Q303) Solve 3x^2+24x+48=0
Ans: x=-4 or x=-4
Q304) Solve 9x^2+22x+8=0
Ans: x=-0.44 or x=-2
Q305) Solve 16x^2+40x+24=0
Ans: x=-1 or x=-1.5
Q306) Solve 43x^2+46x+3=0
Ans: x=-0.07 or x=-1
Q307) Solve 13x^2+48x+35=0
Ans: x=-1 or x=-2.69
Q308) Solve 48x^2+43x+9=0
Ans: x=-0.33 or x=-0.56
Q309) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q310) Solve 46x^2+41x+9=0
Ans: x=-0.39 or x=-0.5
Q311) Solve 2x^2+46x+44=0
Ans: x=-1 or x=-22
Q312) Solve 22x^2+40x+18=0
Ans: x=-0.82 or x=-1
Q313) Solve 4x^2+24x+36=0
Ans: x=-3 or x=-3
Q314) Solve 20x^2+41x+21=0
Ans: x=-1 or x=-1.05
Q315) Solve x^2+8x+7=0
Ans: x=-1 or x=-7
Q316) Solve 16x^2+46x+28=0
Ans: x=-0.87 or x=-2
Q317) Solve 20x^2+22x+6=0
Ans: x=-0.5 or x=-0.6
Q318) Solve 2x^2+32x+30=0
Ans: x=-1 or x=-15
Q319) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q320) Solve 40x^2+49x+9=0
Ans: x=-0.22 or x=-1
Q321) Solve 46x^2+43x+10=0
Ans: x=-0.43 or x=-0.5
Q322) Solve 25x^2+45x+20=0
Ans: x=-0.8 or x=-1
Q323) Solve 17x^2+37x+20=0
Ans: x=-1 or x=-1.18
Q324) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q325) Solve 2x^2+32x+30=0
Ans: x=-1 or x=-15
Q326) Solve 27x^2+48x+16=0
Ans: x=-0.44 or x=-1.33
Q327) Solve 18x^2+48x+32=0
Ans: x=-1.33 or x=-1.33
Q328) Solve 3x^2+10x+7=0
Ans: x=-1 or x=-2.33
Q329) Solve 8x^2+14x+5=0
Ans: x=-0.5 or x=-1.25
Q330) Solve 26x^2+47x+17=0
Ans: x=-0.5 or x=-1.31
Q331) Solve 16x^2+8x+1=0
Ans: x=-0.25 or x=-0.25
Q332) Solve 5x^2+26x+32=0
Ans: x=-2 or x=-3.2
Q333) Solve 2x^2+29x+50=0
Ans: x=-2 or x=-12.5
Q334) Solve 13x^2+48x+27=0
Ans: x=-0.69 or x=-3
Q335) Solve 9x^2+36x+11=0
Ans: x=-0.33 or x=-3.67
Q336) Solve 24x^2+34x+11=0
Ans: x=-0.5 or x=-0.92
Q337) Solve 10x^2+50x+40=0
Ans: x=-1 or x=-4
Q338) Solve 2x^2+21x+45=0
Ans: x=-3 or x=-7.5
Q339) Solve 16x^2+25x+9=0
Ans: x=-0.56 or x=-1
Q340) Solve 43x^2+48x+5=0
Ans: x=-0.12 or x=-1
Q341) Solve 5x^2+50x+45=0
Ans: x=-1 or x=-9
Q342) Solve 21x^2+46x+8=0
Ans: x=-0.19 or x=-2
Q343) Solve 4x^2+8x+4=0
Ans: x=-1 or x=-1
Q344) Solve 2x^2+39x+37=0
Ans: x=-1 or x=-18.5
Q345) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q346) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q347) Solve 9x^2+32x+23=0
Ans: x=-1 or x=-2.56
Q348) Solve 8x^2+40x+42=0
Ans: x=-1.5 or x=-3.5
Q349) Solve 5x^2+26x+33=0
Ans: x=-2.2 or x=-3
Q350) Solve 18x^2+33x+9=0
Ans: x=-0.33 or x=-1.5
Q351) Solve 24x^2+43x+18=0
Ans: x=-0.67 or x=-1.12
Q352) Solve 11x^2+40x+21=0
Ans: x=-0.64 or x=-3
Q353) Solve 8x^2+11x+3=0
Ans: x=-0.37 or x=-1
Q354) Solve 9x^2+37x+4=0
Ans: x=-0.11 or x=-4
Q355) Solve 9x^2+36x+20=0
Ans: x=-0.67 or x=-3.33
Q356) Solve 32x^2+38x+6=0
Ans: x=-0.19 or x=-1
Q357) Solve 36x^2+24x+4=0
Ans: x=-0.33 or x=-0.33
Q358) Solve 48x^2+46x+10=0
Ans: x=-0.33 or x=-0.62
Q359) Solve 15x^2+41x+28=0
Ans: x=-1.33 or x=-1.4
Q360) Solve 27x^2+38x+11=0
Ans: x=-0.41 or x=-1
Q361) Solve 4x^2+18x+18=0
Ans: x=-1.5 or x=-3
Q362) Solve 8x^2+40x+18=0
Ans: x=-0.5 or x=-4.5
Q363) Solve 18x^2+29x+10=0
Ans: x=-0.5 or x=-1.11
Q364) Solve 19x^2+42x+23=0
Ans: x=-1 or x=-1.21
Q365) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q366) Solve x^2+47x+46=0
Ans: x=-1 or x=-46
Q367) Solve 38x^2+41x+3=0
Ans: x=-0.08 or x=-1
Q368) Solve 8x^2+16x+8=0
Ans: x=-1 or x=-1
Q369) Solve 42x^2+13x+1=0
Ans: x=-0.14 or x=-0.17
Q370) Solve 13x^2+19x+6=0
Ans: x=-0.46 or x=-1
Q371) Solve 38x^2+42x+4=0
Ans: x=-0.11 or x=-1
Q372) Solve 3x^2+28x+9=0
Ans: x=-0.33 or x=-9
Q373) Solve 7x^2+45x+18=0
Ans: x=-0.43 or x=-6
Q374) Solve 6x^2+10x+4=0
Ans: x=-0.67 or x=-1
Q375) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q376) Solve 14x^2+49x+42=0
Ans: x=-1.5 or x=-2
Q377) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q378) Solve 7x^2+40x+25=0
Ans: x=-0.71 or x=-5
Q379) Solve 12x^2+10x+2=0
Ans: x=-0.33 or x=-0.5
Q380) Solve x^2+12x+20=0
Ans: x=-2 or x=-10
Q381) Solve 22x^2+37x+6=0
Ans: x=-0.18 or x=-1.5
Q382) Solve 14x^2+43x+20=0
Ans: x=-0.57 or x=-2.5
Q383) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q384) Solve 6x^2+10x+4=0
Ans: x=-0.67 or x=-1
Q385) Solve x^2+14x+13=0
Ans: x=-1 or x=-13
Q386) Solve 5x^2+39x+34=0
Ans: x=-1 or x=-6.8
Q387) Solve 6x^2+29x+33=0
Ans: x=-1.83 or x=-3
Q388) Solve 23x^2+34x+11=0
Ans: x=-0.48 or x=-1
Q389) Solve 8x^2+34x+8=0
Ans: x=-0.25 or x=-4
Q390) Solve 5x^2+22x+24=0
Ans: x=-2 or x=-2.4
Q391) Solve 12x^2+31x+20=0
Ans: x=-1.25 or x=-1.33
Q392) Solve 13x^2+36x+23=0
Ans: x=-1 or x=-1.77
Q393) Solve 39x^2+37x+8=0
Ans: x=-0.33 or x=-0.62
Q394) Solve 22x^2+47x+21=0
Ans: x=-0.64 or x=-1.5
Q395) Solve 28x^2+41x+13=0
Ans: x=-0.46 or x=-1
Q396) Solve 14x^2+37x+23=0
Ans: x=-1 or x=-1.64
Q397) Solve 3x^2+13x+12=0
Ans: x=-1.33 or x=-3
Q398) Solve 39x^2+47x+14=0
Ans: x=-0.54 or x=-0.67
Q399) Solve 34x^2+37x+3=0
Ans: x=-0.09 or x=-1
Q400) Solve 23x^2+50x+27=0
Ans: x=-1 or x=-1.17
Q401) Solve 3x^2+38x+35=0
Ans: x=-1 or x=-11.67
Q402) Solve 9x^2+15x+6=0
Ans: x=-0.67 or x=-1
Q403) Solve 27x^2+33x+6=0
Ans: x=-0.22 or x=-1
Q404) Solve 6x^2+29x+9=0
Ans: x=-0.33 or x=-4.5
Q405) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q406) Solve 20x^2+34x+12=0
Ans: x=-0.5 or x=-1.2
Q407) Solve 12x^2+37x+21=0
Ans: x=-0.75 or x=-2.33
Q408) Solve 2x^2+23x+21=0
Ans: x=-1 or x=-10.5
Q409) Solve 5x^2+30x+45=0
Ans: x=-3 or x=-3
Q410) Solve 45x^2+44x+7=0
Ans: x=-0.2 or x=-0.78
Q411) Solve x^2+45x+44=0
Ans: x=-1 or x=-44
Q412) Solve 10x^2+43x+12=0
Ans: x=-0.3 or x=-4
Q413) Solve 3x^2+14x+15=0
Ans: x=-1.67 or x=-3
Q414) Solve 30x^2+19x+3=0
Ans: x=-0.3 or x=-0.33
Q415) Solve 3x^2+22x+19=0
Ans: x=-1 or x=-6.33
Q416) Solve 4x^2+15x+11=0
Ans: x=-1 or x=-2.75
Q417) Solve 10x^2+32x+22=0
Ans: x=-1 or x=-2.2
Q418) Solve 21x^2+22x+1=0
Ans: x=-0.05 or x=-1
Q419) Solve 10x^2+45x+50=0
Ans: x=-2 or x=-2.5
Q420) Solve 5x^2+28x+39=0
Ans: x=-2.6 or x=-3
Q421) Solve 22x^2+38x+16=0
Ans: x=-0.73 or x=-1
Q422) Solve 6x^2+37x+35=0
Ans: x=-1.17 or x=-5
Q423) Solve 21x^2+44x+20=0
Ans: x=-0.67 or x=-1.43
Q424) Solve 4x^2+22x+28=0
Ans: x=-2 or x=-3.5
Q425) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q426) Solve 36x^2+47x+11=0
Ans: x=-0.31 or x=-1
Q427) Solve 16x^2+42x+5=0
Ans: x=-0.12 or x=-2.5
Q428) Solve x^2+8x+7=0
Ans: x=-1 or x=-7
Q429) Solve 18x^2+36x+18=0
Ans: x=-1 or x=-1
Q430) Solve 5x^2+27x+10=0
Ans: x=-0.4 or x=-5
Q431) Solve 28x^2+41x+13=0
Ans: x=-0.46 or x=-1
Q432) Solve 23x^2+43x+20=0
Ans: x=-0.87 or x=-1
Q433) Solve 29x^2+45x+16=0
Ans: x=-0.55 or x=-1
Q434) Solve 6x^2+40x+50=0
Ans: x=-1.67 or x=-5
Q435) Solve 32x^2+40x+8=0
Ans: x=-0.25 or x=-1
Q436) Solve x^2+9x+14=0
Ans: x=-2 or x=-7
Q437) Solve 16x^2+36x+14=0
Ans: x=-0.5 or x=-1.75
Q438) Solve 14x^2+49x+42=0
Ans: x=-1.5 or x=-2
Q439) Solve 7x^2+32x+16=0
Ans: x=-0.57 or x=-4
Q440) Solve 13x^2+43x+34=0
Ans: x=-1.31 or x=-2
Q441) Solve x^2+24x+23=0
Ans: x=-1 or x=-23
Q442) Solve 5x^2+34x+45=0
Ans: x=-1.8 or x=-5
Q443) Solve 31x^2+42x+11=0
Ans: x=-0.35 or x=-1
Q444) Solve 25x^2+30x+8=0
Ans: x=-0.4 or x=-0.8
Q445) Solve 6x^2+39x+45=0
Ans: x=-1.5 or x=-5
Q446) Solve 32x^2+30x+7=0
Ans: x=-0.44 or x=-0.5
Q447) Solve 4x^2+31x+42=0
Ans: x=-1.75 or x=-6
Q448) Solve 19x^2+32x+13=0
Ans: x=-0.68 or x=-1
Q449) Solve 9x^2+43x+34=0
Ans: x=-1 or x=-3.78
Q450) Solve 16x^2+46x+15=0
Ans: x=-0.37 or x=-2.5
Q451) Solve 14x^2+33x+18=0
Ans: x=-0.86 or x=-1.5
Q452) Solve 8x^2+41x+36=0
Ans: x=-1.12 or x=-4
Q453) Solve 23x^2+40x+17=0
Ans: x=-0.74 or x=-1
Q454) Solve 20x^2+49x+18=0
Ans: x=-0.45 or x=-2
Q455) Solve 6x^2+17x+12=0
Ans: x=-1.33 or x=-1.5
Q456) Solve 4x^2+32x+39=0
Ans: x=-1.5 or x=-6.5
Q457) Solve 6x^2+13x+2=0
Ans: x=-0.17 or x=-2
Q458) Solve 39x^2+46x+11=0
Ans: x=-0.33 or x=-0.85
Q459) Solve 24x^2+42x+9=0
Ans: x=-0.25 or x=-1.5
Q460) Solve 36x^2+36x+9=0
Ans: x=-0.5 or x=-0.5
Q461) Solve 6x^2+38x+32=0
Ans: x=-1 or x=-5.33
Q462) Solve 5x^2+40x+35=0
Ans: x=-1 or x=-7
Q463) Solve 7x^2+44x+12=0
Ans: x=-0.29 or x=-6
Q464) Solve 12x^2+47x+11=0
Ans: x=-0.25 or x=-3.67
Q465) Solve 36x^2+12x+1=0
Ans: x=-0.17 or x=-0.17
Q466) Solve 5x^2+9x+4=0
Ans: x=-0.8 or x=-1
Q467) Solve 6x^2+32x+10=0
Ans: x=-0.33 or x=-5
Q468) Solve 3x^2+19x+26=0
Ans: x=-2 or x=-4.33
Q469) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q470) Solve 10x^2+38x+36=0
Ans: x=-1.8 or x=-2
Q471) Solve 13x^2+21x+8=0
Ans: x=-0.62 or x=-1
Q472) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q473) Solve 7x^2+23x+16=0
Ans: x=-1 or x=-2.29
Q474) Solve 26x^2+47x+21=0
Ans: x=-0.81 or x=-1
Q475) Solve 7x^2+48x+41=0
Ans: x=-1 or x=-5.86
Q476) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q477) Solve 20x^2+38x+14=0
Ans: x=-0.5 or x=-1.4
Q478) Solve 48x^2+14x+1=0
Ans: x=-0.12 or x=-0.17
Q479) Solve 9x^2+9x+2=0
Ans: x=-0.33 or x=-0.67
Q480) Solve 6x^2+29x+30=0
Ans: x=-1.5 or x=-3.33
Q481) Solve 30x^2+23x+2=0
Ans: x=-0.1 or x=-0.67
Q482) Solve 48x^2+38x+5=0
Ans: x=-0.17 or x=-0.62
Q483) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q484) Solve 4x^2+46x+42=0
Ans: x=-1 or x=-10.5
Q485) Solve 29x^2+31x+2=0
Ans: x=-0.07 or x=-1
Q486) Solve 17x^2+43x+26=0
Ans: x=-1 or x=-1.53
Q487) Solve 10x^2+23x+12=0
Ans: x=-0.8 or x=-1.5
Q488) Solve 8x^2+40x+48=0
Ans: x=-2 or x=-3
Q489) Solve 13x^2+38x+24=0
Ans: x=-0.92 or x=-2
Q490) Solve 12x^2+22x+10=0
Ans: x=-0.83 or x=-1
Q491) Solve 12x^2+22x+6=0
Ans: x=-0.33 or x=-1.5
Q492) Solve 16x^2+22x+6=0
Ans: x=-0.37 or x=-1
Q493) Solve 12x^2+36x+27=0
Ans: x=-1.5 or x=-1.5
Q494) Solve 4x^2+31x+46=0
Ans: x=-2 or x=-5.75
Q495) Solve 37x^2+50x+13=0
Ans: x=-0.35 or x=-1
Q496) Solve 11x^2+49x+20=0
Ans: x=-0.45 or x=-4
Q497) Solve 32x^2+44x+5=0
Ans: x=-0.12 or x=-1.25
Q498) Solve 14x^2+42x+28=0
Ans: x=-1 or x=-2
Q499) Solve 7x^2+23x+18=0
Ans: x=-1.29 or x=-2
Q500) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q501) Solve 18x^2+25x+7=0
Ans: x=-0.39 or x=-1
Q502) Solve 40x^2+22x+3=0
Ans: x=-0.25 or x=-0.3
Q503) Solve 16x^2+28x+10=0
Ans: x=-0.5 or x=-1.25
Q504) Solve 13x^2+46x+33=0
Ans: x=-1 or x=-2.54
Q505) Solve 15x^2+49x+12=0
Ans: x=-0.27 or x=-3
Q506) Solve x^2+23x+22=0
Ans: x=-1 or x=-22
Q507) Solve x^2+22x+40=0
Ans: x=-2 or x=-20
Q508) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q509) Solve 40x^2+14x+1=0
Ans: x=-0.1 or x=-0.25
Q510) Solve 5x^2+26x+24=0
Ans: x=-1.2 or x=-4
Q511) Solve 36x^2+36x+9=0
Ans: x=-0.5 or x=-0.5
Q512) Solve 12x^2+41x+24=0
Ans: x=-0.75 or x=-2.67
Q513) Solve 19x^2+49x+22=0
Ans: x=-0.58 or x=-2
Q514) Solve 35x^2+42x+7=0
Ans: x=-0.2 or x=-1
Q515) Solve 6x^2+33x+15=0
Ans: x=-0.5 or x=-5
Q516) Solve 19x^2+47x+18=0
Ans: x=-0.47 or x=-2
Q517) Solve 2x^2+13x+18=0
Ans: x=-2 or x=-4.5
Q518) Solve 40x^2+18x+2=0
Ans: x=-0.2 or x=-0.25
Q519) Solve 17x^2+32x+15=0
Ans: x=-0.88 or x=-1
Q520) Solve 6x^2+38x+32=0
Ans: x=-1 or x=-5.33
Q521) Solve 39x^2+38x+8=0
Ans: x=-0.31 or x=-0.67
Q522) Solve 16x^2+40x+16=0
Ans: x=-0.5 or x=-2
Q523) Solve 7x^2+20x+13=0
Ans: x=-1 or x=-1.86
Q524) Solve 38x^2+41x+11=0
Ans: x=-0.5 or x=-0.58
Q525) Solve 2x^2+9x+10=0
Ans: x=-2 or x=-2.5
Q526) Solve 5x^2+16x+3=0
Ans: x=-0.2 or x=-3
Q527) Solve 4x^2+29x+25=0
Ans: x=-1 or x=-6.25
Q528) Solve 10x^2+17x+7=0
Ans: x=-0.7 or x=-1
Q529) Solve 4x^2+25x+6=0
Ans: x=-0.25 or x=-6
Q530) Solve 29x^2+48x+19=0
Ans: x=-0.66 or x=-1
Q531) Solve 6x^2+43x+26=0
Ans: x=-0.67 or x=-6.5
Q532) Solve 3x^2+13x+12=0
Ans: x=-1.33 or x=-3
Q533) Solve 28x^2+29x+6=0
Ans: x=-0.29 or x=-0.75
Q534) Solve 8x^2+38x+35=0
Ans: x=-1.25 or x=-3.5
Q535) Solve 25x^2+39x+14=0
Ans: x=-0.56 or x=-1
Q536) Solve 24x^2+22x+4=0
Ans: x=-0.25 or x=-0.67
Q537) Solve 12x^2+22x+10=0
Ans: x=-0.83 or x=-1
Q538) Solve 9x^2+37x+28=0
Ans: x=-1 or x=-3.11
Q539) Solve 18x^2+25x+7=0
Ans: x=-0.39 or x=-1
Q540) Solve 3x^2+7x+4=0
Ans: x=-1 or x=-1.33
Q541) Solve 15x^2+28x+12=0
Ans: x=-0.67 or x=-1.2
Q542) Solve 23x^2+27x+4=0
Ans: x=-0.17 or x=-1
Q543) Solve 18x^2+22x+4=0
Ans: x=-0.22 or x=-1
Q544) Solve 6x^2+35x+50=0
Ans: x=-2.5 or x=-3.33
Q545) Solve 44x^2+42x+10=0
Ans: x=-0.45 or x=-0.5
Q546) Solve 22x^2+47x+18=0
Ans: x=-0.5 or x=-1.64
Q547) Solve 8x^2+9x+1=0
Ans: x=-0.12 or x=-1
Q548) Solve 7x^2+29x+4=0
Ans: x=-0.14 or x=-4
Q549) Solve 9x^2+36x+11=0
Ans: x=-0.33 or x=-3.67
Q550) Solve 32x^2+36x+7=0
Ans: x=-0.25 or x=-0.87
Q551) Solve x^2+9x+18=0
Ans: x=-3 or x=-6
Q552) Solve 9x^2+26x+17=0
Ans: x=-1 or x=-1.89
Q553) Solve 10x^2+47x+48=0
Ans: x=-1.5 or x=-3.2
Q554) Solve x^2+25x+46=0
Ans: x=-2 or x=-23
Q555) Solve 21x^2+22x+1=0
Ans: x=-0.05 or x=-1
Q556) Solve 10x^2+28x+16=0
Ans: x=-0.8 or x=-2
Q557) Solve 9x^2+38x+8=0
Ans: x=-0.22 or x=-4
Q558) Solve 24x^2+18x+3=0
Ans: x=-0.25 or x=-0.5
Q559) Solve 20x^2+21x+1=0
Ans: x=-0.05 or x=-1
Q560) Solve 29x^2+34x+5=0
Ans: x=-0.17 or x=-1
Q561) Solve 18x^2+48x+30=0
Ans: x=-1 or x=-1.67
Q562) Solve 12x^2+13x+3=0
Ans: x=-0.33 or x=-0.75
Q563) Solve 6x^2+25x+4=0
Ans: x=-0.17 or x=-4
Q564) Solve 15x^2+49x+26=0
Ans: x=-0.67 or x=-2.6
Q565) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q566) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q567) Solve 21x^2+49x+14=0
Ans: x=-0.33 or x=-2
Q568) Solve 20x^2+36x+16=0
Ans: x=-0.8 or x=-1
Q569) Solve 16x^2+32x+16=0
Ans: x=-1 or x=-1
Q570) Solve 8x^2+28x+20=0
Ans: x=-1 or x=-2.5
Q571) Solve 36x^2+45x+9=0
Ans: x=-0.25 or x=-1
Q572) Solve 10x^2+39x+14=0
Ans: x=-0.4 or x=-3.5
Q573) Solve 9x^2+27x+18=0
Ans: x=-1 or x=-2
Q574) Solve 14x^2+32x+8=0
Ans: x=-0.29 or x=-2
Q575) Solve 40x^2+28x+4=0
Ans: x=-0.2 or x=-0.5
Q576) Solve x^2+44x+43=0
Ans: x=-1 or x=-43
Q577) Solve 12x^2+36x+24=0
Ans: x=-1 or x=-2
Q578) Solve 2x^2+9x+10=0
Ans: x=-2 or x=-2.5
Q579) Solve 16x^2+46x+15=0
Ans: x=-0.37 or x=-2.5
Q580) Solve 22x^2+45x+17=0
Ans: x=-0.5 or x=-1.55
Q581) Solve 19x^2+40x+4=0
Ans: x=-0.11 or x=-2
Q582) Solve 9x^2+26x+17=0
Ans: x=-1 or x=-1.89
Q583) Solve 8x^2+41x+5=0
Ans: x=-0.12 or x=-5
Q584) Solve 13x^2+32x+19=0
Ans: x=-1 or x=-1.46
Q585) Solve 3x^2+44x+41=0
Ans: x=-1 or x=-13.67
Q586) Solve 15x^2+34x+19=0
Ans: x=-1 or x=-1.27
Q587) Solve 12x^2+46x+42=0
Ans: x=-1.5 or x=-2.33
Q588) Solve 34x^2+27x+5=0
Ans: x=-0.29 or x=-0.5
Q589) Solve 8x^2+24x+18=0
Ans: x=-1.5 or x=-1.5
Q590) Solve 2x^2+9x+10=0
Ans: x=-2 or x=-2.5
Q591) Solve 30x^2+41x+7=0
Ans: x=-0.2 or x=-1.17
Q592) Solve 12x^2+50x+38=0
Ans: x=-1 or x=-3.17
Q593) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q594) Solve 8x^2+36x+16=0
Ans: x=-0.5 or x=-4
Q595) Solve 11x^2+23x+2=0
Ans: x=-0.09 or x=-2
Q596) Solve 31x^2+36x+5=0
Ans: x=-0.16 or x=-1
Q597) Solve 36x^2+41x+5=0
Ans: x=-0.14 or x=-1
Q598) Solve 6x^2+36x+30=0
Ans: x=-1 or x=-5
Q599) Solve 12x^2+43x+35=0
Ans: x=-1.25 or x=-2.33
Q600) Solve 44x^2+36x+7=0
Ans: x=-0.32 or x=-0.5
Q601) Solve 43x^2+49x+6=0
Ans: x=-0.14 or x=-1
Q602) Solve 44x^2+24x+1=0
Ans: x=-0.05 or x=-0.5
Q603) Solve 11x^2+15x+4=0
Ans: x=-0.36 or x=-1
Q604) Solve 10x^2+26x+12=0
Ans: x=-0.6 or x=-2
Q605) Solve 19x^2+41x+6=0
Ans: x=-0.16 or x=-2
Q606) Solve 9x^2+32x+23=0
Ans: x=-1 or x=-2.56
Q607) Solve 10x^2+21x+9=0
Ans: x=-0.6 or x=-1.5
Q608) Solve 7x^2+33x+26=0
Ans: x=-1 or x=-3.71
Q609) Solve 26x^2+41x+3=0
Ans: x=-0.08 or x=-1.5
Q610) Solve 8x^2+38x+39=0
Ans: x=-1.5 or x=-3.25
Q611) Solve 9x^2+10x+1=0
Ans: x=-0.11 or x=-1
Q612) Solve 49x^2+49x+12=0
Ans: x=-0.43 or x=-0.57
Q613) Solve 38x^2+23x+2=0
Ans: x=-0.11 or x=-0.5
Q614) Solve 6x^2+19x+10=0
Ans: x=-0.67 or x=-2.5
Q615) Solve 23x^2+37x+14=0
Ans: x=-0.61 or x=-1
Q616) Solve 40x^2+36x+8=0
Ans: x=-0.4 or x=-0.5
Q617) Solve 9x^2+42x+13=0
Ans: x=-0.33 or x=-4.33
Q618) Solve 4x^2+15x+11=0
Ans: x=-1 or x=-2.75
Q619) Solve 12x^2+13x+3=0
Ans: x=-0.33 or x=-0.75
Q620) Solve 34x^2+37x+10=0
Ans: x=-0.5 or x=-0.59
Q621) Solve 30x^2+39x+9=0
Ans: x=-0.3 or x=-1
Q622) Solve 9x^2+41x+20=0
Ans: x=-0.56 or x=-4
Q623) Solve x^2+13x+36=0
Ans: x=-4 or x=-9
Q624) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q625) Solve 5x^2+14x+8=0
Ans: x=-0.8 or x=-2
Q626) Solve 12x^2+40x+32=0
Ans: x=-1.33 or x=-2
Q627) Solve 8x^2+47x+39=0
Ans: x=-1 or x=-4.87
Q628) Solve 7x^2+46x+24=0
Ans: x=-0.57 or x=-6
Q629) Solve 25x^2+44x+19=0
Ans: x=-0.76 or x=-1
Q630) Solve 32x^2+46x+15=0
Ans: x=-0.5 or x=-0.94
Q631) Solve 42x^2+13x+1=0
Ans: x=-0.14 or x=-0.17
Q632) Solve 3x^2+30x+27=0
Ans: x=-1 or x=-9
Q633) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q634) Solve 9x^2+39x+40=0
Ans: x=-1.67 or x=-2.67
Q635) Solve 18x^2+37x+15=0
Ans: x=-0.56 or x=-1.5
Q636) Solve 20x^2+31x+11=0
Ans: x=-0.55 or x=-1
Q637) Solve 38x^2+45x+7=0
Ans: x=-0.18 or x=-1
Q638) Solve 36x^2+39x+9=0
Ans: x=-0.33 or x=-0.75
Q639) Solve 32x^2+28x+5=0
Ans: x=-0.25 or x=-0.62
Q640) Solve 2x^2+22x+36=0
Ans: x=-2 or x=-9
Q641) Solve 18x^2+45x+13=0
Ans: x=-0.33 or x=-2.17
Q642) Solve 6x^2+27x+27=0
Ans: x=-1.5 or x=-3
Q643) Solve 21x^2+43x+20=0
Ans: x=-0.71 or x=-1.33
Q644) Solve 21x^2+47x+22=0
Ans: x=-0.67 or x=-1.57
Q645) Solve 33x^2+34x+1=0
Ans: x=-0.03 or x=-1
Q646) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q647) Solve 48x^2+49x+11=0
Ans: x=-0.33 or x=-0.69
Q648) Solve 13x^2+45x+18=0
Ans: x=-0.46 or x=-3
Q649) Solve 22x^2+25x+3=0
Ans: x=-0.14 or x=-1
Q650) Solve 7x^2+28x+28=0
Ans: x=-2 or x=-2
Q651) Solve 13x^2+46x+21=0
Ans: x=-0.54 or x=-3
Q652) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q653) Solve 16x^2+33x+2=0
Ans: x=-0.06 or x=-2
Q654) Solve 11x^2+40x+21=0
Ans: x=-0.64 or x=-3
Q655) Solve 2x^2+15x+22=0
Ans: x=-2 or x=-5.5
Q656) Solve 22x^2+35x+3=0
Ans: x=-0.09 or x=-1.5
Q657) Solve 4x^2+27x+45=0
Ans: x=-3 or x=-3.75
Q658) Solve x^2+12x+36=0
Ans: x=-6 or x=-6
Q659) Solve 10x^2+31x+13=0
Ans: x=-0.5 or x=-2.6
Q660) Solve 14x^2+49x+21=0
Ans: x=-0.5 or x=-3
Q661) Solve 11x^2+16x+5=0
Ans: x=-0.45 or x=-1
Q662) Solve 3x^2+25x+48=0
Ans: x=-3 or x=-5.33
Q663) Solve 5x^2+28x+15=0
Ans: x=-0.6 or x=-5
Q664) Solve 6x^2+27x+30=0
Ans: x=-2 or x=-2.5
Q665) Solve 25x^2+40x+15=0
Ans: x=-0.6 or x=-1
Q666) Solve 25x^2+29x+4=0
Ans: x=-0.16 or x=-1
Q667) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q668) Solve 17x^2+42x+25=0
Ans: x=-1 or x=-1.47
Q669) Solve 14x^2+39x+16=0
Ans: x=-0.5 or x=-2.29
Q670) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q671) Solve 28x^2+23x+4=0
Ans: x=-0.25 or x=-0.57
Q672) Solve 11x^2+47x+50=0
Ans: x=-2 or x=-2.27
Q673) Solve 3x^2+31x+50=0
Ans: x=-2 or x=-8.33
Q674) Solve x^2+19x+34=0
Ans: x=-2 or x=-17
Q675) Solve 4x^2+40x+19=0
Ans: x=-0.5 or x=-9.5
Q676) Solve 15x^2+50x+40=0
Ans: x=-1.33 or x=-2
Q677) Solve 25x^2+50x+16=0
Ans: x=-0.4 or x=-1.6
Q678) Solve 32x^2+50x+3=0
Ans: x=-0.06 or x=-1.5
Q679) Solve 26x^2+47x+21=0
Ans: x=-0.81 or x=-1
Q680) Solve 14x^2+17x+5=0
Ans: x=-0.5 or x=-0.71
Q681) Solve 30x^2+46x+8=0
Ans: x=-0.2 or x=-1.33
Q682) Solve 16x^2+10x+1=0
Ans: x=-0.12 or x=-0.5
Q683) Solve 20x^2+23x+3=0
Ans: x=-0.15 or x=-1
Q684) Solve 37x^2+46x+9=0
Ans: x=-0.24 or x=-1
Q685) Solve 15x^2+38x+23=0
Ans: x=-1 or x=-1.53
Q686) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q687) Solve 25x^2+42x+17=0
Ans: x=-0.68 or x=-1
Q688) Solve 7x^2+8x+1=0
Ans: x=-0.14 or x=-1
Q689) Solve 42x^2+50x+8=0
Ans: x=-0.19 or x=-1
Q690) Solve 40x^2+44x+12=0
Ans: x=-0.5 or x=-0.6
Q691) Solve 3x^2+38x+35=0
Ans: x=-1 or x=-11.67
Q692) Solve 23x^2+45x+22=0
Ans: x=-0.96 or x=-1
Q693) Solve 10x^2+37x+34=0
Ans: x=-1.7 or x=-2
Q694) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q695) Solve 10x^2+40x+30=0
Ans: x=-1 or x=-3
Q696) Solve 19x^2+29x+10=0
Ans: x=-0.53 or x=-1
Q697) Solve 4x^2+11x+7=0
Ans: x=-1 or x=-1.75
Q698) Solve 7x^2+34x+24=0
Ans: x=-0.86 or x=-4
Q699) Solve 2x^2+49x+47=0
Ans: x=-1 or x=-23.5
Q700) Solve 33x^2+46x+16=0
Ans: x=-0.67 or x=-0.73
Q701) Solve 18x^2+39x+20=0
Ans: x=-0.83 or x=-1.33
Q702) Solve 2x^2+13x+21=0
Ans: x=-3 or x=-3.5
Q703) Solve x^2+14x+13=0
Ans: x=-1 or x=-13
Q704) Solve 8x^2+46x+11=0
Ans: x=-0.25 or x=-5.5
Q705) Solve 45x^2+48x+3=0
Ans: x=-0.07 or x=-1
Q706) Solve 42x^2+38x+8=0
Ans: x=-0.33 or x=-0.57
Q707) Solve 28x^2+22x+4=0
Ans: x=-0.29 or x=-0.5
Q708) Solve 48x^2+35x+2=0
Ans: x=-0.06 or x=-0.67
Q709) Solve 2x^2+14x+24=0
Ans: x=-3 or x=-4
Q710) Solve 2x^2+34x+32=0
Ans: x=-1 or x=-16
Q711) Solve 24x^2+49x+22=0
Ans: x=-0.67 or x=-1.37
Q712) Solve 14x^2+50x+44=0
Ans: x=-1.57 or x=-2
Q713) Solve 6x^2+23x+21=0
Ans: x=-1.5 or x=-2.33
Q714) Solve 14x^2+46x+36=0
Ans: x=-1.29 or x=-2
Q715) Solve 15x^2+37x+18=0
Ans: x=-0.67 or x=-1.8
Q716) Solve 27x^2+49x+22=0
Ans: x=-0.81 or x=-1
Q717) Solve 2x^2+16x+32=0
Ans: x=-4 or x=-4
Q718) Solve 3x^2+43x+14=0
Ans: x=-0.33 or x=-14
Q719) Solve 36x^2+44x+13=0
Ans: x=-0.5 or x=-0.72
Q720) Solve 9x^2+33x+30=0
Ans: x=-1.67 or x=-2
Q721) Solve 17x^2+42x+16=0
Ans: x=-0.47 or x=-2
Q722) Solve 17x^2+19x+2=0
Ans: x=-0.12 or x=-1
Q723) Solve 11x^2+39x+18=0
Ans: x=-0.55 or x=-3
Q724) Solve 20x^2+41x+20=0
Ans: x=-0.8 or x=-1.25
Q725) Solve 10x^2+31x+22=0
Ans: x=-1.1 or x=-2
Q726) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q727) Solve 36x^2+27x+5=0
Ans: x=-0.33 or x=-0.42
Q728) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q729) Solve 25x^2+48x+23=0
Ans: x=-0.92 or x=-1
Q730) Solve 33x^2+46x+13=0
Ans: x=-0.39 or x=-1
Q731) Solve 4x^2+22x+18=0
Ans: x=-1 or x=-4.5
Q732) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q733) Solve 9x^2+37x+4=0
Ans: x=-0.11 or x=-4
Q734) Solve 49x^2+21x+2=0
Ans: x=-0.14 or x=-0.29
Q735) Solve 40x^2+38x+9=0
Ans: x=-0.45 or x=-0.5
Q736) Solve 21x^2+17x+2=0
Ans: x=-0.14 or x=-0.67
Q737) Solve 4x^2+36x+32=0
Ans: x=-1 or x=-8
Q738) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q739) Solve 2x^2+49x+24=0
Ans: x=-0.5 or x=-24
Q740) Solve 25x^2+40x+7=0
Ans: x=-0.2 or x=-1.4
Q741) Solve 20x^2+47x+14=0
Ans: x=-0.35 or x=-2
Q742) Solve 14x^2+20x+6=0
Ans: x=-0.43 or x=-1
Q743) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q744) Solve 14x^2+17x+3=0
Ans: x=-0.21 or x=-1
Q745) Solve 32x^2+44x+15=0
Ans: x=-0.62 or x=-0.75
Q746) Solve 6x^2+14x+4=0
Ans: x=-0.33 or x=-2
Q747) Solve 9x^2+27x+14=0
Ans: x=-0.67 or x=-2.33
Q748) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q749) Solve 34x^2+29x+6=0
Ans: x=-0.35 or x=-0.5
Q750) Solve 6x^2+31x+5=0
Ans: x=-0.17 or x=-5
Q751) Solve 7x^2+50x+7=0
Ans: x=-0.14 or x=-7
Q752) Solve 20x^2+48x+27=0
Ans: x=-0.9 or x=-1.5
Q753) Solve 32x^2+22x+3=0
Ans: x=-0.19 or x=-0.5
Q754) Solve 20x^2+9x+1=0
Ans: x=-0.2 or x=-0.25
Q755) Solve 7x^2+25x+18=0
Ans: x=-1 or x=-2.57
Q756) Solve 12x^2+14x+4=0
Ans: x=-0.5 or x=-0.67
Q757) Solve 3x^2+15x+12=0
Ans: x=-1 or x=-4
Q758) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q759) Solve 2x^2+25x+50=0
Ans: x=-2.5 or x=-10
Q760) Solve 21x^2+38x+16=0
Ans: x=-0.67 or x=-1.14
Q761) Solve 12x^2+49x+30=0
Ans: x=-0.75 or x=-3.33
Q762) Solve 35x^2+48x+16=0
Ans: x=-0.57 or x=-0.8
Q763) Solve 15x^2+47x+32=0
Ans: x=-1 or x=-2.13
Q764) Solve 25x^2+40x+7=0
Ans: x=-0.2 or x=-1.4
Q765) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q766) Solve 16x^2+46x+30=0
Ans: x=-1 or x=-1.87
Q767) Solve 19x^2+30x+11=0
Ans: x=-0.58 or x=-1
Q768) Solve 28x^2+32x+9=0
Ans: x=-0.5 or x=-0.64
Q769) Solve 25x^2+42x+17=0
Ans: x=-0.68 or x=-1
Q770) Solve 5x^2+46x+48=0
Ans: x=-1.2 or x=-8
Q771) Solve 32x^2+39x+7=0
Ans: x=-0.22 or x=-1
Q772) Solve 22x^2+43x+15=0
Ans: x=-0.45 or x=-1.5
Q773) Solve 12x^2+32x+16=0
Ans: x=-0.67 or x=-2
Q774) Solve 30x^2+28x+6=0
Ans: x=-0.33 or x=-0.6
Q775) Solve 45x^2+30x+5=0
Ans: x=-0.33 or x=-0.33
Q776) Solve 9x^2+26x+16=0
Ans: x=-0.89 or x=-2
Q777) Solve x^2+16x+28=0
Ans: x=-2 or x=-14
Q778) Solve 21x^2+25x+4=0
Ans: x=-0.19 or x=-1
Q779) Solve 29x^2+38x+9=0
Ans: x=-0.31 or x=-1
Q780) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q781) Solve 3x^2+48x+45=0
Ans: x=-1 or x=-15
Q782) Solve 14x^2+39x+22=0
Ans: x=-0.79 or x=-2
Q783) Solve 2x^2+35x+33=0
Ans: x=-1 or x=-16.5
Q784) Solve 27x^2+33x+10=0
Ans: x=-0.56 or x=-0.67
Q785) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q786) Solve 9x^2+44x+35=0
Ans: x=-1 or x=-3.89
Q787) Solve 4x^2+16x+7=0
Ans: x=-0.5 or x=-3.5
Q788) Solve 25x^2+27x+2=0
Ans: x=-0.08 or x=-1
Q789) Solve 3x^2+17x+10=0
Ans: x=-0.67 or x=-5
Q790) Solve 27x^2+27x+6=0
Ans: x=-0.33 or x=-0.67
Q791) Solve 2x^2+49x+24=0
Ans: x=-0.5 or x=-24
Q792) Solve 10x^2+46x+24=0
Ans: x=-0.6 or x=-4
Q793) Solve 16x^2+26x+3=0
Ans: x=-0.12 or x=-1.5
Q794) Solve 40x^2+43x+3=0
Ans: x=-0.07 or x=-1
Q795) Solve 10x^2+37x+34=0
Ans: x=-1.7 or x=-2
Q796) Solve 8x^2+40x+48=0
Ans: x=-2 or x=-3
Q797) Solve 4x^2+8x+4=0
Ans: x=-1 or x=-1
Q798) Solve 13x^2+43x+30=0
Ans: x=-1 or x=-2.31
Q799) Solve 12x^2+26x+10=0
Ans: x=-0.5 or x=-1.67
Q800) Solve 13x^2+47x+24=0
Ans: x=-0.62 or x=-3
Q801) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q802) Solve 48x^2+40x+8=0
Ans: x=-0.33 or x=-0.5
Q803) Solve 10x^2+45x+35=0
Ans: x=-1 or x=-3.5
Q804) Solve 31x^2+48x+17=0
Ans: x=-0.55 or x=-1
Q805) Solve 2x^2+31x+42=0
Ans: x=-1.5 or x=-14
Q806) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q807) Solve 30x^2+22x+4=0
Ans: x=-0.33 or x=-0.4
Q808) Solve 20x^2+47x+27=0
Ans: x=-1 or x=-1.35
Q809) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q810) Solve 29x^2+42x+13=0
Ans: x=-0.45 or x=-1
Q811) Solve 14x^2+31x+6=0
Ans: x=-0.21 or x=-2
Q812) Solve 20x^2+47x+27=0
Ans: x=-1 or x=-1.35
Q813) Solve 20x^2+43x+14=0
Ans: x=-0.4 or x=-1.75
Q814) Solve 16x^2+27x+11=0
Ans: x=-0.69 or x=-1
Q815) Solve 32x^2+42x+13=0
Ans: x=-0.5 or x=-0.81
Q816) Solve 6x^2+32x+26=0
Ans: x=-1 or x=-4.33
Q817) Solve x^2+30x+29=0
Ans: x=-1 or x=-29
Q818) Solve 12x^2+44x+24=0
Ans: x=-0.67 or x=-3
Q819) Solve 16x^2+49x+3=0
Ans: x=-0.06 or x=-3
Q820) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q821) Solve 17x^2+36x+4=0
Ans: x=-0.12 or x=-2
Q822) Solve 18x^2+47x+22=0
Ans: x=-0.61 or x=-2
Q823) Solve 37x^2+49x+12=0
Ans: x=-0.32 or x=-1
Q824) Solve 12x^2+17x+5=0
Ans: x=-0.42 or x=-1
Q825) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q826) Solve 12x^2+44x+35=0
Ans: x=-1.17 or x=-2.5
Q827) Solve 4x^2+28x+40=0
Ans: x=-2 or x=-5
Q828) Solve 9x^2+21x+10=0
Ans: x=-0.67 or x=-1.67
Q829) Solve 5x^2+24x+28=0
Ans: x=-2 or x=-2.8
Q830) Solve 9x^2+40x+16=0
Ans: x=-0.44 or x=-4
Q831) Solve 24x^2+23x+5=0
Ans: x=-0.33 or x=-0.62
Q832) Solve 9x^2+38x+40=0
Ans: x=-2 or x=-2.22
Q833) Solve 6x^2+32x+40=0
Ans: x=-2 or x=-3.33
Q834) Solve 3x^2+47x+44=0
Ans: x=-1 or x=-14.67
Q835) Solve 18x^2+41x+21=0
Ans: x=-0.78 or x=-1.5
Q836) Solve 13x^2+26x+13=0
Ans: x=-1 or x=-1
Q837) Solve 4x^2+25x+39=0
Ans: x=-3 or x=-3.25
Q838) Solve 8x^2+14x+5=0
Ans: x=-0.5 or x=-1.25
Q839) Solve 12x^2+22x+8=0
Ans: x=-0.5 or x=-1.33
Q840) Solve 36x^2+48x+16=0
Ans: x=-0.67 or x=-0.67
Q841) Solve 8x^2+18x+9=0
Ans: x=-0.75 or x=-1.5
Q842) Solve 42x^2+34x+4=0
Ans: x=-0.14 or x=-0.67
Q843) Solve 15x^2+37x+20=0
Ans: x=-0.8 or x=-1.67
Q844) Solve 33x^2+35x+2=0
Ans: x=-0.06 or x=-1
Q845) Solve 2x^2+23x+21=0
Ans: x=-1 or x=-10.5
Q846) Solve 13x^2+40x+27=0
Ans: x=-1 or x=-2.08
Q847) Solve 11x^2+29x+18=0
Ans: x=-1 or x=-1.64
Q848) Solve 3x^2+29x+46=0
Ans: x=-2 or x=-7.67
Q849) Solve 18x^2+33x+14=0
Ans: x=-0.67 or x=-1.17
Q850) Solve 18x^2+15x+2=0
Ans: x=-0.17 or x=-0.67
Q851) Solve 13x^2+45x+18=0
Ans: x=-0.46 or x=-3
Q852) Solve 31x^2+48x+17=0
Ans: x=-0.55 or x=-1
Q853) Solve 40x^2+46x+13=0
Ans: x=-0.5 or x=-0.65
Q854) Solve 40x^2+47x+12=0
Ans: x=-0.37 or x=-0.8
Q855) Solve 4x^2+37x+40=0
Ans: x=-1.25 or x=-8
Q856) Solve 25x^2+40x+12=0
Ans: x=-0.4 or x=-1.2
Q857) Solve 48x^2+40x+8=0
Ans: x=-0.33 or x=-0.5
Q858) Solve 14x^2+29x+11=0
Ans: x=-0.5 or x=-1.57
Q859) Solve 20x^2+45x+25=0
Ans: x=-1 or x=-1.25
Q860) Solve 7x^2+36x+44=0
Ans: x=-2 or x=-3.14
Q861) Solve 4x^2+36x+17=0
Ans: x=-0.5 or x=-8.5
Q862) Solve 48x^2+20x+2=0
Ans: x=-0.17 or x=-0.25
Q863) Solve 11x^2+41x+24=0
Ans: x=-0.73 or x=-3
Q864) Solve 6x^2+25x+4=0
Ans: x=-0.17 or x=-4
Q865) Solve 5x^2+31x+26=0
Ans: x=-1 or x=-5.2
Q866) Solve 15x^2+41x+12=0
Ans: x=-0.33 or x=-2.4
Q867) Solve 35x^2+47x+12=0
Ans: x=-0.34 or x=-1
Q868) Solve 4x^2+23x+28=0
Ans: x=-1.75 or x=-4
Q869) Solve 5x^2+22x+24=0
Ans: x=-2 or x=-2.4
Q870) Solve 6x^2+35x+16=0
Ans: x=-0.5 or x=-5.33
Q871) Solve 9x^2+45x+26=0
Ans: x=-0.67 or x=-4.33
Q872) Solve 30x^2+41x+13=0
Ans: x=-0.5 or x=-0.87
Q873) Solve 4x^2+29x+25=0
Ans: x=-1 or x=-6.25
Q874) Solve 4x^2+21x+26=0
Ans: x=-2 or x=-3.25
Q875) Solve x^2+18x+17=0
Ans: x=-1 or x=-17
Q876) Solve 10x^2+49x+39=0
Ans: x=-1 or x=-3.9
Q877) Solve 9x^2+43x+34=0
Ans: x=-1 or x=-3.78
Q878) Solve 10x^2+43x+19=0
Ans: x=-0.5 or x=-3.8
Q879) Solve 5x^2+33x+46=0
Ans: x=-2 or x=-4.6
Q880) Solve 6x^2+46x+28=0
Ans: x=-0.67 or x=-7
Q881) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q882) Solve x^2+11x+30=0
Ans: x=-5 or x=-6
Q883) Solve 2x^2+47x+45=0
Ans: x=-1 or x=-22.5
Q884) Solve 3x^2+23x+14=0
Ans: x=-0.67 or x=-7
Q885) Solve 30x^2+43x+4=0
Ans: x=-0.1 or x=-1.33
Q886) Solve x^2+18x+17=0
Ans: x=-1 or x=-17
Q887) Solve 11x^2+32x+20=0
Ans: x=-0.91 or x=-2
Q888) Solve 27x^2+34x+7=0
Ans: x=-0.26 or x=-1
Q889) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q890) Solve 11x^2+27x+10=0
Ans: x=-0.45 or x=-2
Q891) Solve 14x^2+38x+20=0
Ans: x=-0.71 or x=-2
Q892) Solve 31x^2+48x+17=0
Ans: x=-0.55 or x=-1
Q893) Solve 7x^2+12x+5=0
Ans: x=-0.71 or x=-1
Q894) Solve 39x^2+49x+10=0
Ans: x=-0.26 or x=-1
Q895) Solve 17x^2+42x+25=0
Ans: x=-1 or x=-1.47
Q896) Solve 11x^2+40x+36=0
Ans: x=-1.64 or x=-2
Q897) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q898) Solve 30x^2+41x+7=0
Ans: x=-0.2 or x=-1.17
Q899) Solve 5x^2+11x+2=0
Ans: x=-0.2 or x=-2
Q900) Solve 38x^2+37x+9=0
Ans: x=-0.47 or x=-0.5
Q901) Solve 14x^2+40x+24=0
Ans: x=-0.86 or x=-2
Q902) Solve 15x^2+37x+20=0
Ans: x=-0.8 or x=-1.67
Q903) Solve 40x^2+28x+4=0
Ans: x=-0.2 or x=-0.5
Q904) Solve 34x^2+46x+12=0
Ans: x=-0.35 or x=-1
Q905) Solve x^2+22x+40=0
Ans: x=-2 or x=-20
Q906) Solve 4x^2+22x+24=0
Ans: x=-1.5 or x=-4
Q907) Solve 3x^2+22x+39=0
Ans: x=-3 or x=-4.33
Q908) Solve x^2+13x+12=0
Ans: x=-1 or x=-12
Q909) Solve 44x^2+23x+3=0
Ans: x=-0.25 or x=-0.27
Q910) Solve 18x^2+43x+14=0
Ans: x=-0.39 or x=-2
Q911) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q912) Solve 9x^2+42x+45=0
Ans: x=-1.67 or x=-3
Q913) Solve 26x^2+15x+1=0
Ans: x=-0.08 or x=-0.5
Q914) Solve x^2+17x+42=0
Ans: x=-3 or x=-14
Q915) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q916) Solve 40x^2+47x+7=0
Ans: x=-0.17 or x=-1
Q917) Solve 21x^2+44x+15=0
Ans: x=-0.43 or x=-1.67
Q918) Solve 32x^2+48x+10=0
Ans: x=-0.25 or x=-1.25
Q919) Solve 48x^2+50x+7=0
Ans: x=-0.17 or x=-0.87
Q920) Solve 40x^2+43x+3=0
Ans: x=-0.07 or x=-1
Q921) Solve 22x^2+37x+6=0
Ans: x=-0.18 or x=-1.5
Q922) Solve 46x^2+43x+10=0
Ans: x=-0.43 or x=-0.5
Q923) Solve 4x^2+20x+16=0
Ans: x=-1 or x=-4
Q924) Solve 7x^2+31x+24=0
Ans: x=-1 or x=-3.43
Q925) Solve 20x^2+37x+15=0
Ans: x=-0.6 or x=-1.25
Q926) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q927) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q928) Solve 9x^2+29x+20=0
Ans: x=-1 or x=-2.22
Q929) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q930) Solve 46x^2+47x+1=0
Ans: x=-0.02 or x=-1
Q931) Solve 7x^2+28x+21=0
Ans: x=-1 or x=-3
Q932) Solve 6x^2+19x+14=0
Ans: x=-1.17 or x=-2
Q933) Solve 6x^2+28x+30=0
Ans: x=-1.67 or x=-3
Q934) Solve 21x^2+42x+21=0
Ans: x=-1 or x=-1
Q935) Solve 20x^2+44x+17=0
Ans: x=-0.5 or x=-1.7
Q936) Solve 28x^2+45x+17=0
Ans: x=-0.61 or x=-1
Q937) Solve 14x^2+39x+16=0
Ans: x=-0.5 or x=-2.29
Q938) Solve 42x^2+37x+6=0
Ans: x=-0.21 or x=-0.67
Q939) Solve 20x^2+23x+6=0
Ans: x=-0.4 or x=-0.75
Q940) Solve 41x^2+48x+7=0
Ans: x=-0.17 or x=-1
Q941) Solve 33x^2+23x+4=0
Ans: x=-0.33 or x=-0.36
Q942) Solve 32x^2+36x+9=0
Ans: x=-0.37 or x=-0.75
Q943) Solve 48x^2+49x+11=0
Ans: x=-0.33 or x=-0.69
Q944) Solve 13x^2+24x+11=0
Ans: x=-0.85 or x=-1
Q945) Solve 39x^2+50x+16=0
Ans: x=-0.62 or x=-0.67
Q946) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q947) Solve 8x^2+48x+40=0
Ans: x=-1 or x=-5
Q948) Solve 16x^2+42x+20=0
Ans: x=-0.62 or x=-2
Q949) Solve 4x^2+27x+44=0
Ans: x=-2.75 or x=-4
Q950) Solve 12x^2+12x+3=0
Ans: x=-0.5 or x=-0.5
Q951) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q952) Solve 40x^2+32x+6=0
Ans: x=-0.3 or x=-0.5
Q953) Solve 15x^2+43x+28=0
Ans: x=-1 or x=-1.87
Q954) Solve 14x^2+36x+22=0
Ans: x=-1 or x=-1.57
Q955) Solve 39x^2+40x+1=0
Ans: x=-0.03 or x=-1
Q956) Solve 14x^2+13x+3=0
Ans: x=-0.43 or x=-0.5
Q957) Solve 19x^2+46x+16=0
Ans: x=-0.42 or x=-2
Q958) Solve 20x^2+29x+9=0
Ans: x=-0.45 or x=-1
Q959) Solve 40x^2+26x+4=0
Ans: x=-0.25 or x=-0.4
Q960) Solve 7x^2+40x+33=0
Ans: x=-1 or x=-4.71
Q961) Solve 24x^2+38x+3=0
Ans: x=-0.08 or x=-1.5
Q962) Solve 8x^2+46x+45=0
Ans: x=-1.25 or x=-4.5
Q963) Solve 6x^2+11x+5=0
Ans: x=-0.83 or x=-1
Q964) Solve 13x^2+19x+6=0
Ans: x=-0.46 or x=-1
Q965) Solve 24x^2+42x+15=0
Ans: x=-0.5 or x=-1.25
Q966) Solve 18x^2+49x+20=0
Ans: x=-0.5 or x=-2.22
Q967) Solve 7x^2+33x+26=0
Ans: x=-1 or x=-3.71
Q968) Solve 35x^2+49x+14=0
Ans: x=-0.4 or x=-1
Q969) Solve 5x^2+26x+5=0
Ans: x=-0.2 or x=-5
Q970) Solve 48x^2+44x+6=0
Ans: x=-0.17 or x=-0.75
Q971) Solve 49x^2+42x+5=0
Ans: x=-0.14 or x=-0.71
Q972) Solve 44x^2+42x+10=0
Ans: x=-0.45 or x=-0.5
Q973) Solve 17x^2+18x+1=0
Ans: x=-0.06 or x=-1
Q974) Solve 4x^2+18x+14=0
Ans: x=-1 or x=-3.5
Q975) Solve 5x^2+12x+7=0
Ans: x=-1 or x=-1.4
Q976) Solve 14x^2+41x+26=0
Ans: x=-0.93 or x=-2
Q977) Solve 7x^2+39x+44=0
Ans: x=-1.57 or x=-4
Q978) Solve 35x^2+47x+6=0
Ans: x=-0.14 or x=-1.2
Q979) Solve 13x^2+45x+18=0
Ans: x=-0.46 or x=-3
Q980) Solve 27x^2+48x+16=0
Ans: x=-0.44 or x=-1.33
Q981) Solve x^2+11x+28=0
Ans: x=-4 or x=-7
Q982) Solve 10x^2+32x+24=0
Ans: x=-1.2 or x=-2
Q983) Solve 11x^2+33x+22=0
Ans: x=-1 or x=-2
Q984) Solve 36x^2+45x+11=0
Ans: x=-0.33 or x=-0.92
Q985) Solve 4x^2+21x+20=0
Ans: x=-1.25 or x=-4
Q986) Solve 48x^2+44x+10=0
Ans: x=-0.42 or x=-0.5
Q987) Solve 4x^2+9x+5=0
Ans: x=-1 or x=-1.25
Q988) Solve 7x^2+27x+20=0
Ans: x=-1 or x=-2.86
Q989) Solve 18x^2+33x+9=0
Ans: x=-0.33 or x=-1.5
Q990) Solve x^2+27x+50=0
Ans: x=-2 or x=-25
Q991) Solve 4x^2+24x+32=0
Ans: x=-2 or x=-4
Q992) Solve 2x^2+18x+16=0
Ans: x=-1 or x=-8
Q993) Solve 48x^2+14x+1=0
Ans: x=-0.12 or x=-0.17
Q994) Solve 10x^2+42x+8=0
Ans: x=-0.2 or x=-4
Q995) Solve 30x^2+25x+5=0
Ans: x=-0.33 or x=-0.5
Q996) Solve 26x^2+47x+17=0
Ans: x=-0.5 or x=-1.31
Q997) Solve 26x^2+46x+20=0
Ans: x=-0.77 or x=-1
Q998) Solve x^2+8x+12=0
Ans: x=-2 or x=-6
Q999) Solve 5x^2+34x+24=0
Ans: x=-0.8 or x=-6
Q1000) Solve 9x^2+45x+26=0
Ans: x=-0.67 or x=-4.33
Q1001) Solve 4x^2+29x+42=0
Ans: x=-2 or x=-5.25
Q1002) Solve 3x^2+21x+30=0
Ans: x=-2 or x=-5
Q1003) Solve 32x^2+42x+13=0
Ans: x=-0.5 or x=-0.81
Q1004) Solve 2x^2+9x+9=0
Ans: x=-1.5 or x=-3
Q1005) Solve 22x^2+25x+3=0
Ans: x=-0.14 or x=-1
Q1006) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q1007) Solve 3x^2+16x+13=0
Ans: x=-1 or x=-4.33
Q1008) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q1009) Solve 8x^2+22x+9=0
Ans: x=-0.5 or x=-2.25
Q1010) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q1011) Solve 32x^2+46x+15=0
Ans: x=-0.5 or x=-0.94
Q1012) Solve 29x^2+33x+4=0
Ans: x=-0.14 or x=-1
Q1013) Solve 4x^2+29x+7=0
Ans: x=-0.25 or x=-7
Q1014) Solve 9x^2+31x+22=0
Ans: x=-1 or x=-2.44
Q1015) Solve 12x^2+44x+40=0
Ans: x=-1.67 or x=-2
Q1016) Solve 12x^2+22x+8=0
Ans: x=-0.5 or x=-1.33
Q1017) Solve 2x^2+39x+19=0
Ans: x=-0.5 or x=-19
Q1018) Solve 44x^2+49x+5=0
Ans: x=-0.11 or x=-1
Q1019) Solve 19x^2+33x+14=0
Ans: x=-0.74 or x=-1
Q1020) Solve 26x^2+27x+1=0
Ans: x=-0.04 or x=-1
Q1021) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q1022) Solve 4x^2+18x+20=0
Ans: x=-2 or x=-2.5
Q1023) Solve 10x^2+48x+32=0
Ans: x=-0.8 or x=-4
Q1024) Solve 33x^2+37x+10=0
Ans: x=-0.45 or x=-0.67
Q1025) Solve 14x^2+43x+29=0
Ans: x=-1 or x=-2.07
Q1026) Solve 20x^2+25x+5=0
Ans: x=-0.25 or x=-1
Q1027) Solve 3x^2+13x+4=0
Ans: x=-0.33 or x=-4
Q1028) Solve 11x^2+35x+24=0
Ans: x=-1 or x=-2.18
Q1029) Solve 6x^2+35x+11=0
Ans: x=-0.33 or x=-5.5
Q1030) Solve 30x^2+41x+13=0
Ans: x=-0.5 or x=-0.87
Q1031) Solve 20x^2+49x+11=0
Ans: x=-0.25 or x=-2.2
Q1032) Solve 9x^2+38x+29=0
Ans: x=-1 or x=-3.22
Q1033) Solve 25x^2+31x+6=0
Ans: x=-0.24 or x=-1
Q1034) Solve x^2+19x+34=0
Ans: x=-2 or x=-17
Q1035) Solve 15x^2+41x+28=0
Ans: x=-1.33 or x=-1.4
Q1036) Solve 13x^2+27x+2=0
Ans: x=-0.08 or x=-2
Q1037) Solve 15x^2+38x+24=0
Ans: x=-1.2 or x=-1.33
Q1038) Solve 17x^2+22x+5=0
Ans: x=-0.29 or x=-1
Q1039) Solve 4x^2+28x+49=0
Ans: x=-3.5 or x=-3.5
Q1040) Solve 17x^2+43x+18=0
Ans: x=-0.53 or x=-2
Q1041) Solve 16x^2+39x+14=0
Ans: x=-0.44 or x=-2
Q1042) Solve 22x^2+33x+11=0
Ans: x=-0.5 or x=-1
Q1043) Solve 6x^2+29x+34=0
Ans: x=-2 or x=-2.83
Q1044) Solve 28x^2+33x+5=0
Ans: x=-0.18 or x=-1
Q1045) Solve 37x^2+49x+12=0
Ans: x=-0.32 or x=-1
Q1046) Solve 21x^2+25x+4=0
Ans: x=-0.19 or x=-1
Q1047) Solve 42x^2+49x+14=0
Ans: x=-0.5 or x=-0.67
Q1048) Solve 4x^2+30x+50=0
Ans: x=-2.5 or x=-5
Q1049) Solve 39x^2+47x+8=0
Ans: x=-0.21 or x=-1
Q1050) Solve 32x^2+16x+2=0
Ans: x=-0.25 or x=-0.25
Q1051) Solve 3x^2+14x+11=0
Ans: x=-1 or x=-3.67
Q1052) Solve 8x^2+37x+29=0
Ans: x=-1 or x=-3.62
Q1053) Solve 32x^2+22x+3=0
Ans: x=-0.19 or x=-0.5
Q1054) Solve 12x^2+15x+3=0
Ans: x=-0.25 or x=-1
Q1055) Solve 3x^2+32x+20=0
Ans: x=-0.67 or x=-10
Q1056) Solve 9x^2+43x+50=0
Ans: x=-2 or x=-2.78
Q1057) Solve 8x^2+9x+1=0
Ans: x=-0.12 or x=-1
Q1058) Solve 6x^2+16x+8=0
Ans: x=-0.67 or x=-2
Q1059) Solve 28x^2+38x+10=0
Ans: x=-0.36 or x=-1
Q1060) Solve 12x^2+33x+18=0
Ans: x=-0.75 or x=-2
Q1061) Solve 21x^2+29x+10=0
Ans: x=-0.67 or x=-0.71
Q1062) Solve 8x^2+9x+1=0
Ans: x=-0.12 or x=-1
Q1063) Solve 6x^2+37x+45=0
Ans: x=-1.67 or x=-4.5
Q1064) Solve 20x^2+21x+1=0
Ans: x=-0.05 or x=-1
Q1065) Solve 19x^2+26x+7=0
Ans: x=-0.37 or x=-1
Q1066) Solve 35x^2+24x+4=0
Ans: x=-0.29 or x=-0.4
Q1067) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q1068) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q1069) Solve x^2+9x+8=0
Ans: x=-1 or x=-8
Q1070) Solve 20x^2+39x+19=0
Ans: x=-0.95 or x=-1
Q1071) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q1072) Solve 35x^2+29x+6=0
Ans: x=-0.4 or x=-0.43
Q1073) Solve 27x^2+45x+12=0
Ans: x=-0.33 or x=-1.33
Q1074) Solve 39x^2+41x+2=0
Ans: x=-0.05 or x=-1
Q1075) Solve 17x^2+39x+22=0
Ans: x=-1 or x=-1.29
Q1076) Solve 28x^2+29x+1=0
Ans: x=-0.04 or x=-1
Q1077) Solve 13x^2+26x+13=0
Ans: x=-1 or x=-1
Q1078) Solve 6x^2+25x+11=0
Ans: x=-0.5 or x=-3.67
Q1079) Solve 3x^2+22x+19=0
Ans: x=-1 or x=-6.33
Q1080) Solve 24x^2+40x+6=0
Ans: x=-0.17 or x=-1.5
Q1081) Solve 32x^2+36x+10=0
Ans: x=-0.5 or x=-0.62
Q1082) Solve 12x^2+47x+26=0
Ans: x=-0.67 or x=-3.25
Q1083) Solve 18x^2+19x+1=0
Ans: x=-0.06 or x=-1
Q1084) Solve 3x^2+16x+13=0
Ans: x=-1 or x=-4.33
Q1085) Solve 3x^2+14x+15=0
Ans: x=-1.67 or x=-3
Q1086) Solve 2x^2+11x+12=0
Ans: x=-1.5 or x=-4
Q1087) Solve 11x^2+14x+3=0
Ans: x=-0.27 or x=-1
Q1088) Solve 10x^2+31x+15=0
Ans: x=-0.6 or x=-2.5
Q1089) Solve 15x^2+43x+26=0
Ans: x=-0.87 or x=-2
Q1090) Solve 16x^2+23x+7=0
Ans: x=-0.44 or x=-1
Q1091) Solve 20x^2+17x+3=0
Ans: x=-0.25 or x=-0.6
Q1092) Solve 3x^2+26x+16=0
Ans: x=-0.67 or x=-8
Q1093) Solve 10x^2+45x+50=0
Ans: x=-2 or x=-2.5
Q1094) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q1095) Solve 30x^2+47x+16=0
Ans: x=-0.5 or x=-1.07
Q1096) Solve 9x^2+28x+20=0
Ans: x=-1.11 or x=-2
Q1097) Solve 12x^2+37x+21=0
Ans: x=-0.75 or x=-2.33
Q1098) Solve 45x^2+46x+1=0
Ans: x=-0.02 or x=-1
Q1099) Solve 21x^2+24x+3=0
Ans: x=-0.14 or x=-1
Q1100) Solve 15x^2+41x+12=0
Ans: x=-0.33 or x=-2.4
Q1101) Solve 5x^2+29x+24=0
Ans: x=-1 or x=-4.8
Q1102) Solve 13x^2+48x+44=0
Ans: x=-1.69 or x=-2
Q1103) Solve 2x^2+12x+18=0
Ans: x=-3 or x=-3
Q1104) Solve 7x^2+30x+23=0
Ans: x=-1 or x=-3.29
Q1105) Solve 39x^2+37x+8=0
Ans: x=-0.33 or x=-0.62
Q1106) Solve 26x^2+31x+5=0
Ans: x=-0.19 or x=-1
Q1107) Solve 7x^2+12x+5=0
Ans: x=-0.71 or x=-1
Q1108) Solve 21x^2+26x+5=0
Ans: x=-0.24 or x=-1
Q1109) Solve 40x^2+31x+6=0
Ans: x=-0.37 or x=-0.4
Q1110) Solve 12x^2+48x+45=0
Ans: x=-1.5 or x=-2.5
Q1111) Solve 30x^2+34x+4=0
Ans: x=-0.13 or x=-1
Q1112) Solve 36x^2+48x+12=0
Ans: x=-0.33 or x=-1
Q1113) Solve 30x^2+17x+2=0
Ans: x=-0.17 or x=-0.4
Q1114) Solve 2x^2+19x+44=0
Ans: x=-4 or x=-5.5
Q1115) Solve 16x^2+40x+21=0
Ans: x=-0.75 or x=-1.75
Q1116) Solve 9x^2+45x+14=0
Ans: x=-0.33 or x=-4.67
Q1117) Solve 4x^2+28x+49=0
Ans: x=-3.5 or x=-3.5
Q1118) Solve 6x^2+23x+20=0
Ans: x=-1.33 or x=-2.5
Q1119) Solve 8x^2+26x+11=0
Ans: x=-0.5 or x=-2.75
Q1120) Solve x^2+45x+44=0
Ans: x=-1 or x=-44
Q1121) Solve 9x^2+32x+28=0
Ans: x=-1.56 or x=-2
Q1122) Solve 24x^2+18x+3=0
Ans: x=-0.25 or x=-0.5
Q1123) Solve 5x^2+43x+24=0
Ans: x=-0.6 or x=-8
Q1124) Solve 33x^2+38x+9=0
Ans: x=-0.33 or x=-0.82
Q1125) Solve 5x^2+25x+30=0
Ans: x=-2 or x=-3
Q1126) Solve 2x^2+19x+35=0
Ans: x=-2.5 or x=-7
Q1127) Solve 24x^2+42x+9=0
Ans: x=-0.25 or x=-1.5
Q1128) Solve 3x^2+25x+42=0
Ans: x=-2.33 or x=-6
Q1129) Solve 42x^2+31x+4=0
Ans: x=-0.17 or x=-0.57
Q1130) Solve 15x^2+19x+4=0
Ans: x=-0.27 or x=-1
Q1131) Solve 12x^2+36x+24=0
Ans: x=-1 or x=-2
Q1132) Solve 28x^2+47x+19=0
Ans: x=-0.68 or x=-1
Q1133) Solve 36x^2+44x+13=0
Ans: x=-0.5 or x=-0.72
Q1134) Solve 5x^2+50x+45=0
Ans: x=-1 or x=-9
Q1135) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q1136) Solve 10x^2+22x+4=0
Ans: x=-0.2 or x=-2
Q1137) Solve 9x^2+45x+44=0
Ans: x=-1.33 or x=-3.67
Q1138) Solve 12x^2+37x+3=0
Ans: x=-0.08 or x=-3
Q1139) Solve 24x^2+18x+3=0
Ans: x=-0.25 or x=-0.5
Q1140) Solve 45x^2+39x+8=0
Ans: x=-0.33 or x=-0.53
Q1141) Solve 7x^2+37x+48=0
Ans: x=-2.29 or x=-3
Q1142) Solve 6x^2+29x+9=0
Ans: x=-0.33 or x=-4.5
Q1143) Solve 49x^2+42x+9=0
Ans: x=-0.43 or x=-0.43
Q1144) Solve 8x^2+43x+15=0
Ans: x=-0.37 or x=-5
Q1145) Solve 34x^2+38x+4=0
Ans: x=-0.12 or x=-1
Q1146) Solve 3x^2+40x+13=0
Ans: x=-0.33 or x=-13
Q1147) Solve 9x^2+39x+40=0
Ans: x=-1.67 or x=-2.67
Q1148) Solve 26x^2+35x+9=0
Ans: x=-0.35 or x=-1
Q1149) Solve 14x^2+31x+6=0
Ans: x=-0.21 or x=-2
Q1150) Solve 16x^2+10x+1=0
Ans: x=-0.12 or x=-0.5
Q1151) Solve 48x^2+40x+7=0
Ans: x=-0.25 or x=-0.58
Q1152) Solve 12x^2+24x+9=0
Ans: x=-0.5 or x=-1.5
Q1153) Solve 7x^2+36x+32=0
Ans: x=-1.14 or x=-4
Q1154) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q1155) Solve 20x^2+34x+6=0
Ans: x=-0.2 or x=-1.5
Q1156) Solve 4x^2+24x+32=0
Ans: x=-2 or x=-4
Q1157) Solve 26x^2+43x+15=0
Ans: x=-0.5 or x=-1.15
Q1158) Solve 5x^2+20x+20=0
Ans: x=-2 or x=-2
Q1159) Solve 28x^2+50x+12=0
Ans: x=-0.29 or x=-1.5
Q1160) Solve 18x^2+20x+2=0
Ans: x=-0.11 or x=-1
Q1161) Solve 42x^2+13x+1=0
Ans: x=-0.14 or x=-0.17
Q1162) Solve 3x^2+25x+50=0
Ans: x=-3.33 or x=-5
Q1163) Solve 42x^2+32x+6=0
Ans: x=-0.33 or x=-0.43
Q1164) Solve 32x^2+50x+17=0
Ans: x=-0.5 or x=-1.06
Q1165) Solve 22x^2+35x+13=0
Ans: x=-0.59 or x=-1
Q1166) Solve 21x^2+16x+3=0
Ans: x=-0.33 or x=-0.43
Q1167) Solve 28x^2+50x+12=0
Ans: x=-0.29 or x=-1.5
Q1168) Solve 18x^2+45x+18=0
Ans: x=-0.5 or x=-2
Q1169) Solve 5x^2+32x+12=0
Ans: x=-0.4 or x=-6
Q1170) Solve 47x^2+49x+2=0
Ans: x=-0.04 or x=-1
Q1171) Solve 44x^2+15x+1=0
Ans: x=-0.09 or x=-0.25
Q1172) Solve 33x^2+23x+4=0
Ans: x=-0.33 or x=-0.36
Q1173) Solve 10x^2+37x+21=0
Ans: x=-0.7 or x=-3
Q1174) Solve 22x^2+30x+8=0
Ans: x=-0.36 or x=-1
Q1175) Solve 8x^2+29x+26=0
Ans: x=-1.62 or x=-2
Q1176) Solve 30x^2+46x+12=0
Ans: x=-0.33 or x=-1.2
Q1177) Solve 26x^2+40x+14=0
Ans: x=-0.54 or x=-1
Q1178) Solve 28x^2+49x+21=0
Ans: x=-0.75 or x=-1
Q1179) Solve 4x^2+31x+21=0
Ans: x=-0.75 or x=-7
Q1180) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q1181) Solve 3x^2+13x+4=0
Ans: x=-0.33 or x=-4
Q1182) Solve 12x^2+48x+36=0
Ans: x=-1 or x=-3
Q1183) Solve 10x^2+39x+38=0
Ans: x=-1.9 or x=-2
Q1184) Solve 22x^2+35x+3=0
Ans: x=-0.09 or x=-1.5
Q1185) Solve 10x^2+43x+12=0
Ans: x=-0.3 or x=-4
Q1186) Solve 4x^2+19x+15=0
Ans: x=-1 or x=-3.75
Q1187) Solve 24x^2+10x+1=0
Ans: x=-0.17 or x=-0.25
Q1188) Solve 18x^2+33x+5=0
Ans: x=-0.17 or x=-1.67
Q1189) Solve 20x^2+43x+6=0
Ans: x=-0.15 or x=-2
Q1190) Solve 11x^2+39x+34=0
Ans: x=-1.55 or x=-2
Q1191) Solve 17x^2+35x+18=0
Ans: x=-1 or x=-1.06
Q1192) Solve 10x^2+39x+27=0
Ans: x=-0.9 or x=-3
Q1193) Solve 8x^2+18x+10=0
Ans: x=-1 or x=-1.25
Q1194) Solve 46x^2+37x+7=0
Ans: x=-0.3 or x=-0.5
Q1195) Solve 14x^2+38x+24=0
Ans: x=-1 or x=-1.71
Q1196) Solve 42x^2+31x+5=0
Ans: x=-0.24 or x=-0.5
Q1197) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q1198) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q1199) Solve 2x^2+14x+20=0
Ans: x=-2 or x=-5
Q1200) Solve 3x^2+27x+42=0
Ans: x=-2 or x=-7
Q1201) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Q1202) Solve 30x^2+49x+17=0
Ans: x=-0.5 or x=-1.13
Q1203) Solve 40x^2+34x+3=0
Ans: x=-0.1 or x=-0.75
Q1204) Solve 11x^2+25x+14=0
Ans: x=-1 or x=-1.27
Q1205) Solve 32x^2+32x+8=0
Ans: x=-0.5 or x=-0.5
Q1206) Solve 25x^2+30x+5=0
Ans: x=-0.2 or x=-1
Q1207) Solve 8x^2+39x+28=0
Ans: x=-0.87 or x=-4
Q1208) Solve 5x^2+35x+30=0
Ans: x=-1 or x=-6
Q1209) Solve 5x^2+23x+18=0
Ans: x=-1 or x=-3.6
Q1210) Solve 14x^2+33x+13=0
Ans: x=-0.5 or x=-1.86
Q1211) Solve 2x^2+10x+12=0
Ans: x=-2 or x=-3
Q1212) Solve 40x^2+22x+1=0
Ans: x=-0.05 or x=-0.5
Q1213) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q1214) Solve 36x^2+32x+7=0
Ans: x=-0.39 or x=-0.5
Q1215) Solve 34x^2+21x+2=0
Ans: x=-0.12 or x=-0.5
Q1216) Solve 46x^2+39x+8=0
Ans: x=-0.35 or x=-0.5
Q1217) Solve 20x^2+49x+18=0
Ans: x=-0.45 or x=-2
Q1218) Solve 10x^2+41x+40=0
Ans: x=-1.6 or x=-2.5
Q1219) Solve 30x^2+41x+6=0
Ans: x=-0.17 or x=-1.2
Q1220) Solve 15x^2+47x+32=0
Ans: x=-1 or x=-2.13
Q1221) Solve 7x^2+36x+5=0
Ans: x=-0.14 or x=-5
Q1222) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q1223) Solve 35x^2+33x+4=0
Ans: x=-0.14 or x=-0.8
Q1224) Solve 10x^2+29x+21=0
Ans: x=-1.4 or x=-1.5
Q1225) Solve 48x^2+35x+2=0
Ans: x=-0.06 or x=-0.67
Q1226) Solve 6x^2+19x+8=0
Ans: x=-0.5 or x=-2.67
Q1227) Solve 50x^2+47x+11=0
Ans: x=-0.44 or x=-0.5
Q1228) Solve 9x^2+21x+6=0
Ans: x=-0.33 or x=-2
Q1229) Solve 4x^2+26x+42=0
Ans: x=-3 or x=-3.5
Q1230) Solve x^2+10x+25=0
Ans: x=-5 or x=-5
Q1231) Solve 21x^2+30x+9=0
Ans: x=-0.43 or x=-1
Q1232) Solve 3x^2+20x+12=0
Ans: x=-0.67 or x=-6
Q1233) Solve 11x^2+38x+32=0
Ans: x=-1.45 or x=-2
Q1234) Solve 4x^2+21x+5=0
Ans: x=-0.25 or x=-5
Q1235) Solve 6x^2+23x+10=0
Ans: x=-0.5 or x=-3.33
Q1236) Solve 5x^2+20x+20=0
Ans: x=-2 or x=-2
Q1237) Solve 30x^2+44x+14=0
Ans: x=-0.47 or x=-1
Q1238) Solve 18x^2+47x+30=0
Ans: x=-1.11 or x=-1.5
Q1239) Solve 5x^2+34x+48=0
Ans: x=-2 or x=-4.8
Q1240) Solve 9x^2+48x+15=0
Ans: x=-0.33 or x=-5
Q1241) Solve 50x^2+41x+8=0
Ans: x=-0.32 or x=-0.5
Q1242) Solve 44x^2+32x+5=0
Ans: x=-0.23 or x=-0.5
Q1243) Solve 20x^2+9x+1=0
Ans: x=-0.2 or x=-0.25
Q1244) Solve 19x^2+28x+9=0
Ans: x=-0.47 or x=-1
Q1245) Solve 21x^2+44x+23=0
Ans: x=-1 or x=-1.1
Q1246) Solve 7x^2+14x+7=0
Ans: x=-1 or x=-1
Q1247) Solve 12x^2+46x+34=0
Ans: x=-1 or x=-2.83
Q1248) Solve 47x^2+49x+2=0
Ans: x=-0.04 or x=-1
Q1249) Solve 12x^2+33x+21=0
Ans: x=-1 or x=-1.75
Q1250) Solve 17x^2+21x+4=0
Ans: x=-0.24 or x=-1
Q1251) Solve 12x^2+32x+21=0
Ans: x=-1.17 or x=-1.5
Q1252) Solve 9x^2+50x+25=0
Ans: x=-0.56 or x=-5
Q1253) Solve 26x^2+35x+9=0
Ans: x=-0.35 or x=-1
Q1254) Solve 4x^2+13x+3=0
Ans: x=-0.25 or x=-3
Q1255) Solve 20x^2+43x+6=0
Ans: x=-0.15 or x=-2
Q1256) Solve 42x^2+26x+4=0
Ans: x=-0.29 or x=-0.33
Q1257) Solve 28x^2+29x+6=0
Ans: x=-0.29 or x=-0.75
Q1258) Solve 32x^2+44x+5=0
Ans: x=-0.12 or x=-1.25
Q1259) Solve 16x^2+22x+7=0
Ans: x=-0.5 or x=-0.87
Q1260) Solve 38x^2+50x+12=0
Ans: x=-0.32 or x=-1
Q1261) Solve 22x^2+39x+14=0
Ans: x=-0.5 or x=-1.27
Q1262) Solve 26x^2+33x+7=0
Ans: x=-0.27 or x=-1
Q1263) Solve 16x^2+35x+6=0
Ans: x=-0.19 or x=-2
Q1264) Solve 9x^2+19x+2=0
Ans: x=-0.11 or x=-2
Q1265) Solve 4x^2+49x+45=0
Ans: x=-1 or x=-11.25
Q1266) Solve 7x^2+50x+7=0
Ans: x=-0.14 or x=-7
Q1267) Solve 12x^2+47x+35=0
Ans: x=-1 or x=-2.92
Q1268) Solve 17x^2+38x+8=0
Ans: x=-0.24 or x=-2
Q1269) Solve 2x^2+27x+46=0
Ans: x=-2 or x=-11.5
Q1270) Solve 32x^2+42x+10=0
Ans: x=-0.31 or x=-1
Q1271) Solve 11x^2+32x+20=0
Ans: x=-0.91 or x=-2
Q1272) Solve 6x^2+29x+28=0
Ans: x=-1.33 or x=-3.5
Q1273) Solve 12x^2+38x+16=0
Ans: x=-0.5 or x=-2.67
Q1274) Solve 20x^2+31x+12=0
Ans: x=-0.75 or x=-0.8
Q1275) Solve 15x^2+14x+3=0
Ans: x=-0.33 or x=-0.6
Q1276) Solve 20x^2+19x+3=0
Ans: x=-0.2 or x=-0.75
Q1277) Solve 15x^2+47x+28=0
Ans: x=-0.8 or x=-2.33
Q1278) Solve 40x^2+49x+15=0
Ans: x=-0.6 or x=-0.62
Q1279) Solve 9x^2+38x+29=0
Ans: x=-1 or x=-3.22
Q1280) Solve 3x^2+40x+48=0
Ans: x=-1.33 or x=-12
Q1281) Solve 28x^2+43x+15=0
Ans: x=-0.54 or x=-1
Q1282) Solve 14x^2+45x+34=0
Ans: x=-1.21 or x=-2
Q1283) Solve x^2+33x+32=0
Ans: x=-1 or x=-32
Q1284) Solve 12x^2+38x+26=0
Ans: x=-1 or x=-2.17
Q1285) Solve 46x^2+29x+3=0
Ans: x=-0.13 or x=-0.5
Q1286) Solve 8x^2+46x+21=0
Ans: x=-0.5 or x=-5.25
Q1287) Solve 4x^2+21x+17=0
Ans: x=-1 or x=-4.25
Q1288) Solve 43x^2+50x+7=0
Ans: x=-0.16 or x=-1
Q1289) Solve 2x^2+11x+9=0
Ans: x=-1 or x=-4.5
Q1290) Solve 12x^2+31x+20=0
Ans: x=-1.25 or x=-1.33
Q1291) Solve 11x^2+46x+48=0
Ans: x=-2 or x=-2.18
Q1292) Solve 14x^2+47x+15=0
Ans: x=-0.36 or x=-3
Q1293) Solve 36x^2+48x+12=0
Ans: x=-0.33 or x=-1
Q1294) Solve 2x^2+24x+40=0
Ans: x=-2 or x=-10
Q1295) Solve 9x^2+41x+46=0
Ans: x=-2 or x=-2.56
Q1296) Solve 12x^2+40x+17=0
Ans: x=-0.5 or x=-2.83
Q1297) Solve 5x^2+32x+48=0
Ans: x=-2.4 or x=-4
Q1298) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q1299) Solve 32x^2+45x+13=0
Ans: x=-0.41 or x=-1
Q1300) Solve 9x^2+45x+44=0
Ans: x=-1.33 or x=-3.67
Q1301) Solve 11x^2+34x+3=0
Ans: x=-0.09 or x=-3
Q1302) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q1303) Solve 40x^2+49x+15=0
Ans: x=-0.6 or x=-0.62
Q1304) Solve 23x^2+37x+14=0
Ans: x=-0.61 or x=-1
Q1305) Solve 27x^2+31x+4=0
Ans: x=-0.15 or x=-1
Q1306) Solve 32x^2+32x+8=0
Ans: x=-0.5 or x=-0.5
Q1307) Solve 4x^2+25x+36=0
Ans: x=-2.25 or x=-4
Q1308) Solve 49x^2+50x+1=0
Ans: x=-0.02 or x=-1
Q1309) Solve 10x^2+34x+28=0
Ans: x=-1.4 or x=-2
Q1310) Solve 4x^2+16x+12=0
Ans: x=-1 or x=-3
Q1311) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q1312) Solve 30x^2+37x+9=0
Ans: x=-0.33 or x=-0.9
Q1313) Solve 13x^2+45x+38=0
Ans: x=-1.46 or x=-2
Q1314) Solve 32x^2+48x+10=0
Ans: x=-0.25 or x=-1.25
Q1315) Solve 46x^2+47x+12=0
Ans: x=-0.5 or x=-0.52
Q1316) Solve 12x^2+47x+46=0
Ans: x=-1.92 or x=-2
Q1317) Solve 2x^2+15x+7=0
Ans: x=-0.5 or x=-7
Q1318) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q1319) Solve 18x^2+21x+5=0
Ans: x=-0.33 or x=-0.83
Q1320) Solve 4x^2+31x+21=0
Ans: x=-0.75 or x=-7
Q1321) Solve 18x^2+35x+13=0
Ans: x=-0.5 or x=-1.44
Q1322) Solve 25x^2+26x+1=0
Ans: x=-0.04 or x=-1
Q1323) Solve 12x^2+47x+11=0
Ans: x=-0.25 or x=-3.67
Q1324) Solve 12x^2+47x+40=0
Ans: x=-1.25 or x=-2.67
Q1325) Solve 16x^2+43x+22=0
Ans: x=-0.69 or x=-2
Q1326) Solve 15x^2+38x+7=0
Ans: x=-0.2 or x=-2.33
Q1327) Solve 25x^2+30x+9=0
Ans: x=-0.6 or x=-0.6
Q1328) Solve 12x^2+19x+5=0
Ans: x=-0.33 or x=-1.25
Q1329) Solve 20x^2+23x+3=0
Ans: x=-0.15 or x=-1
Q1330) Solve 36x^2+42x+6=0
Ans: x=-0.17 or x=-1
Q1331) Solve 6x^2+37x+6=0
Ans: x=-0.17 or x=-6
Q1332) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q1333) Solve 18x^2+42x+12=0
Ans: x=-0.33 or x=-2
Q1334) Solve 5x^2+14x+9=0
Ans: x=-1 or x=-1.8
Q1335) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q1336) Solve 16x^2+12x+2=0
Ans: x=-0.25 or x=-0.5
Q1337) Solve x^2+48x+47=0
Ans: x=-1 or x=-47
Q1338) Solve 4x^2+26x+12=0
Ans: x=-0.5 or x=-6
Q1339) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q1340) Solve 6x^2+25x+11=0
Ans: x=-0.5 or x=-3.67
Q1341) Solve 32x^2+44x+12=0
Ans: x=-0.37 or x=-1
Q1342) Solve 13x^2+17x+4=0
Ans: x=-0.31 or x=-1
Q1343) Solve x^2+15x+14=0
Ans: x=-1 or x=-14
Q1344) Solve 2x^2+45x+43=0
Ans: x=-1 or x=-21.5
Q1345) Solve 3x^2+23x+30=0
Ans: x=-1.67 or x=-6
Q1346) Solve 15x^2+41x+12=0
Ans: x=-0.33 or x=-2.4
Q1347) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q1348) Solve 8x^2+35x+33=0
Ans: x=-1.37 or x=-3
Q1349) Solve 3x^2+23x+34=0
Ans: x=-2 or x=-5.67
Q1350) Solve 13x^2+40x+28=0
Ans: x=-1.08 or x=-2
Q1351) Solve 2x^2+39x+37=0
Ans: x=-1 or x=-18.5
Q1352) Solve 9x^2+13x+4=0
Ans: x=-0.44 or x=-1
Q1353) Solve 12x^2+28x+15=0
Ans: x=-0.83 or x=-1.5
Q1354) Solve 11x^2+35x+26=0
Ans: x=-1.18 or x=-2
Q1355) Solve 23x^2+28x+5=0
Ans: x=-0.22 or x=-1
Q1356) Solve 18x^2+37x+15=0
Ans: x=-0.56 or x=-1.5
Q1357) Solve 3x^2+22x+40=0
Ans: x=-3.33 or x=-4
Q1358) Solve 4x^2+9x+5=0
Ans: x=-1 or x=-1.25
Q1359) Solve 14x^2+43x+20=0
Ans: x=-0.57 or x=-2.5
Q1360) Solve 11x^2+24x+4=0
Ans: x=-0.18 or x=-2
Q1361) Solve 28x^2+50x+12=0
Ans: x=-0.29 or x=-1.5
Q1362) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q1363) Solve 14x^2+31x+6=0
Ans: x=-0.21 or x=-2
Q1364) Solve 21x^2+37x+10=0
Ans: x=-0.33 or x=-1.43
Q1365) Solve 10x^2+41x+21=0
Ans: x=-0.6 or x=-3.5
Q1366) Solve 5x^2+11x+6=0
Ans: x=-1 or x=-1.2
Q1367) Solve 20x^2+33x+7=0
Ans: x=-0.25 or x=-1.4
Q1368) Solve 35x^2+27x+4=0
Ans: x=-0.2 or x=-0.57
Q1369) Solve 9x^2+27x+14=0
Ans: x=-0.67 or x=-2.33
Q1370) Solve 28x^2+39x+8=0
Ans: x=-0.25 or x=-1.14
Q1371) Solve 3x^2+19x+30=0
Ans: x=-3 or x=-3.33
Q1372) Solve 22x^2+13x+1=0
Ans: x=-0.09 or x=-0.5
Q1373) Solve 28x^2+35x+7=0
Ans: x=-0.25 or x=-1
Q1374) Solve 34x^2+43x+9=0
Ans: x=-0.26 or x=-1
Q1375) Solve 14x^2+46x+12=0
Ans: x=-0.29 or x=-3
Q1376) Solve 8x^2+22x+15=0
Ans: x=-1.25 or x=-1.5
Q1377) Solve 8x^2+27x+9=0
Ans: x=-0.37 or x=-3
Q1378) Solve 14x^2+31x+15=0
Ans: x=-0.71 or x=-1.5
Q1379) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q1380) Solve 22x^2+47x+6=0
Ans: x=-0.14 or x=-2
Q1381) Solve 42x^2+29x+4=0
Ans: x=-0.19 or x=-0.5
Q1382) Solve 5x^2+17x+12=0
Ans: x=-1 or x=-2.4
Q1383) Solve 6x^2+43x+26=0
Ans: x=-0.67 or x=-6.5
Q1384) Solve 39x^2+41x+10=0
Ans: x=-0.38 or x=-0.67
Q1385) Solve 15x^2+36x+21=0
Ans: x=-1 or x=-1.4
Q1386) Solve 20x^2+22x+2=0
Ans: x=-0.1 or x=-1
Q1387) Solve 5x^2+27x+10=0
Ans: x=-0.4 or x=-5
Q1388) Solve 39x^2+31x+6=0
Ans: x=-0.33 or x=-0.46
Q1389) Solve 5x^2+26x+5=0
Ans: x=-0.2 or x=-5
Q1390) Solve 22x^2+23x+6=0
Ans: x=-0.5 or x=-0.55
Q1391) Solve 2x^2+25x+42=0
Ans: x=-2 or x=-10.5
Q1392) Solve 36x^2+39x+10=0
Ans: x=-0.42 or x=-0.67
Q1393) Solve x^2+25x+46=0
Ans: x=-2 or x=-23
Q1394) Solve 14x^2+44x+6=0
Ans: x=-0.14 or x=-3
Q1395) Solve 9x^2+48x+28=0
Ans: x=-0.67 or x=-4.67
Q1396) Solve 12x^2+26x+14=0
Ans: x=-1 or x=-1.17
Q1397) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q1398) Solve 2x^2+16x+14=0
Ans: x=-1 or x=-7
Q1399) Solve 18x^2+18x+4=0
Ans: x=-0.33 or x=-0.67
Q1400) Solve 8x^2+46x+11=0
Ans: x=-0.25 or x=-5.5
Q1401) Solve 15x^2+37x+22=0
Ans: x=-1 or x=-1.47
Q1402) Solve 4x^2+20x+21=0
Ans: x=-1.5 or x=-3.5
Q1403) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q1404) Solve 12x^2+36x+27=0
Ans: x=-1.5 or x=-1.5
Q1405) Solve 23x^2+29x+6=0
Ans: x=-0.26 or x=-1
Q1406) Solve 24x^2+38x+13=0
Ans: x=-0.5 or x=-1.08
Q1407) Solve 21x^2+32x+12=0
Ans: x=-0.67 or x=-0.86
Q1408) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q1409) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q1410) Solve 3x^2+15x+18=0
Ans: x=-2 or x=-3
Q1411) Solve 14x^2+26x+12=0
Ans: x=-0.86 or x=-1
Q1412) Solve 18x^2+39x+20=0
Ans: x=-0.83 or x=-1.33
Q1413) Solve 17x^2+21x+4=0
Ans: x=-0.24 or x=-1
Q1414) Solve 18x^2+33x+5=0
Ans: x=-0.17 or x=-1.67
Q1415) Solve 4x^2+42x+20=0
Ans: x=-0.5 or x=-10
Q1416) Solve x^2+13x+12=0
Ans: x=-1 or x=-12
Q1417) Solve 13x^2+17x+4=0
Ans: x=-0.31 or x=-1
Q1418) Solve 3x^2+11x+10=0
Ans: x=-1.67 or x=-2
Q1419) Solve 24x^2+23x+5=0
Ans: x=-0.33 or x=-0.62
Q1420) Solve 4x^2+28x+33=0
Ans: x=-1.5 or x=-5.5
Q1421) Solve 6x^2+32x+32=0
Ans: x=-1.33 or x=-4
Q1422) Solve 24x^2+41x+12=0
Ans: x=-0.37 or x=-1.33
Q1423) Solve 8x^2+39x+46=0
Ans: x=-2 or x=-2.87
Q1424) Solve 24x^2+25x+1=0
Ans: x=-0.04 or x=-1
Q1425) Solve 22x^2+50x+28=0
Ans: x=-1 or x=-1.27
Q1426) Solve x^2+8x+7=0
Ans: x=-1 or x=-7
Q1427) Solve 7x^2+27x+26=0
Ans: x=-1.86 or x=-2
Q1428) Solve 10x^2+20x+10=0
Ans: x=-1 or x=-1
Q1429) Solve x^2+34x+33=0
Ans: x=-1 or x=-33
Q1430) Solve 6x^2+7x+1=0
Ans: x=-0.17 or x=-1
Q1431) Solve 6x^2+46x+40=0
Ans: x=-1 or x=-6.67
Q1432) Solve 20x^2+28x+9=0
Ans: x=-0.5 or x=-0.9
Q1433) Solve 6x^2+35x+25=0
Ans: x=-0.83 or x=-5
Q1434) Solve 12x^2+46x+40=0
Ans: x=-1.33 or x=-2.5
Q1435) Solve 44x^2+38x+8=0
Ans: x=-0.36 or x=-0.5
Q1436) Solve 32x^2+36x+4=0
Ans: x=-0.12 or x=-1
Q1437) Solve 3x^2+40x+48=0
Ans: x=-1.33 or x=-12
Q1438) Solve 35x^2+41x+12=0
Ans: x=-0.57 or x=-0.6
Q1439) Solve 5x^2+29x+36=0
Ans: x=-1.8 or x=-4
Q1440) Solve 4x^2+43x+30=0
Ans: x=-0.75 or x=-10
Q1441) Solve 22x^2+41x+12=0
Ans: x=-0.36 or x=-1.5
Q1442) Solve 3x^2+30x+48=0
Ans: x=-2 or x=-8
Q1443) Solve 12x^2+41x+35=0
Ans: x=-1.67 or x=-1.75
Q1444) Solve 31x^2+38x+7=0
Ans: x=-0.23 or x=-1
Q1445) Solve 24x^2+50x+11=0
Ans: x=-0.25 or x=-1.83
Q1446) Solve 26x^2+49x+15=0
Ans: x=-0.38 or x=-1.5
Q1447) Solve 38x^2+47x+14=0
Ans: x=-0.5 or x=-0.74
Q1448) Solve 11x^2+31x+20=0
Ans: x=-1 or x=-1.82
Q1449) Solve 25x^2+20x+4=0
Ans: x=-0.4 or x=-0.4
Q1450) Solve 40x^2+50x+10=0
Ans: x=-0.25 or x=-1
Q1451) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q1452) Solve 33x^2+50x+17=0
Ans: x=-0.52 or x=-1
Q1453) Solve 14x^2+31x+17=0
Ans: x=-1 or x=-1.21
Q1454) Solve 2x^2+13x+11=0
Ans: x=-1 or x=-5.5
Q1455) Solve 3x^2+20x+25=0
Ans: x=-1.67 or x=-5
Q1456) Solve 8x^2+20x+12=0
Ans: x=-1 or x=-1.5
Q1457) Solve 14x^2+19x+6=0
Ans: x=-0.5 or x=-0.86
Q1458) Solve 45x^2+39x+8=0
Ans: x=-0.33 or x=-0.53
Q1459) Solve 34x^2+48x+14=0
Ans: x=-0.41 or x=-1
Q1460) Solve 37x^2+45x+8=0
Ans: x=-0.22 or x=-1
Q1461) Solve 30x^2+35x+10=0
Ans: x=-0.5 or x=-0.67
Q1462) Solve 40x^2+32x+6=0
Ans: x=-0.3 or x=-0.5
Q1463) Solve 40x^2+41x+10=0
Ans: x=-0.4 or x=-0.62
Q1464) Solve 15x^2+49x+26=0
Ans: x=-0.67 or x=-2.6
Q1465) Solve 5x^2+41x+36=0
Ans: x=-1 or x=-7.2
Q1466) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q1467) Solve 13x^2+27x+14=0
Ans: x=-1 or x=-1.08
Q1468) Solve 7x^2+14x+7=0
Ans: x=-1 or x=-1
Q1469) Solve 14x^2+41x+17=0
Ans: x=-0.5 or x=-2.43
Q1470) Solve 42x^2+37x+8=0
Ans: x=-0.38 or x=-0.5
Q1471) Solve 6x^2+16x+8=0
Ans: x=-0.67 or x=-2
Q1472) Solve 27x^2+48x+20=0
Ans: x=-0.67 or x=-1.11
Q1473) Solve 11x^2+41x+38=0
Ans: x=-1.73 or x=-2
Q1474) Solve 44x^2+38x+8=0
Ans: x=-0.36 or x=-0.5
Q1475) Solve 9x^2+39x+42=0
Ans: x=-2 or x=-2.33
Q1476) Solve 16x^2+20x+4=0
Ans: x=-0.25 or x=-1
Q1477) Solve 6x^2+13x+2=0
Ans: x=-0.17 or x=-2
Q1478) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q1479) Solve 42x^2+50x+12=0
Ans: x=-0.33 or x=-0.86
Q1480) Solve 4x^2+18x+20=0
Ans: x=-2 or x=-2.5
Q1481) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q1482) Solve 12x^2+28x+16=0
Ans: x=-1 or x=-1.33
Q1483) Solve 2x^2+22x+36=0
Ans: x=-2 or x=-9
Q1484) Solve 4x^2+4x+1=0
Ans: x=-0.5 or x=-0.5
Q1485) Solve 9x^2+33x+10=0
Ans: x=-0.33 or x=-3.33
Q1486) Solve 32x^2+48x+10=0
Ans: x=-0.25 or x=-1.25
Q1487) Solve 42x^2+44x+2=0
Ans: x=-0.05 or x=-1
Q1488) Solve 20x^2+30x+10=0
Ans: x=-0.5 or x=-1
Q1489) Solve 36x^2+43x+7=0
Ans: x=-0.19 or x=-1
Q1490) Solve 26x^2+46x+20=0
Ans: x=-0.77 or x=-1
Q1491) Solve 4x^2+50x+24=0
Ans: x=-0.5 or x=-12
Q1492) Solve 6x^2+21x+15=0
Ans: x=-1 or x=-2.5
Q1493) Solve 11x^2+44x+44=0
Ans: x=-2 or x=-2
Q1494) Solve 38x^2+47x+14=0
Ans: x=-0.5 or x=-0.74
Q1495) Solve 47x^2+50x+3=0
Ans: x=-0.06 or x=-1
Q1496) Solve 8x^2+14x+5=0
Ans: x=-0.5 or x=-1.25
Q1497) Solve 19x^2+42x+8=0
Ans: x=-0.21 or x=-2
Q1498) Solve 6x^2+45x+21=0
Ans: x=-0.5 or x=-7
Q1499) Solve 21x^2+24x+3=0
Ans: x=-0.14 or x=-1
Q1500) Solve 19x^2+36x+17=0
Ans: x=-0.89 or x=-1
Q1501) Solve 20x^2+29x+9=0
Ans: x=-0.45 or x=-1
Q1502) Solve 14x^2+47x+38=0
Ans: x=-1.36 or x=-2
Q1503) Solve 15x^2+21x+6=0
Ans: x=-0.4 or x=-1
Q1504) Solve 42x^2+20x+2=0
Ans: x=-0.14 or x=-0.33
Q1505) Solve 10x^2+39x+38=0
Ans: x=-1.9 or x=-2
Q1506) Solve 16x^2+35x+6=0
Ans: x=-0.19 or x=-2
Q1507) Solve 15x^2+38x+16=0
Ans: x=-0.53 or x=-2
Q1508) Solve 40x^2+30x+5=0
Ans: x=-0.25 or x=-0.5
Q1509) Solve 36x^2+27x+2=0
Ans: x=-0.08 or x=-0.67
Q1510) Solve 40x^2+41x+1=0
Ans: x=-0.02 or x=-1
Q1511) Solve 10x^2+39x+14=0
Ans: x=-0.4 or x=-3.5
Q1512) Solve 21x^2+23x+2=0
Ans: x=-0.1 or x=-1
Q1513) Solve 14x^2+25x+9=0
Ans: x=-0.5 or x=-1.29
Q1514) Solve 24x^2+34x+7=0
Ans: x=-0.25 or x=-1.17
Q1515) Solve 46x^2+47x+12=0
Ans: x=-0.5 or x=-0.52
Q1516) Solve 16x^2+49x+33=0
Ans: x=-1 or x=-2.06
Q1517) Solve 3x^2+13x+14=0
Ans: x=-2 or x=-2.33
Q1518) Solve 45x^2+46x+8=0
Ans: x=-0.22 or x=-0.8
Q1519) Solve 35x^2+39x+4=0
Ans: x=-0.11 or x=-1
Q1520) Solve 24x^2+33x+9=0
Ans: x=-0.37 or x=-1
Q1521) Solve 7x^2+19x+12=0
Ans: x=-1 or x=-1.71
Q1522) Solve 24x^2+43x+18=0
Ans: x=-0.67 or x=-1.12
Q1523) Solve 9x^2+48x+28=0
Ans: x=-0.67 or x=-4.67
Q1524) Solve 13x^2+40x+27=0
Ans: x=-1 or x=-2.08
Q1525) Solve 12x^2+34x+24=0
Ans: x=-1.33 or x=-1.5
Q1526) Solve 8x^2+39x+31=0
Ans: x=-1 or x=-3.87
Q1527) Solve 40x^2+47x+7=0
Ans: x=-0.17 or x=-1
Q1528) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q1529) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q1530) Solve 13x^2+50x+48=0
Ans: x=-1.85 or x=-2
Q1531) Solve 13x^2+46x+21=0
Ans: x=-0.54 or x=-3
Q1532) Solve 20x^2+49x+18=0
Ans: x=-0.45 or x=-2
Q1533) Solve 9x^2+45x+44=0
Ans: x=-1.33 or x=-3.67
Q1534) Solve 12x^2+49x+37=0
Ans: x=-1 or x=-3.08
Q1535) Solve 16x^2+26x+10=0
Ans: x=-0.62 or x=-1
Q1536) Solve 22x^2+47x+21=0
Ans: x=-0.64 or x=-1.5
Q1537) Solve 9x^2+48x+48=0
Ans: x=-1.33 or x=-4
Q1538) Solve 7x^2+38x+40=0
Ans: x=-1.43 or x=-4
Q1539) Solve 8x^2+26x+18=0
Ans: x=-1 or x=-2.25
Q1540) Solve 13x^2+45x+32=0
Ans: x=-1 or x=-2.46
Q1541) Solve 43x^2+47x+4=0
Ans: x=-0.09 or x=-1
Q1542) Solve 11x^2+22x+11=0
Ans: x=-1 or x=-1
Q1543) Solve 11x^2+26x+15=0
Ans: x=-1 or x=-1.36
Q1544) Solve 3x^2+30x+48=0
Ans: x=-2 or x=-8
Q1545) Solve 10x^2+43x+12=0
Ans: x=-0.3 or x=-4
Q1546) Solve 2x^2+41x+39=0
Ans: x=-1 or x=-19.5
Q1547) Solve 4x^2+29x+42=0
Ans: x=-2 or x=-5.25
Q1548) Solve 16x^2+47x+30=0
Ans: x=-0.94 or x=-2
Q1549) Solve 6x^2+14x+4=0
Ans: x=-0.33 or x=-2
Q1550) Solve 30x^2+17x+1=0
Ans: x=-0.07 or x=-0.5
Q1551) Solve 2x^2+13x+11=0
Ans: x=-1 or x=-5.5
Q1552) Solve 12x^2+38x+28=0
Ans: x=-1.17 or x=-2
Q1553) Solve 32x^2+44x+9=0
Ans: x=-0.25 or x=-1.12
Q1554) Solve 6x^2+13x+2=0
Ans: x=-0.17 or x=-2
Q1555) Solve 9x^2+29x+6=0
Ans: x=-0.22 or x=-3
Q1556) Solve 39x^2+41x+2=0
Ans: x=-0.05 or x=-1
Q1557) Solve 4x^2+36x+32=0
Ans: x=-1 or x=-8
Q1558) Solve 28x^2+46x+16=0
Ans: x=-0.5 or x=-1.14
Q1559) Solve 30x^2+41x+14=0
Ans: x=-0.67 or x=-0.7
Q1560) Solve 3x^2+28x+32=0
Ans: x=-1.33 or x=-8
Q1561) Solve 13x^2+43x+30=0
Ans: x=-1 or x=-2.31
Q1562) Solve 40x^2+50x+10=0
Ans: x=-0.25 or x=-1
Q1563) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q1564) Solve 6x^2+28x+22=0
Ans: x=-1 or x=-3.67
Q1565) Solve 18x^2+25x+7=0
Ans: x=-0.39 or x=-1
Q1566) Solve 4x^2+45x+11=0
Ans: x=-0.25 or x=-11
Q1567) Solve 21x^2+24x+3=0
Ans: x=-0.14 or x=-1
Q1568) Solve 35x^2+39x+10=0
Ans: x=-0.4 or x=-0.71
Q1569) Solve 37x^2+50x+13=0
Ans: x=-0.35 or x=-1
Q1570) Solve 40x^2+46x+6=0
Ans: x=-0.15 or x=-1
Q1571) Solve 5x^2+41x+36=0
Ans: x=-1 or x=-7.2
Q1572) Solve 20x^2+41x+2=0
Ans: x=-0.05 or x=-2
Q1573) Solve 18x^2+45x+25=0
Ans: x=-0.83 or x=-1.67
Q1574) Solve 20x^2+50x+30=0
Ans: x=-1 or x=-1.5
Q1575) Solve 50x^2+39x+7=0
Ans: x=-0.28 or x=-0.5
Q1576) Solve 16x^2+32x+15=0
Ans: x=-0.75 or x=-1.25
Q1577) Solve 25x^2+40x+16=0
Ans: x=-0.8 or x=-0.8
Q1578) Solve 3x^2+44x+28=0
Ans: x=-0.67 or x=-14
Q1579) Solve 50x^2+45x+4=0
Ans: x=-0.1 or x=-0.8
Q1580) Solve 36x^2+36x+5=0
Ans: x=-0.17 or x=-0.83
Q1581) Solve 12x^2+10x+2=0
Ans: x=-0.33 or x=-0.5
Q1582) Solve 5x^2+38x+33=0
Ans: x=-1 or x=-6.6
Q1583) Solve 12x^2+31x+20=0
Ans: x=-1.25 or x=-1.33
Q1584) Solve 4x^2+19x+12=0
Ans: x=-0.75 or x=-4
Q1585) Solve 20x^2+33x+7=0
Ans: x=-0.25 or x=-1.4
Q1586) Solve 25x^2+37x+12=0
Ans: x=-0.48 or x=-1
Q1587) Solve 8x^2+24x+18=0
Ans: x=-1.5 or x=-1.5
Q1588) Solve 18x^2+36x+16=0
Ans: x=-0.67 or x=-1.33
Q1589) Solve 14x^2+46x+12=0
Ans: x=-0.29 or x=-3
Q1590) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q1591) Solve 7x^2+37x+36=0
Ans: x=-1.29 or x=-4
Q1592) Solve 18x^2+49x+20=0
Ans: x=-0.5 or x=-2.22
Q1593) Solve 5x^2+22x+8=0
Ans: x=-0.4 or x=-4
Q1594) Solve 36x^2+33x+6=0
Ans: x=-0.25 or x=-0.67
Q1595) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q1596) Solve 42x^2+34x+4=0
Ans: x=-0.14 or x=-0.67
Q1597) Solve 44x^2+39x+7=0
Ans: x=-0.25 or x=-0.64
Q1598) Solve 5x^2+38x+33=0
Ans: x=-1 or x=-6.6
Q1599) Solve 20x^2+36x+13=0
Ans: x=-0.5 or x=-1.3
Q1600) Solve 28x^2+38x+12=0
Ans: x=-0.5 or x=-0.86
Q1601) Solve 12x^2+44x+39=0
Ans: x=-1.5 or x=-2.17
Q1602) Solve 8x^2+18x+10=0
Ans: x=-1 or x=-1.25
Q1603) Solve 3x^2+25x+22=0
Ans: x=-1 or x=-7.33
Q1604) Solve x^2+11x+18=0
Ans: x=-2 or x=-9
Q1605) Solve 11x^2+46x+8=0
Ans: x=-0.18 or x=-4
Q1606) Solve 18x^2+44x+26=0
Ans: x=-1 or x=-1.44
Q1607) Solve x^2+36x+35=0
Ans: x=-1 or x=-35
Q1608) Solve 45x^2+45x+10=0
Ans: x=-0.33 or x=-0.67
Q1609) Solve 8x^2+35x+38=0
Ans: x=-2 or x=-2.37
Q1610) Solve 2x^2+35x+33=0
Ans: x=-1 or x=-16.5
Q1611) Solve 7x^2+41x+30=0
Ans: x=-0.86 or x=-5
Q1612) Solve 20x^2+48x+16=0
Ans: x=-0.4 or x=-2
Q1613) Solve 6x^2+49x+23=0
Ans: x=-0.5 or x=-7.67
Q1614) Solve 11x^2+24x+4=0
Ans: x=-0.18 or x=-2
Q1615) Solve 4x^2+30x+44=0
Ans: x=-2 or x=-5.5
Q1616) Solve 27x^2+41x+14=0
Ans: x=-0.52 or x=-1
Q1617) Solve x^2+8x+7=0
Ans: x=-1 or x=-7
Q1618) Solve 13x^2+28x+15=0
Ans: x=-1 or x=-1.15
Q1619) Solve 35x^2+47x+6=0
Ans: x=-0.14 or x=-1.2
Q1620) Solve 15x^2+26x+8=0
Ans: x=-0.4 or x=-1.33
Q1621) Solve 7x^2+38x+15=0
Ans: x=-0.43 or x=-5
Q1622) Solve 43x^2+44x+1=0
Ans: x=-0.02 or x=-1
Q1623) Solve 6x^2+34x+28=0
Ans: x=-1 or x=-4.67
Q1624) Solve 3x^2+26x+23=0
Ans: x=-1 or x=-7.67
Q1625) Solve 21x^2+25x+4=0
Ans: x=-0.19 or x=-1
Q1626) Solve 22x^2+43x+16=0
Ans: x=-0.5 or x=-1.45
Q1627) Solve 14x^2+17x+5=0
Ans: x=-0.5 or x=-0.71
Q1628) Solve 15x^2+26x+8=0
Ans: x=-0.4 or x=-1.33
Q1629) Solve 18x^2+9x+1=0
Ans: x=-0.17 or x=-0.33
Q1630) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q1631) Solve 20x^2+40x+20=0
Ans: x=-1 or x=-1
Q1632) Solve x^2+14x+45=0
Ans: x=-5 or x=-9
Q1633) Solve 4x^2+21x+26=0
Ans: x=-2 or x=-3.25
Q1634) Solve 26x^2+45x+9=0
Ans: x=-0.23 or x=-1.5
Q1635) Solve 39x^2+38x+8=0
Ans: x=-0.31 or x=-0.67
Q1636) Solve 12x^2+48x+21=0
Ans: x=-0.5 or x=-3.5
Q1637) Solve 8x^2+16x+8=0
Ans: x=-1 or x=-1
Q1638) Solve 34x^2+44x+10=0
Ans: x=-0.29 or x=-1
Q1639) Solve 4x^2+22x+24=0
Ans: x=-1.5 or x=-4
Q1640) Solve 7x^2+27x+18=0
Ans: x=-0.86 or x=-3
Q1641) Solve 16x^2+28x+10=0
Ans: x=-0.5 or x=-1.25
Q1642) Solve 39x^2+38x+8=0
Ans: x=-0.31 or x=-0.67
Q1643) Solve 12x^2+46x+14=0
Ans: x=-0.33 or x=-3.5
Q1644) Solve 14x^2+45x+25=0
Ans: x=-0.71 or x=-2.5
Q1645) Solve 17x^2+23x+6=0
Ans: x=-0.35 or x=-1
Q1646) Solve 12x^2+39x+27=0
Ans: x=-1 or x=-2.25
Q1647) Solve x^2+9x+8=0
Ans: x=-1 or x=-8
Q1648) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q1649) Solve 36x^2+24x+3=0
Ans: x=-0.17 or x=-0.5
Q1650) Solve 25x^2+35x+6=0
Ans: x=-0.2 or x=-1.2
Q1651) Solve 8x^2+33x+34=0
Ans: x=-2 or x=-2.12
Q1652) Solve 19x^2+34x+15=0
Ans: x=-0.79 or x=-1
Q1653) Solve 32x^2+32x+8=0
Ans: x=-0.5 or x=-0.5
Q1654) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q1655) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q1656) Solve 12x^2+26x+14=0
Ans: x=-1 or x=-1.17
Q1657) Solve 10x^2+7x+1=0
Ans: x=-0.2 or x=-0.5
Q1658) Solve 24x^2+50x+4=0
Ans: x=-0.08 or x=-2
Q1659) Solve 12x^2+32x+5=0
Ans: x=-0.17 or x=-2.5
Q1660) Solve 28x^2+33x+9=0
Ans: x=-0.43 or x=-0.75
Q1661) Solve 6x^2+35x+44=0
Ans: x=-1.83 or x=-4
Q1662) Solve 42x^2+45x+12=0
Ans: x=-0.5 or x=-0.57
Q1663) Solve 16x^2+38x+12=0
Ans: x=-0.37 or x=-2
Q1664) Solve 39x^2+28x+5=0
Ans: x=-0.33 or x=-0.38
Q1665) Solve 32x^2+32x+6=0
Ans: x=-0.25 or x=-0.75
Q1666) Solve 5x^2+34x+48=0
Ans: x=-2 or x=-4.8
Q1667) Solve 11x^2+28x+12=0
Ans: x=-0.55 or x=-2
Q1668) Solve 4x^2+18x+18=0
Ans: x=-1.5 or x=-3
Q1669) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q1670) Solve 24x^2+38x+10=0
Ans: x=-0.33 or x=-1.25
Q1671) Solve 31x^2+49x+18=0
Ans: x=-0.58 or x=-1
Q1672) Solve 10x^2+9x+2=0
Ans: x=-0.4 or x=-0.5
Q1673) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q1674) Solve 13x^2+50x+48=0
Ans: x=-1.85 or x=-2
Q1675) Solve 9x^2+32x+23=0
Ans: x=-1 or x=-2.56
Q1676) Solve 24x^2+34x+11=0
Ans: x=-0.5 or x=-0.92
Q1677) Solve 48x^2+44x+6=0
Ans: x=-0.17 or x=-0.75
Q1678) Solve 14x^2+43x+29=0
Ans: x=-1 or x=-2.07
Q1679) Solve 2x^2+23x+21=0
Ans: x=-1 or x=-10.5
Q1680) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q1681) Solve 5x^2+28x+23=0
Ans: x=-1 or x=-4.6
Q1682) Solve 28x^2+48x+17=0
Ans: x=-0.5 or x=-1.21
Q1683) Solve 2x^2+20x+18=0
Ans: x=-1 or x=-9
Q1684) Solve 3x^2+22x+24=0
Ans: x=-1.33 or x=-6
Q1685) Solve 3x^2+47x+44=0
Ans: x=-1 or x=-14.67
Q1686) Solve 2x^2+29x+14=0
Ans: x=-0.5 or x=-14
Q1687) Solve 11x^2+46x+8=0
Ans: x=-0.18 or x=-4
Q1688) Solve 12x^2+35x+25=0
Ans: x=-1.25 or x=-1.67
Q1689) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q1690) Solve 4x^2+11x+7=0
Ans: x=-1 or x=-1.75
Q1691) Solve 22x^2+32x+10=0
Ans: x=-0.45 or x=-1
Q1692) Solve 11x^2+45x+46=0
Ans: x=-2 or x=-2.09
Q1693) Solve 4x^2+45x+11=0
Ans: x=-0.25 or x=-11
Q1694) Solve 33x^2+40x+7=0
Ans: x=-0.21 or x=-1
Q1695) Solve 4x^2+14x+10=0
Ans: x=-1 or x=-2.5
Q1696) Solve 30x^2+34x+4=0
Ans: x=-0.13 or x=-1
Q1697) Solve 15x^2+44x+29=0
Ans: x=-1 or x=-1.93
Q1698) Solve 30x^2+46x+8=0
Ans: x=-0.2 or x=-1.33
Q1699) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q1700) Solve 19x^2+48x+29=0
Ans: x=-1 or x=-1.53
Q1701) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q1702) Solve 24x^2+29x+7=0
Ans: x=-0.33 or x=-0.87
Q1703) Solve 20x^2+47x+21=0
Ans: x=-0.6 or x=-1.75
Q1704) Solve 21x^2+32x+11=0
Ans: x=-0.52 or x=-1
Q1705) Solve 12x^2+46x+14=0
Ans: x=-0.33 or x=-3.5
Q1706) Solve 6x^2+25x+19=0
Ans: x=-1 or x=-3.17
Q1707) Solve 35x^2+17x+2=0
Ans: x=-0.2 or x=-0.29
Q1708) Solve 26x^2+45x+19=0
Ans: x=-0.73 or x=-1
Q1709) Solve 15x^2+37x+18=0
Ans: x=-0.67 or x=-1.8
Q1710) Solve 5x^2+22x+21=0
Ans: x=-1.4 or x=-3
Q1711) Solve 4x^2+32x+39=0
Ans: x=-1.5 or x=-6.5
Q1712) Solve 39x^2+46x+7=0
Ans: x=-0.18 or x=-1
Q1713) Solve 20x^2+23x+3=0
Ans: x=-0.15 or x=-1
Q1714) Solve 11x^2+26x+8=0
Ans: x=-0.36 or x=-2
Q1715) Solve 6x^2+50x+16=0
Ans: x=-0.33 or x=-8
Q1716) Solve 3x^2+31x+10=0
Ans: x=-0.33 or x=-10
Q1717) Solve 8x^2+26x+11=0
Ans: x=-0.5 or x=-2.75
Q1718) Solve 29x^2+50x+21=0
Ans: x=-0.72 or x=-1
Q1719) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q1720) Solve 6x^2+20x+6=0
Ans: x=-0.33 or x=-3
Q1721) Solve 42x^2+23x+1=0
Ans: x=-0.05 or x=-0.5
Q1722) Solve 3x^2+12x+9=0
Ans: x=-1 or x=-3
Q1723) Solve 26x^2+45x+16=0
Ans: x=-0.5 or x=-1.23
Q1724) Solve 30x^2+43x+13=0
Ans: x=-0.43 or x=-1
Q1725) Solve 10x^2+39x+17=0
Ans: x=-0.5 or x=-3.4
Q1726) Solve 15x^2+40x+20=0
Ans: x=-0.67 or x=-2
Q1727) Solve 5x^2+6x+1=0
Ans: x=-0.2 or x=-1
Q1728) Solve 12x^2+40x+28=0
Ans: x=-1 or x=-2.33
Q1729) Solve 16x^2+42x+26=0
Ans: x=-1 or x=-1.62
Q1730) Solve 45x^2+42x+8=0
Ans: x=-0.27 or x=-0.67
Q1731) Solve 14x^2+20x+6=0
Ans: x=-0.43 or x=-1
Q1732) Solve 3x^2+26x+48=0
Ans: x=-2.67 or x=-6
Q1733) Solve 5x^2+47x+18=0
Ans: x=-0.4 or x=-9
Q1734) Solve 13x^2+33x+20=0
Ans: x=-1 or x=-1.54
Q1735) Solve 4x^2+27x+38=0
Ans: x=-2 or x=-4.75
Q1736) Solve 4x^2+32x+28=0
Ans: x=-1 or x=-7
Q1737) Solve 12x^2+26x+4=0
Ans: x=-0.17 or x=-2
Q1738) Solve 2x^2+50x+48=0
Ans: x=-1 or x=-24
Q1739) Solve 44x^2+46x+12=0
Ans: x=-0.5 or x=-0.55
Q1740) Solve 24x^2+49x+22=0
Ans: x=-0.67 or x=-1.37
Q1741) Solve 6x^2+26x+24=0
Ans: x=-1.33 or x=-3
Q1742) Solve 12x^2+42x+18=0
Ans: x=-0.5 or x=-3
Q1743) Solve 17x^2+33x+16=0
Ans: x=-0.94 or x=-1
Q1744) Solve 7x^2+26x+19=0
Ans: x=-1 or x=-2.71
Q1745) Solve 17x^2+33x+16=0
Ans: x=-0.94 or x=-1
Q1746) Solve 19x^2+45x+26=0
Ans: x=-1 or x=-1.37
Q1747) Solve 35x^2+22x+3=0
Ans: x=-0.2 or x=-0.43
Q1748) Solve x^2+12x+11=0
Ans: x=-1 or x=-11
Q1749) Solve 7x^2+23x+16=0
Ans: x=-1 or x=-2.29
Q1750) Solve 2x^2+22x+48=0
Ans: x=-3 or x=-8
Q1751) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q1752) Solve 33x^2+31x+6=0
Ans: x=-0.27 or x=-0.67
Q1753) Solve 4x^2+46x+22=0
Ans: x=-0.5 or x=-11
Q1754) Solve 27x^2+38x+11=0
Ans: x=-0.41 or x=-1
Q1755) Solve 24x^2+47x+13=0
Ans: x=-0.33 or x=-1.62
Q1756) Solve 22x^2+17x+3=0
Ans: x=-0.27 or x=-0.5
Q1757) Solve 20x^2+17x+3=0
Ans: x=-0.25 or x=-0.6
Q1758) Solve 3x^2+38x+35=0
Ans: x=-1 or x=-11.67
Q1759) Solve 22x^2+36x+14=0
Ans: x=-0.64 or x=-1
Q1760) Solve 16x^2+49x+33=0
Ans: x=-1 or x=-2.06
Q1761) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q1762) Solve 10x^2+39x+17=0
Ans: x=-0.5 or x=-3.4
Q1763) Solve 19x^2+49x+22=0
Ans: x=-0.58 or x=-2
Q1764) Solve 10x^2+48x+38=0
Ans: x=-1 or x=-3.8
Q1765) Solve 17x^2+50x+32=0
Ans: x=-0.94 or x=-2
Q1766) Solve 9x^2+24x+7=0
Ans: x=-0.33 or x=-2.33
Q1767) Solve 15x^2+44x+29=0
Ans: x=-1 or x=-1.93
Q1768) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q1769) Solve 46x^2+33x+5=0
Ans: x=-0.22 or x=-0.5
Q1770) Solve 28x^2+43x+10=0
Ans: x=-0.29 or x=-1.25
Q1771) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q1772) Solve 4x^2+28x+48=0
Ans: x=-3 or x=-4
Q1773) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q1774) Solve 28x^2+46x+16=0
Ans: x=-0.5 or x=-1.14
Q1775) Solve 11x^2+31x+18=0
Ans: x=-0.82 or x=-2
Q1776) Solve 28x^2+43x+10=0
Ans: x=-0.29 or x=-1.25
Q1777) Solve 12x^2+15x+3=0
Ans: x=-0.25 or x=-1
Q1778) Solve 4x^2+24x+20=0
Ans: x=-1 or x=-5
Q1779) Solve 10x^2+16x+6=0
Ans: x=-0.6 or x=-1
Q1780) Solve 11x^2+47x+12=0
Ans: x=-0.27 or x=-4
Q1781) Solve 5x^2+19x+12=0
Ans: x=-0.8 or x=-3
Q1782) Solve 2x^2+49x+24=0
Ans: x=-0.5 or x=-24
Q1783) Solve 28x^2+48x+17=0
Ans: x=-0.5 or x=-1.21
Q1784) Solve 10x^2+41x+18=0
Ans: x=-0.5 or x=-3.6
Q1785) Solve 40x^2+34x+7=0
Ans: x=-0.35 or x=-0.5
Q1786) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q1787) Solve 32x^2+34x+2=0
Ans: x=-0.06 or x=-1
Q1788) Solve 50x^2+35x+6=0
Ans: x=-0.3 or x=-0.4
Q1789) Solve 38x^2+50x+12=0
Ans: x=-0.32 or x=-1
Q1790) Solve 32x^2+36x+10=0
Ans: x=-0.5 or x=-0.62
Q1791) Solve 5x^2+31x+42=0
Ans: x=-2 or x=-4.2
Q1792) Solve 5x^2+27x+36=0
Ans: x=-2.4 or x=-3
Q1793) Solve 8x^2+8x+2=0
Ans: x=-0.5 or x=-0.5
Q1794) Solve 6x^2+27x+12=0
Ans: x=-0.5 or x=-4
Q1795) Solve 15x^2+38x+23=0
Ans: x=-1 or x=-1.53
Q1796) Solve 12x^2+39x+27=0
Ans: x=-1 or x=-2.25
Q1797) Solve 30x^2+38x+12=0
Ans: x=-0.6 or x=-0.67
Q1798) Solve 2x^2+29x+14=0
Ans: x=-0.5 or x=-14
Q1799) Solve 18x^2+40x+22=0
Ans: x=-1 or x=-1.22
Q1800) Solve 12x^2+49x+15=0
Ans: x=-0.33 or x=-3.75
Q1801) Solve 8x^2+25x+18=0
Ans: x=-1.12 or x=-2
Q1802) Solve 3x^2+26x+23=0
Ans: x=-1 or x=-7.67
Q1803) Solve 7x^2+37x+10=0
Ans: x=-0.29 or x=-5
Q1804) Solve 24x^2+49x+22=0
Ans: x=-0.67 or x=-1.37
Q1805) Solve 46x^2+49x+13=0
Ans: x=-0.5 or x=-0.57
Q1806) Solve 12x^2+34x+24=0
Ans: x=-1.33 or x=-1.5
Q1807) Solve 11x^2+41x+24=0
Ans: x=-0.73 or x=-3
Q1808) Solve 2x^2+22x+48=0
Ans: x=-3 or x=-8
Q1809) Solve 10x^2+21x+9=0
Ans: x=-0.6 or x=-1.5
Q1810) Solve 16x^2+32x+7=0
Ans: x=-0.25 or x=-1.75
Q1811) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q1812) Solve 38x^2+40x+2=0
Ans: x=-0.05 or x=-1
Q1813) Solve 3x^2+20x+33=0
Ans: x=-3 or x=-3.67
Q1814) Solve 8x^2+36x+36=0
Ans: x=-1.5 or x=-3
Q1815) Solve 15x^2+33x+18=0
Ans: x=-1 or x=-1.2
Q1816) Solve 5x^2+16x+12=0
Ans: x=-1.2 or x=-2
Q1817) Solve 12x^2+32x+13=0
Ans: x=-0.5 or x=-2.17
Q1818) Solve 29x^2+32x+3=0
Ans: x=-0.1 or x=-1
Q1819) Solve 8x^2+30x+7=0
Ans: x=-0.25 or x=-3.5
Q1820) Solve 10x^2+21x+11=0
Ans: x=-1 or x=-1.1
Q1821) Solve x^2+19x+18=0
Ans: x=-1 or x=-18
Q1822) Solve 6x^2+37x+6=0
Ans: x=-0.17 or x=-6
Q1823) Solve 6x^2+19x+13=0
Ans: x=-1 or x=-2.17
Q1824) Solve 11x^2+42x+40=0
Ans: x=-1.82 or x=-2
Q1825) Solve 27x^2+12x+1=0
Ans: x=-0.11 or x=-0.33
Q1826) Solve 7x^2+35x+42=0
Ans: x=-2 or x=-3
Q1827) Solve 48x^2+46x+10=0
Ans: x=-0.33 or x=-0.62
Q1828) Solve 40x^2+41x+10=0
Ans: x=-0.4 or x=-0.62
Q1829) Solve 15x^2+21x+6=0
Ans: x=-0.4 or x=-1
Q1830) Solve 27x^2+15x+2=0
Ans: x=-0.22 or x=-0.33
Q1831) Solve 10x^2+50x+40=0
Ans: x=-1 or x=-4
Q1832) Solve 8x^2+39x+46=0
Ans: x=-2 or x=-2.87
Q1833) Solve 33x^2+46x+13=0
Ans: x=-0.39 or x=-1
Q1834) Solve 43x^2+46x+3=0
Ans: x=-0.07 or x=-1
Q1835) Solve 26x^2+27x+7=0
Ans: x=-0.5 or x=-0.54
Q1836) Solve 30x^2+49x+17=0
Ans: x=-0.5 or x=-1.13
Q1837) Solve 18x^2+9x+1=0
Ans: x=-0.17 or x=-0.33
Q1838) Solve 8x^2+40x+18=0
Ans: x=-0.5 or x=-4.5
Q1839) Solve 10x^2+42x+36=0
Ans: x=-1.2 or x=-3
Q1840) Solve 3x^2+38x+24=0
Ans: x=-0.67 or x=-12
Q1841) Solve 30x^2+38x+8=0
Ans: x=-0.27 or x=-1
Q1842) Solve 40x^2+46x+6=0
Ans: x=-0.15 or x=-1
Q1843) Solve 4x^2+22x+10=0
Ans: x=-0.5 or x=-5
Q1844) Solve 6x^2+19x+3=0
Ans: x=-0.17 or x=-3
Q1845) Solve 31x^2+46x+15=0
Ans: x=-0.48 or x=-1
Q1846) Solve 9x^2+39x+42=0
Ans: x=-2 or x=-2.33
Q1847) Solve 29x^2+41x+12=0
Ans: x=-0.41 or x=-1
Q1848) Solve 38x^2+46x+8=0
Ans: x=-0.21 or x=-1
Q1849) Solve 38x^2+50x+12=0
Ans: x=-0.32 or x=-1
Q1850) Solve 22x^2+49x+10=0
Ans: x=-0.23 or x=-2
Q1851) Solve 8x^2+18x+10=0
Ans: x=-1 or x=-1.25
Q1852) Solve 49x^2+28x+4=0
Ans: x=-0.29 or x=-0.29
Q1853) Solve 10x^2+33x+14=0
Ans: x=-0.5 or x=-2.8
Q1854) Solve 20x^2+9x+1=0
Ans: x=-0.2 or x=-0.25
Q1855) Solve 19x^2+50x+24=0
Ans: x=-0.63 or x=-2
Q1856) Solve 13x^2+36x+20=0
Ans: x=-0.77 or x=-2
Q1857) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q1858) Solve 4x^2+32x+28=0
Ans: x=-1 or x=-7
Q1859) Solve 14x^2+49x+35=0
Ans: x=-1 or x=-2.5
Q1860) Solve 21x^2+27x+6=0
Ans: x=-0.29 or x=-1
Q1861) Solve 5x^2+22x+8=0
Ans: x=-0.4 or x=-4
Q1862) Solve 9x^2+12x+3=0
Ans: x=-0.33 or x=-1
Q1863) Solve 10x^2+41x+18=0
Ans: x=-0.5 or x=-3.6
Q1864) Solve 2x^2+9x+4=0
Ans: x=-0.5 or x=-4
Q1865) Solve 3x^2+11x+10=0
Ans: x=-1.67 or x=-2
Q1866) Solve 3x^2+20x+33=0
Ans: x=-3 or x=-3.67
Q1867) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q1868) Solve 7x^2+43x+36=0
Ans: x=-1 or x=-5.14
Q1869) Solve 3x^2+27x+24=0
Ans: x=-1 or x=-8
Q1870) Solve 8x^2+46x+11=0
Ans: x=-0.25 or x=-5.5
Q1871) Solve 9x^2+24x+7=0
Ans: x=-0.33 or x=-2.33
Q1872) Solve 27x^2+38x+11=0
Ans: x=-0.41 or x=-1
Q1873) Solve x^2+18x+45=0
Ans: x=-3 or x=-15
Q1874) Solve 8x^2+18x+9=0
Ans: x=-0.75 or x=-1.5
Q1875) Solve 20x^2+48x+19=0
Ans: x=-0.5 or x=-1.9
Q1876) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q1877) Solve 42x^2+31x+5=0
Ans: x=-0.24 or x=-0.5
Q1878) Solve 6x^2+25x+24=0
Ans: x=-1.5 or x=-2.67
Q1879) Solve 12x^2+40x+25=0
Ans: x=-0.83 or x=-2.5
Q1880) Solve 6x^2+19x+13=0
Ans: x=-1 or x=-2.17
Q1881) Solve 6x^2+49x+43=0
Ans: x=-1 or x=-7.17
Q1882) Solve 8x^2+34x+8=0
Ans: x=-0.25 or x=-4
Q1883) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q1884) Solve 16x^2+46x+19=0
Ans: x=-0.5 or x=-2.37
Q1885) Solve 44x^2+41x+6=0
Ans: x=-0.18 or x=-0.75
Q1886) Solve 12x^2+38x+28=0
Ans: x=-1.17 or x=-2
Q1887) Solve 4x^2+22x+18=0
Ans: x=-1 or x=-4.5
Q1888) Solve 3x^2+35x+22=0
Ans: x=-0.67 or x=-11
Q1889) Solve 3x^2+31x+50=0
Ans: x=-2 or x=-8.33
Q1890) Solve 19x^2+39x+2=0
Ans: x=-0.05 or x=-2
Q1891) Solve 13x^2+50x+37=0
Ans: x=-1 or x=-2.85
Q1892) Solve 42x^2+50x+8=0
Ans: x=-0.19 or x=-1
Q1893) Solve 6x^2+23x+7=0
Ans: x=-0.33 or x=-3.5
Q1894) Solve 28x^2+44x+15=0
Ans: x=-0.5 or x=-1.07
Q1895) Solve 18x^2+27x+4=0
Ans: x=-0.17 or x=-1.33
Q1896) Solve 7x^2+31x+30=0
Ans: x=-1.43 or x=-3
Q1897) Solve 5x^2+30x+40=0
Ans: x=-2 or x=-4
Q1898) Solve 4x^2+32x+28=0
Ans: x=-1 or x=-7
Q1899) Solve 23x^2+34x+11=0
Ans: x=-0.48 or x=-1
Q1900) Solve 11x^2+26x+8=0
Ans: x=-0.36 or x=-2
Q1901) Solve 8x^2+30x+22=0
Ans: x=-1 or x=-2.75
Q1902) Solve 3x^2+7x+2=0
Ans: x=-0.33 or x=-2
Q1903) Solve 30x^2+41x+14=0
Ans: x=-0.67 or x=-0.7
Q1904) Solve 50x^2+45x+4=0
Ans: x=-0.1 or x=-0.8
Q1905) Solve 21x^2+31x+10=0
Ans: x=-0.48 or x=-1
Q1906) Solve 22x^2+13x+1=0
Ans: x=-0.09 or x=-0.5
Q1907) Solve 10x^2+47x+9=0
Ans: x=-0.2 or x=-4.5
Q1908) Solve 6x^2+37x+35=0
Ans: x=-1.17 or x=-5
Q1909) Solve 18x^2+35x+17=0
Ans: x=-0.94 or x=-1
Q1910) Solve 12x^2+46x+40=0
Ans: x=-1.33 or x=-2.5
Q1911) Solve x^2+40x+39=0
Ans: x=-1 or x=-39
Q1912) Solve 5x^2+21x+22=0
Ans: x=-2 or x=-2.2
Q1913) Solve 12x^2+37x+28=0
Ans: x=-1.33 or x=-1.75
Q1914) Solve 6x^2+27x+27=0
Ans: x=-1.5 or x=-3
Q1915) Solve 18x^2+21x+5=0
Ans: x=-0.33 or x=-0.83
Q1916) Solve 18x^2+48x+24=0
Ans: x=-0.67 or x=-2
Q1917) Solve 13x^2+38x+24=0
Ans: x=-0.92 or x=-2
Q1918) Solve 45x^2+47x+2=0
Ans: x=-0.04 or x=-1
Q1919) Solve 26x^2+45x+9=0
Ans: x=-0.23 or x=-1.5
Q1920) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q1921) Solve 7x^2+29x+24=0
Ans: x=-1.14 or x=-3
Q1922) Solve 8x^2+19x+11=0
Ans: x=-1 or x=-1.37
Q1923) Solve 6x^2+25x+21=0
Ans: x=-1.17 or x=-3
Q1924) Solve 19x^2+40x+21=0
Ans: x=-1 or x=-1.11
Q1925) Solve 4x^2+24x+11=0
Ans: x=-0.5 or x=-5.5
Q1926) Solve 10x^2+19x+9=0
Ans: x=-0.9 or x=-1
Q1927) Solve 4x^2+43x+30=0
Ans: x=-0.75 or x=-10
Q1928) Solve 15x^2+30x+15=0
Ans: x=-1 or x=-1
Q1929) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q1930) Solve 10x^2+39x+35=0
Ans: x=-1.4 or x=-2.5
Q1931) Solve 18x^2+49x+20=0
Ans: x=-0.5 or x=-2.22
Q1932) Solve 16x^2+49x+34=0
Ans: x=-1.06 or x=-2
Q1933) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q1934) Solve 36x^2+24x+3=0
Ans: x=-0.17 or x=-0.5
Q1935) Solve x^2+38x+37=0
Ans: x=-1 or x=-37
Q1936) Solve 44x^2+27x+4=0
Ans: x=-0.25 or x=-0.36
Q1937) Solve 48x^2+25x+3=0
Ans: x=-0.19 or x=-0.33
Q1938) Solve x^2+11x+18=0
Ans: x=-2 or x=-9
Q1939) Solve 33x^2+31x+6=0
Ans: x=-0.27 or x=-0.67
Q1940) Solve 23x^2+45x+22=0
Ans: x=-0.96 or x=-1
Q1941) Solve 24x^2+38x+8=0
Ans: x=-0.25 or x=-1.33
Q1942) Solve 12x^2+12x+3=0
Ans: x=-0.5 or x=-0.5
Q1943) Solve 10x^2+22x+12=0
Ans: x=-1 or x=-1.2
Q1944) Solve 5x^2+13x+6=0
Ans: x=-0.6 or x=-2
Q1945) Solve x^2+15x+14=0
Ans: x=-1 or x=-14
Q1946) Solve 11x^2+42x+31=0
Ans: x=-1 or x=-2.82
Q1947) Solve 11x^2+28x+12=0
Ans: x=-0.55 or x=-2
Q1948) Solve 12x^2+23x+10=0
Ans: x=-0.67 or x=-1.25
Q1949) Solve 26x^2+44x+18=0
Ans: x=-0.69 or x=-1
Q1950) Solve 15x^2+46x+31=0
Ans: x=-1 or x=-2.07
Q1951) Solve 2x^2+25x+50=0
Ans: x=-2.5 or x=-10
Q1952) Solve 35x^2+37x+2=0
Ans: x=-0.06 or x=-1
Q1953) Solve 6x^2+21x+15=0
Ans: x=-1 or x=-2.5
Q1954) Solve 26x^2+28x+2=0
Ans: x=-0.08 or x=-1
Q1955) Solve 26x^2+37x+12=0
Ans: x=-0.5 or x=-0.92
Q1956) Solve 24x^2+36x+12=0
Ans: x=-0.5 or x=-1
Q1957) Solve 41x^2+46x+5=0
Ans: x=-0.12 or x=-1
Q1958) Solve 10x^2+33x+20=0
Ans: x=-0.8 or x=-2.5
Q1959) Solve 12x^2+38x+30=0
Ans: x=-1.5 or x=-1.67
Q1960) Solve 48x^2+28x+2=0
Ans: x=-0.08 or x=-0.5
Q1961) Solve 28x^2+48x+17=0
Ans: x=-0.5 or x=-1.21
Q1962) Solve 29x^2+39x+10=0
Ans: x=-0.34 or x=-1
Q1963) Solve 2x^2+27x+13=0
Ans: x=-0.5 or x=-13
Q1964) Solve 18x^2+44x+16=0
Ans: x=-0.44 or x=-2
Q1965) Solve 15x^2+43x+22=0
Ans: x=-0.67 or x=-2.2
Q1966) Solve 22x^2+50x+12=0
Ans: x=-0.27 or x=-2
Q1967) Solve 3x^2+45x+42=0
Ans: x=-1 or x=-14
Q1968) Solve 10x^2+43x+42=0
Ans: x=-1.5 or x=-2.8
Q1969) Solve 5x^2+29x+24=0
Ans: x=-1 or x=-4.8
Q1970) Solve 31x^2+46x+15=0
Ans: x=-0.48 or x=-1
Q1971) Solve 18x^2+29x+11=0
Ans: x=-0.61 or x=-1
Q1972) Solve 30x^2+48x+18=0
Ans: x=-0.6 or x=-1
Q1973) Solve 39x^2+25x+4=0
Ans: x=-0.31 or x=-0.33
Q1974) Solve 15x^2+17x+2=0
Ans: x=-0.13 or x=-1
Q1975) Solve 35x^2+38x+3=0
Ans: x=-0.09 or x=-1
Q1976) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q1977) Solve 5x^2+38x+48=0
Ans: x=-1.6 or x=-6
Q1978) Solve 18x^2+45x+27=0
Ans: x=-1 or x=-1.5
Q1979) Solve x^2+14x+13=0
Ans: x=-1 or x=-13
Q1980) Solve 20x^2+49x+9=0
Ans: x=-0.2 or x=-2.25
Q1981) Solve 33x^2+25x+2=0
Ans: x=-0.09 or x=-0.67
Q1982) Solve 21x^2+50x+29=0
Ans: x=-1 or x=-1.38
Q1983) Solve 2x^2+5x+3=0
Ans: x=-1 or x=-1.5
Q1984) Solve 15x^2+18x+3=0
Ans: x=-0.2 or x=-1
Q1985) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q1986) Solve 4x^2+20x+9=0
Ans: x=-0.5 or x=-4.5
Q1987) Solve 20x^2+47x+14=0
Ans: x=-0.35 or x=-2
Q1988) Solve 16x^2+46x+30=0
Ans: x=-1 or x=-1.87
Q1989) Solve 4x^2+26x+30=0
Ans: x=-1.5 or x=-5
Q1990) Solve 2x^2+9x+7=0
Ans: x=-1 or x=-3.5
Q1991) Solve 33x^2+17x+2=0
Ans: x=-0.18 or x=-0.33
Q1992) Solve 8x^2+32x+32=0
Ans: x=-2 or x=-2
Q1993) Solve 11x^2+50x+24=0
Ans: x=-0.55 or x=-4
Q1994) Solve 24x^2+44x+20=0
Ans: x=-0.83 or x=-1
Q1995) Solve 6x^2+19x+3=0
Ans: x=-0.17 or x=-3
Q1996) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q1997) Solve 28x^2+38x+10=0
Ans: x=-0.36 or x=-1
Q1998) Solve 45x^2+42x+8=0
Ans: x=-0.27 or x=-0.67
Q1999) Solve 5x^2+31x+6=0
Ans: x=-0.2 or x=-6
Q2000) Solve 25x^2+42x+17=0
Ans: x=-0.68 or x=-1
Q2001) Solve 8x^2+19x+6=0
Ans: x=-0.37 or x=-2
Q2002) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q2003) Solve 13x^2+32x+19=0
Ans: x=-1 or x=-1.46
Q2004) Solve 9x^2+33x+30=0
Ans: x=-1.67 or x=-2
Q2005) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q2006) Solve 12x^2+13x+3=0
Ans: x=-0.33 or x=-0.75
Q2007) Solve 6x^2+30x+36=0
Ans: x=-2 or x=-3
Q2008) Solve 4x^2+18x+8=0
Ans: x=-0.5 or x=-4
Q2009) Solve 8x^2+28x+20=0
Ans: x=-1 or x=-2.5
Q2010) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q2011) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q2012) Solve 33x^2+48x+15=0
Ans: x=-0.45 or x=-1
Q2013) Solve 11x^2+47x+12=0
Ans: x=-0.27 or x=-4
Q2014) Solve 28x^2+48x+9=0
Ans: x=-0.21 or x=-1.5
Q2015) Solve 6x^2+20x+16=0
Ans: x=-1.33 or x=-2
Q2016) Solve 10x^2+35x+15=0
Ans: x=-0.5 or x=-3
Q2017) Solve 6x^2+29x+34=0
Ans: x=-2 or x=-2.83
Q2018) Solve 4x^2+19x+21=0
Ans: x=-1.75 or x=-3
Q2019) Solve 40x^2+13x+1=0
Ans: x=-0.12 or x=-0.2
Q2020) Solve x^2+15x+50=0
Ans: x=-5 or x=-10
Q2021) Solve 40x^2+31x+6=0
Ans: x=-0.37 or x=-0.4
Q2022) Solve 6x^2+20x+16=0
Ans: x=-1.33 or x=-2
Q2023) Solve 19x^2+23x+4=0
Ans: x=-0.21 or x=-1
Q2024) Solve 25x^2+49x+24=0
Ans: x=-0.96 or x=-1
Q2025) Solve 8x^2+32x+14=0
Ans: x=-0.5 or x=-3.5
Q2026) Solve 3x^2+25x+48=0
Ans: x=-3 or x=-5.33
Q2027) Solve 44x^2+28x+3=0
Ans: x=-0.14 or x=-0.5
Q2028) Solve 6x^2+47x+15=0
Ans: x=-0.33 or x=-7.5
Q2029) Solve 5x^2+37x+32=0
Ans: x=-1 or x=-6.4
Q2030) Solve 4x^2+34x+16=0
Ans: x=-0.5 or x=-8
Q2031) Solve 6x^2+31x+14=0
Ans: x=-0.5 or x=-4.67
Q2032) Solve 11x^2+40x+21=0
Ans: x=-0.64 or x=-3
Q2033) Solve 18x^2+45x+25=0
Ans: x=-0.83 or x=-1.67
Q2034) Solve 12x^2+48x+45=0
Ans: x=-1.5 or x=-2.5
Q2035) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q2036) Solve 15x^2+50x+15=0
Ans: x=-0.33 or x=-3
Q2037) Solve 17x^2+50x+33=0
Ans: x=-1 or x=-1.94
Q2038) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q2039) Solve 12x^2+42x+30=0
Ans: x=-1 or x=-2.5
Q2040) Solve 4x^2+39x+27=0
Ans: x=-0.75 or x=-9
Q2041) Solve 28x^2+11x+1=0
Ans: x=-0.14 or x=-0.25
Q2042) Solve 49x^2+28x+4=0
Ans: x=-0.29 or x=-0.29
Q2043) Solve 35x^2+37x+6=0
Ans: x=-0.2 or x=-0.86
Q2044) Solve 50x^2+30x+4=0
Ans: x=-0.2 or x=-0.4
Q2045) Solve 4x^2+30x+26=0
Ans: x=-1 or x=-6.5
Q2046) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q2047) Solve x^2+14x+33=0
Ans: x=-3 or x=-11
Q2048) Solve 9x^2+21x+10=0
Ans: x=-0.67 or x=-1.67
Q2049) Solve 35x^2+38x+3=0
Ans: x=-0.09 or x=-1
Q2050) Solve 8x^2+42x+19=0
Ans: x=-0.5 or x=-4.75
Q2051) Solve 30x^2+41x+13=0
Ans: x=-0.5 or x=-0.87
Q2052) Solve 11x^2+45x+4=0
Ans: x=-0.09 or x=-4
Q2053) Solve 21x^2+40x+16=0
Ans: x=-0.57 or x=-1.33
Q2054) Solve 2x^2+29x+14=0
Ans: x=-0.5 or x=-14
Q2055) Solve 20x^2+38x+14=0
Ans: x=-0.5 or x=-1.4
Q2056) Solve 20x^2+14x+2=0
Ans: x=-0.2 or x=-0.5
Q2057) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q2058) Solve 9x^2+41x+20=0
Ans: x=-0.56 or x=-4
Q2059) Solve 9x^2+12x+3=0
Ans: x=-0.33 or x=-1
Q2060) Solve 9x^2+41x+46=0
Ans: x=-2 or x=-2.56
Q2061) Solve 5x^2+43x+24=0
Ans: x=-0.6 or x=-8
Q2062) Solve 10x^2+32x+6=0
Ans: x=-0.2 or x=-3
Q2063) Solve 25x^2+50x+21=0
Ans: x=-0.6 or x=-1.4
Q2064) Solve 15x^2+46x+31=0
Ans: x=-1 or x=-2.07
Q2065) Solve 13x^2+40x+3=0
Ans: x=-0.08 or x=-3
Q2066) Solve 12x^2+46x+44=0
Ans: x=-1.83 or x=-2
Q2067) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q2068) Solve 33x^2+37x+4=0
Ans: x=-0.12 or x=-1
Q2069) Solve 48x^2+46x+5=0
Ans: x=-0.12 or x=-0.83
Q2070) Solve 8x^2+24x+18=0
Ans: x=-1.5 or x=-1.5
Q2071) Solve 3x^2+8x+5=0
Ans: x=-1 or x=-1.67
Q2072) Solve 21x^2+46x+13=0
Ans: x=-0.33 or x=-1.86
Q2073) Solve 10x^2+11x+1=0
Ans: x=-0.1 or x=-1
Q2074) Solve 15x^2+47x+28=0
Ans: x=-0.8 or x=-2.33
Q2075) Solve 38x^2+23x+2=0
Ans: x=-0.11 or x=-0.5
Q2076) Solve 7x^2+30x+32=0
Ans: x=-2 or x=-2.29
Q2077) Solve x^2+10x+21=0
Ans: x=-3 or x=-7
Q2078) Solve 15x^2+41x+26=0
Ans: x=-1 or x=-1.73
Q2079) Solve 17x^2+34x+17=0
Ans: x=-1 or x=-1
Q2080) Solve 25x^2+42x+17=0
Ans: x=-0.68 or x=-1
Q2081) Solve 13x^2+35x+22=0
Ans: x=-1 or x=-1.69
Q2082) Solve 42x^2+47x+5=0
Ans: x=-0.12 or x=-1
Q2083) Solve 31x^2+38x+7=0
Ans: x=-0.23 or x=-1
Q2084) Solve 2x^2+32x+30=0
Ans: x=-1 or x=-15
Q2085) Solve 5x^2+26x+5=0
Ans: x=-0.2 or x=-5
Q2086) Solve 21x^2+29x+8=0
Ans: x=-0.38 or x=-1
Q2087) Solve 13x^2+33x+14=0
Ans: x=-0.54 or x=-2
Q2088) Solve 16x^2+49x+33=0
Ans: x=-1 or x=-2.06
Q2089) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q2090) Solve 42x^2+32x+6=0
Ans: x=-0.33 or x=-0.43
Q2091) Solve 11x^2+48x+16=0
Ans: x=-0.36 or x=-4
Q2092) Solve 12x^2+26x+4=0
Ans: x=-0.17 or x=-2
Q2093) Solve 3x^2+8x+5=0
Ans: x=-1 or x=-1.67
Q2094) Solve 44x^2+39x+7=0
Ans: x=-0.25 or x=-0.64
Q2095) Solve 49x^2+49x+6=0
Ans: x=-0.14 or x=-0.86
Q2096) Solve 13x^2+48x+44=0
Ans: x=-1.69 or x=-2
Q2097) Solve 11x^2+38x+15=0
Ans: x=-0.45 or x=-3
Q2098) Solve 35x^2+31x+6=0
Ans: x=-0.29 or x=-0.6
Q2099) Solve 36x^2+36x+5=0
Ans: x=-0.17 or x=-0.83
Q2100) Solve 7x^2+32x+33=0
Ans: x=-1.57 or x=-3
Q2101) Solve 20x^2+36x+16=0
Ans: x=-0.8 or x=-1
Q2102) Solve 6x^2+11x+3=0
Ans: x=-0.33 or x=-1.5
Q2103) Solve 14x^2+46x+12=0
Ans: x=-0.29 or x=-3
Q2104) Solve 14x^2+15x+1=0
Ans: x=-0.07 or x=-1
Q2105) Solve 4x^2+35x+31=0
Ans: x=-1 or x=-7.75
Q2106) Solve 2x^2+7x+5=0
Ans: x=-1 or x=-2.5
Q2107) Solve 11x^2+38x+32=0
Ans: x=-1.45 or x=-2
Q2108) Solve 26x^2+35x+9=0
Ans: x=-0.35 or x=-1
Q2109) Solve 6x^2+12x+6=0
Ans: x=-1 or x=-1
Q2110) Solve 3x^2+8x+5=0
Ans: x=-1 or x=-1.67
Q2111) Solve 9x^2+38x+33=0
Ans: x=-1.22 or x=-3
Q2112) Solve 2x^2+9x+7=0
Ans: x=-1 or x=-3.5
Q2113) Solve 9x^2+48x+48=0
Ans: x=-1.33 or x=-4
Q2114) Solve 10x^2+14x+4=0
Ans: x=-0.4 or x=-1
Q2115) Solve 3x^2+42x+39=0
Ans: x=-1 or x=-13
Q2116) Solve 28x^2+24x+5=0
Ans: x=-0.36 or x=-0.5
Q2117) Solve 10x^2+41x+33=0
Ans: x=-1.1 or x=-3
Q2118) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q2119) Solve 40x^2+41x+1=0
Ans: x=-0.02 or x=-1
Q2120) Solve 22x^2+48x+26=0
Ans: x=-1 or x=-1.18
Q2121) Solve 10x^2+18x+8=0
Ans: x=-0.8 or x=-1
Q2122) Solve 35x^2+17x+2=0
Ans: x=-0.2 or x=-0.29
Q2123) Solve 26x^2+39x+13=0
Ans: x=-0.5 or x=-1
Q2124) Solve 7x^2+45x+38=0
Ans: x=-1 or x=-5.43
Q2125) Solve 6x^2+27x+21=0
Ans: x=-1 or x=-3.5
Q2126) Solve 33x^2+45x+12=0
Ans: x=-0.36 or x=-1
Q2127) Solve 40x^2+42x+8=0
Ans: x=-0.25 or x=-0.8
Q2128) Solve 28x^2+33x+9=0
Ans: x=-0.43 or x=-0.75
Q2129) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q2130) Solve 2x^2+13x+6=0
Ans: x=-0.5 or x=-6
Q2131) Solve 18x^2+21x+6=0
Ans: x=-0.5 or x=-0.67
Q2132) Solve 4x^2+17x+13=0
Ans: x=-1 or x=-3.25
Q2133) Solve 18x^2+25x+7=0
Ans: x=-0.39 or x=-1
Q2134) Solve 10x^2+36x+32=0
Ans: x=-1.6 or x=-2
Q2135) Solve 16x^2+48x+36=0
Ans: x=-1.5 or x=-1.5
Q2136) Solve 24x^2+45x+21=0
Ans: x=-0.87 or x=-1
Q2137) Solve 3x^2+17x+14=0
Ans: x=-1 or x=-4.67
Q2138) Solve 13x^2+26x+13=0
Ans: x=-1 or x=-1
Q2139) Solve 36x^2+18x+2=0
Ans: x=-0.17 or x=-0.33
Q2140) Solve 4x^2+11x+6=0
Ans: x=-0.75 or x=-2
Q2141) Solve 16x^2+30x+14=0
Ans: x=-0.87 or x=-1
Q2142) Solve 6x^2+23x+7=0
Ans: x=-0.33 or x=-3.5
Q2143) Solve 3x^2+22x+7=0
Ans: x=-0.33 or x=-7
Q2144) Solve 9x^2+24x+12=0
Ans: x=-0.67 or x=-2
Q2145) Solve 28x^2+34x+10=0
Ans: x=-0.5 or x=-0.71
Q2146) Solve 11x^2+37x+26=0
Ans: x=-1 or x=-2.36
Q2147) Solve 30x^2+11x+1=0
Ans: x=-0.17 or x=-0.2
Q2148) Solve 9x^2+41x+20=0
Ans: x=-0.56 or x=-4
Q2149) Solve 6x^2+34x+48=0
Ans: x=-2.67 or x=-3
Q2150) Solve 7x^2+23x+6=0
Ans: x=-0.29 or x=-3
Q2151) Solve 48x^2+36x+6=0
Ans: x=-0.25 or x=-0.5
Q2152) Solve 49x^2+49x+12=0
Ans: x=-0.43 or x=-0.57
Q2153) Solve 18x^2+41x+21=0
Ans: x=-0.78 or x=-1.5
Q2154) Solve 15x^2+13x+2=0
Ans: x=-0.2 or x=-0.67
Q2155) Solve 5x^2+6x+1=0
Ans: x=-0.2 or x=-1
Q2156) Solve 16x^2+35x+19=0
Ans: x=-1 or x=-1.19
Q2157) Solve 47x^2+49x+2=0
Ans: x=-0.04 or x=-1
Q2158) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q2159) Solve 8x^2+46x+45=0
Ans: x=-1.25 or x=-4.5
Q2160) Solve 22x^2+45x+23=0
Ans: x=-1 or x=-1.05
Q2161) Solve x^2+14x+49=0
Ans: x=-7 or x=-7
Q2162) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q2163) Solve 16x^2+46x+28=0
Ans: x=-0.87 or x=-2
Q2164) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q2165) Solve 9x^2+28x+3=0
Ans: x=-0.11 or x=-3
Q2166) Solve 27x^2+18x+3=0
Ans: x=-0.33 or x=-0.33
Q2167) Solve 9x^2+22x+13=0
Ans: x=-1 or x=-1.44
Q2168) Solve 12x^2+32x+20=0
Ans: x=-1 or x=-1.67
Q2169) Solve 3x^2+26x+35=0
Ans: x=-1.67 or x=-7
Q2170) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q2171) Solve 33x^2+25x+2=0
Ans: x=-0.09 or x=-0.67
Q2172) Solve 45x^2+46x+1=0
Ans: x=-0.02 or x=-1
Q2173) Solve 12x^2+28x+15=0
Ans: x=-0.83 or x=-1.5
Q2174) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q2175) Solve 10x^2+39x+38=0
Ans: x=-1.9 or x=-2
Q2176) Solve 20x^2+33x+13=0
Ans: x=-0.65 or x=-1
Q2177) Solve 28x^2+47x+15=0
Ans: x=-0.43 or x=-1.25
Q2178) Solve 30x^2+13x+1=0
Ans: x=-0.1 or x=-0.33
Q2179) Solve 9x^2+12x+4=0
Ans: x=-0.67 or x=-0.67
Q2180) Solve 42x^2+41x+10=0
Ans: x=-0.48 or x=-0.5
Q2181) Solve 18x^2+35x+17=0
Ans: x=-0.94 or x=-1
Q2182) Solve x^2+21x+20=0
Ans: x=-1 or x=-20
Q2183) Solve 14x^2+19x+6=0
Ans: x=-0.5 or x=-0.86
Q2184) Solve 15x^2+46x+3=0
Ans: x=-0.07 or x=-3
Q2185) Solve 24x^2+50x+11=0
Ans: x=-0.25 or x=-1.83
Q2186) Solve 8x^2+17x+2=0
Ans: x=-0.12 or x=-2
Q2187) Solve 30x^2+31x+8=0
Ans: x=-0.5 or x=-0.53
Q2188) Solve 45x^2+42x+8=0
Ans: x=-0.27 or x=-0.67
Q2189) Solve 15x^2+36x+21=0
Ans: x=-1 or x=-1.4
Q2190) Solve 22x^2+45x+2=0
Ans: x=-0.05 or x=-2
Q2191) Solve 3x^2+21x+36=0
Ans: x=-3 or x=-4
Q2192) Solve 34x^2+27x+5=0
Ans: x=-0.29 or x=-0.5
Q2193) Solve 10x^2+41x+40=0
Ans: x=-1.6 or x=-2.5
Q2194) Solve 26x^2+40x+14=0
Ans: x=-0.54 or x=-1
Q2195) Solve 35x^2+48x+13=0
Ans: x=-0.37 or x=-1
Q2196) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q2197) Solve 31x^2+40x+9=0
Ans: x=-0.29 or x=-1
Q2198) Solve 12x^2+28x+8=0
Ans: x=-0.33 or x=-2
Q2199) Solve 8x^2+44x+20=0
Ans: x=-0.5 or x=-5
Q2200) Solve 25x^2+45x+20=0
Ans: x=-0.8 or x=-1
Q2201) Solve 40x^2+49x+9=0
Ans: x=-0.22 or x=-1
Q2202) Solve 40x^2+26x+3=0
Ans: x=-0.15 or x=-0.5
Q2203) Solve 10x^2+29x+12=0
Ans: x=-0.5 or x=-2.4
Q2204) Solve 18x^2+28x+10=0
Ans: x=-0.56 or x=-1
Q2205) Solve 5x^2+32x+27=0
Ans: x=-1 or x=-5.4
Q2206) Solve 44x^2+48x+13=0
Ans: x=-0.5 or x=-0.59
Q2207) Solve 18x^2+41x+21=0
Ans: x=-0.78 or x=-1.5
Q2208) Solve 28x^2+31x+3=0
Ans: x=-0.11 or x=-1
Q2209) Solve 20x^2+9x+1=0
Ans: x=-0.2 or x=-0.25
Q2210) Solve 2x^2+5x+3=0
Ans: x=-1 or x=-1.5
Q2211) Solve 20x^2+13x+2=0
Ans: x=-0.25 or x=-0.4
Q2212) Solve 9x^2+42x+33=0
Ans: x=-1 or x=-3.67
Q2213) Solve 2x^2+13x+15=0
Ans: x=-1.5 or x=-5
Q2214) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q2215) Solve 48x^2+26x+3=0
Ans: x=-0.17 or x=-0.37
Q2216) Solve 43x^2+48x+5=0
Ans: x=-0.12 or x=-1
Q2217) Solve 22x^2+49x+24=0
Ans: x=-0.73 or x=-1.5
Q2218) Solve 25x^2+30x+8=0
Ans: x=-0.4 or x=-0.8
Q2219) Solve 12x^2+49x+44=0
Ans: x=-1.33 or x=-2.75
Q2220) Solve 40x^2+22x+1=0
Ans: x=-0.05 or x=-0.5
Q2221) Solve 11x^2+48x+45=0
Ans: x=-1.36 or x=-3
Q2222) Solve 48x^2+35x+2=0
Ans: x=-0.06 or x=-0.67
Q2223) Solve 24x^2+11x+1=0
Ans: x=-0.12 or x=-0.33
Q2224) Solve 27x^2+30x+8=0
Ans: x=-0.44 or x=-0.67
Q2225) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q2226) Solve 9x^2+36x+27=0
Ans: x=-1 or x=-3
Q2227) Solve 24x^2+48x+18=0
Ans: x=-0.5 or x=-1.5
Q2228) Solve 2x^2+27x+46=0
Ans: x=-2 or x=-11.5
Q2229) Solve 18x^2+39x+21=0
Ans: x=-1 or x=-1.17
Q2230) Solve 40x^2+13x+1=0
Ans: x=-0.12 or x=-0.2
Q2231) Solve 9x^2+15x+6=0
Ans: x=-0.67 or x=-1
Q2232) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q2233) Solve x^2+14x+48=0
Ans: x=-6 or x=-8
Q2234) Solve 45x^2+30x+5=0
Ans: x=-0.33 or x=-0.33
Q2235) Solve 50x^2+40x+8=0
Ans: x=-0.4 or x=-0.4
Q2236) Solve 25x^2+28x+3=0
Ans: x=-0.12 or x=-1
Q2237) Solve 14x^2+45x+9=0
Ans: x=-0.21 or x=-3
Q2238) Solve 2x^2+37x+18=0
Ans: x=-0.5 or x=-18
Q2239) Solve 24x^2+46x+20=0
Ans: x=-0.67 or x=-1.25
Q2240) Solve 8x^2+37x+20=0
Ans: x=-0.62 or x=-4
Q2241) Solve 20x^2+28x+9=0
Ans: x=-0.5 or x=-0.9
Q2242) Solve 4x^2+30x+26=0
Ans: x=-1 or x=-6.5
Q2243) Solve 4x^2+26x+36=0
Ans: x=-2 or x=-4.5
Q2244) Solve 14x^2+33x+19=0
Ans: x=-1 or x=-1.36
Q2245) Solve 9x^2+47x+38=0
Ans: x=-1 or x=-4.22
Q2246) Solve 44x^2+24x+1=0
Ans: x=-0.05 or x=-0.5
Q2247) Solve 28x^2+38x+10=0
Ans: x=-0.36 or x=-1
Q2248) Solve 16x^2+14x+3=0
Ans: x=-0.37 or x=-0.5
Q2249) Solve 4x^2+20x+25=0
Ans: x=-2.5 or x=-2.5
Q2250) Solve 38x^2+45x+7=0
Ans: x=-0.18 or x=-1
Q2251) Solve 39x^2+50x+16=0
Ans: x=-0.62 or x=-0.67
Q2252) Solve 32x^2+48x+10=0
Ans: x=-0.25 or x=-1.25
Q2253) Solve 16x^2+35x+19=0
Ans: x=-1 or x=-1.19
Q2254) Solve 10x^2+38x+24=0
Ans: x=-0.8 or x=-3
Q2255) Solve 7x^2+50x+7=0
Ans: x=-0.14 or x=-7
Q2256) Solve 32x^2+28x+3=0
Ans: x=-0.12 or x=-0.75
Q2257) Solve 4x^2+25x+6=0
Ans: x=-0.25 or x=-6
Q2258) Solve 26x^2+27x+1=0
Ans: x=-0.04 or x=-1
Q2259) Solve 10x^2+30x+20=0
Ans: x=-1 or x=-2
Q2260) Solve 21x^2+48x+12=0
Ans: x=-0.29 or x=-2
Q2261) Solve 11x^2+29x+14=0
Ans: x=-0.64 or x=-2
Q2262) Solve 24x^2+49x+25=0
Ans: x=-1 or x=-1.04
Q2263) Solve 8x^2+33x+27=0
Ans: x=-1.12 or x=-3
Q2264) Solve 4x^2+9x+2=0
Ans: x=-0.25 or x=-2
Q2265) Solve 10x^2+24x+8=0
Ans: x=-0.4 or x=-2
Q2266) Solve 45x^2+21x+2=0
Ans: x=-0.13 or x=-0.33
Q2267) Solve 5x^2+27x+22=0
Ans: x=-1 or x=-4.4
Q2268) Solve 28x^2+47x+19=0
Ans: x=-0.68 or x=-1
Q2269) Solve 12x^2+29x+10=0
Ans: x=-0.42 or x=-2
Q2270) Solve 28x^2+26x+6=0
Ans: x=-0.43 or x=-0.5
Q2271) Solve 4x^2+24x+35=0
Ans: x=-2.5 or x=-3.5
Q2272) Solve 29x^2+50x+21=0
Ans: x=-0.72 or x=-1
Q2273) Solve 24x^2+44x+20=0
Ans: x=-0.83 or x=-1
Q2274) Solve 9x^2+50x+41=0
Ans: x=-1 or x=-4.56
Q2275) Solve 7x^2+27x+26=0
Ans: x=-1.86 or x=-2
Q2276) Solve 16x^2+42x+17=0
Ans: x=-0.5 or x=-2.12
Q2277) Solve 23x^2+25x+2=0
Ans: x=-0.09 or x=-1
Q2278) Solve 14x^2+13x+3=0
Ans: x=-0.43 or x=-0.5
Q2279) Solve 20x^2+12x+1=0
Ans: x=-0.1 or x=-0.5
Q2280) Solve 13x^2+29x+6=0
Ans: x=-0.23 or x=-2
Q2281) Solve 9x^2+40x+31=0
Ans: x=-1 or x=-3.44
Q2282) Solve 9x^2+36x+32=0
Ans: x=-1.33 or x=-2.67
Q2283) Solve 9x^2+16x+7=0
Ans: x=-0.78 or x=-1
Q2284) Solve 14x^2+43x+33=0
Ans: x=-1.5 or x=-1.57
Q2285) Solve 13x^2+33x+14=0
Ans: x=-0.54 or x=-2
Q2286) Solve 6x^2+34x+40=0
Ans: x=-1.67 or x=-4
Q2287) Solve 20x^2+23x+3=0
Ans: x=-0.15 or x=-1
Q2288) Solve 33x^2+49x+16=0
Ans: x=-0.48 or x=-1
Q2289) Solve 9x^2+45x+50=0
Ans: x=-1.67 or x=-3.33
Q2290) Solve 5x^2+41x+36=0
Ans: x=-1 or x=-7.2
Q2291) Solve 32x^2+48x+18=0
Ans: x=-0.75 or x=-0.75
Q2292) Solve 28x^2+25x+3=0
Ans: x=-0.14 or x=-0.75
Q2293) Solve 9x^2+48x+39=0
Ans: x=-1 or x=-4.33
Q2294) Solve 40x^2+34x+7=0
Ans: x=-0.35 or x=-0.5
Q2295) Solve 12x^2+22x+6=0
Ans: x=-0.33 or x=-1.5
Q2296) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q2297) Solve 6x^2+45x+21=0
Ans: x=-0.5 or x=-7
Q2298) Solve 3x^2+17x+20=0
Ans: x=-1.67 or x=-4
Q2299) Solve 3x^2+28x+25=0
Ans: x=-1 or x=-8.33
Q2300) Solve 19x^2+23x+4=0
Ans: x=-0.21 or x=-1
Q2301) Solve 2x^2+23x+38=0
Ans: x=-2 or x=-9.5
Q2302) Solve 36x^2+49x+10=0
Ans: x=-0.25 or x=-1.11
Q2303) Solve 46x^2+39x+8=0
Ans: x=-0.35 or x=-0.5
Q2304) Solve 9x^2+46x+5=0
Ans: x=-0.11 or x=-5
Q2305) Solve 24x^2+44x+16=0
Ans: x=-0.5 or x=-1.33
Q2306) Solve 14x^2+43x+20=0
Ans: x=-0.57 or x=-2.5
Q2307) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q2308) Solve 42x^2+43x+11=0
Ans: x=-0.5 or x=-0.52
Q2309) Solve 22x^2+49x+27=0
Ans: x=-1 or x=-1.23
Q2310) Solve 14x^2+27x+13=0
Ans: x=-0.93 or x=-1
Q2311) Solve 15x^2+43x+28=0
Ans: x=-1 or x=-1.87
Q2312) Solve 10x^2+18x+8=0
Ans: x=-0.8 or x=-1
Q2313) Solve 13x^2+49x+46=0
Ans: x=-1.77 or x=-2
Q2314) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q2315) Solve x^2+49x+48=0
Ans: x=-1 or x=-48
Q2316) Solve 24x^2+35x+9=0
Ans: x=-0.33 or x=-1.12
Q2317) Solve 30x^2+43x+14=0
Ans: x=-0.5 or x=-0.93
Q2318) Solve 13x^2+26x+13=0
Ans: x=-1 or x=-1
Q2319) Solve 26x^2+46x+20=0
Ans: x=-0.77 or x=-1
Q2320) Solve 8x^2+48x+22=0
Ans: x=-0.5 or x=-5.5
Q2321) Solve 44x^2+43x+8=0
Ans: x=-0.25 or x=-0.73
Q2322) Solve 2x^2+7x+3=0
Ans: x=-0.5 or x=-3
Q2323) Solve 6x^2+37x+42=0
Ans: x=-1.5 or x=-4.67
Q2324) Solve 6x^2+49x+43=0
Ans: x=-1 or x=-7.17
Q2325) Solve 26x^2+45x+19=0
Ans: x=-0.73 or x=-1
Q2326) Solve 7x^2+23x+6=0
Ans: x=-0.29 or x=-3
Q2327) Solve 16x^2+32x+7=0
Ans: x=-0.25 or x=-1.75
Q2328) Solve 6x^2+37x+17=0
Ans: x=-0.5 or x=-5.67
Q2329) Solve 25x^2+50x+24=0
Ans: x=-0.8 or x=-1.2
Q2330) Solve 6x^2+23x+15=0
Ans: x=-0.83 or x=-3
Q2331) Solve 34x^2+44x+10=0
Ans: x=-0.29 or x=-1
Q2332) Solve 3x^2+17x+14=0
Ans: x=-1 or x=-4.67
Q2333) Solve 40x^2+41x+10=0
Ans: x=-0.4 or x=-0.62
Q2334) Solve 3x^2+19x+6=0
Ans: x=-0.33 or x=-6
Q2335) Solve 4x^2+17x+13=0
Ans: x=-1 or x=-3.25
Q2336) Solve 48x^2+41x+6=0
Ans: x=-0.19 or x=-0.67
Q2337) Solve 36x^2+48x+16=0
Ans: x=-0.67 or x=-0.67
Q2338) Solve 9x^2+11x+2=0
Ans: x=-0.22 or x=-1
Q2339) Solve 8x^2+31x+21=0
Ans: x=-0.87 or x=-3
Q2340) Solve x^2+37x+36=0
Ans: x=-1 or x=-36
Q2341) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q2342) Solve 48x^2+46x+11=0
Ans: x=-0.46 or x=-0.5
Q2343) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q2344) Solve 40x^2+47x+7=0
Ans: x=-0.17 or x=-1
Q2345) Solve 27x^2+42x+11=0
Ans: x=-0.33 or x=-1.22
Q2346) Solve 8x^2+40x+42=0
Ans: x=-1.5 or x=-3.5
Q2347) Solve 6x^2+31x+25=0
Ans: x=-1 or x=-4.17
Q2348) Solve 9x^2+29x+20=0
Ans: x=-1 or x=-2.22
Q2349) Solve 34x^2+21x+2=0
Ans: x=-0.12 or x=-0.5
Q2350) Solve 35x^2+22x+3=0
Ans: x=-0.2 or x=-0.43
Q2351) Solve 28x^2+33x+5=0
Ans: x=-0.18 or x=-1
Q2352) Solve 11x^2+41x+38=0
Ans: x=-1.73 or x=-2
Q2353) Solve 36x^2+34x+8=0
Ans: x=-0.44 or x=-0.5
Q2354) Solve 10x^2+37x+7=0
Ans: x=-0.2 or x=-3.5
Q2355) Solve 18x^2+31x+6=0
Ans: x=-0.22 or x=-1.5
Q2356) Solve 5x^2+43x+38=0
Ans: x=-1 or x=-7.6
Q2357) Solve 17x^2+47x+26=0
Ans: x=-0.76 or x=-2
Q2358) Solve 2x^2+37x+18=0
Ans: x=-0.5 or x=-18
Q2359) Solve 33x^2+38x+5=0
Ans: x=-0.15 or x=-1
Q2360) Solve 15x^2+49x+34=0
Ans: x=-1 or x=-2.27
Q2361) Solve 2x^2+35x+48=0
Ans: x=-1.5 or x=-16
Q2362) Solve 32x^2+36x+4=0
Ans: x=-0.12 or x=-1
Q2363) Solve 10x^2+43x+45=0
Ans: x=-1.8 or x=-2.5
Q2364) Solve 10x^2+34x+28=0
Ans: x=-1.4 or x=-2
Q2365) Solve 36x^2+28x+5=0
Ans: x=-0.28 or x=-0.5
Q2366) Solve 2x^2+11x+12=0
Ans: x=-1.5 or x=-4
Q2367) Solve 6x^2+35x+50=0
Ans: x=-2.5 or x=-3.33
Q2368) Solve 19x^2+30x+11=0
Ans: x=-0.58 or x=-1
Q2369) Solve 15x^2+46x+3=0
Ans: x=-0.07 or x=-3
Q2370) Solve 8x^2+9x+1=0
Ans: x=-0.12 or x=-1
Q2371) Solve 30x^2+46x+12=0
Ans: x=-0.33 or x=-1.2
Q2372) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q2373) Solve 8x^2+28x+24=0
Ans: x=-1.5 or x=-2
Q2374) Solve 26x^2+28x+2=0
Ans: x=-0.08 or x=-1
Q2375) Solve 9x^2+36x+27=0
Ans: x=-1 or x=-3
Q2376) Solve 3x^2+29x+26=0
Ans: x=-1 or x=-8.67
Q2377) Solve 9x^2+43x+34=0
Ans: x=-1 or x=-3.78
Q2378) Solve 48x^2+19x+1=0
Ans: x=-0.06 or x=-0.33
Q2379) Solve 3x^2+31x+10=0
Ans: x=-0.33 or x=-10
Q2380) Solve 48x^2+48x+12=0
Ans: x=-0.5 or x=-0.5
Q2381) Solve 48x^2+40x+3=0
Ans: x=-0.08 or x=-0.75
Q2382) Solve 11x^2+43x+42=0
Ans: x=-1.91 or x=-2
Q2383) Solve 10x^2+34x+12=0
Ans: x=-0.4 or x=-3
Q2384) Solve 38x^2+39x+10=0
Ans: x=-0.5 or x=-0.53
Q2385) Solve 21x^2+32x+12=0
Ans: x=-0.67 or x=-0.86
Q2386) Solve 14x^2+27x+10=0
Ans: x=-0.5 or x=-1.43
Q2387) Solve 4x^2+7x+3=0
Ans: x=-0.75 or x=-1
Q2388) Solve 8x^2+31x+30=0
Ans: x=-1.87 or x=-2
Q2389) Solve 4x^2+19x+15=0
Ans: x=-1 or x=-3.75
Q2390) Solve 18x^2+35x+12=0
Ans: x=-0.44 or x=-1.5
Q2391) Solve 11x^2+48x+37=0
Ans: x=-1 or x=-3.36
Q2392) Solve 8x^2+25x+18=0
Ans: x=-1.12 or x=-2
Q2393) Solve 8x^2+22x+9=0
Ans: x=-0.5 or x=-2.25
Q2394) Solve 10x^2+47x+42=0
Ans: x=-1.2 or x=-3.5
Q2395) Solve 44x^2+40x+9=0
Ans: x=-0.41 or x=-0.5
Q2396) Solve 14x^2+37x+15=0
Ans: x=-0.5 or x=-2.14
Q2397) Solve 6x^2+11x+3=0
Ans: x=-0.33 or x=-1.5
Q2398) Solve x^2+34x+33=0
Ans: x=-1 or x=-33
Q2399) Solve 4x^2+28x+40=0
Ans: x=-2 or x=-5
Q2400) Solve 12x^2+42x+30=0
Ans: x=-1 or x=-2.5
Q2401) Solve 12x^2+50x+28=0
Ans: x=-0.67 or x=-3.5
Q2402) Solve 15x^2+43x+26=0
Ans: x=-0.87 or x=-2
Q2403) Solve 5x^2+10x+5=0
Ans: x=-1 or x=-1
Q2404) Solve 5x^2+29x+24=0
Ans: x=-1 or x=-4.8
Q2405) Solve 5x^2+24x+19=0
Ans: x=-1 or x=-3.8
Q2406) Solve 38x^2+45x+7=0
Ans: x=-0.18 or x=-1
Q2407) Solve 24x^2+18x+3=0
Ans: x=-0.25 or x=-0.5
Q2408) Solve x^2+12x+36=0
Ans: x=-6 or x=-6
Q2409) Solve 19x^2+30x+11=0
Ans: x=-0.58 or x=-1
Q2410) Solve 22x^2+35x+12=0
Ans: x=-0.5 or x=-1.09
Q2411) Solve 8x^2+25x+18=0
Ans: x=-1.12 or x=-2
Q2412) Solve 4x^2+40x+19=0
Ans: x=-0.5 or x=-9.5
Q2413) Solve 3x^2+49x+16=0
Ans: x=-0.33 or x=-16
Q2414) Solve 9x^2+40x+39=0
Ans: x=-1.44 or x=-3
Q2415) Solve 6x^2+22x+12=0
Ans: x=-0.67 or x=-3
Q2416) Solve 8x^2+42x+40=0
Ans: x=-1.25 or x=-4
Q2417) Solve 38x^2+31x+6=0
Ans: x=-0.32 or x=-0.5
Q2418) Solve 2x^2+7x+3=0
Ans: x=-0.5 or x=-3
Q2419) Solve 39x^2+25x+4=0
Ans: x=-0.31 or x=-0.33
Q2420) Solve 39x^2+34x+7=0
Ans: x=-0.33 or x=-0.54
Q2421) Solve 6x^2+29x+34=0
Ans: x=-2 or x=-2.83
Q2422) Solve 38x^2+50x+12=0
Ans: x=-0.32 or x=-1
Q2423) Solve 26x^2+29x+3=0
Ans: x=-0.12 or x=-1
Q2424) Solve 14x^2+22x+8=0
Ans: x=-0.57 or x=-1
Q2425) Solve 20x^2+37x+17=0
Ans: x=-0.85 or x=-1
Q2426) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q2427) Solve 24x^2+35x+9=0
Ans: x=-0.33 or x=-1.12
Q2428) Solve 45x^2+46x+1=0
Ans: x=-0.02 or x=-1
Q2429) Solve 8x^2+22x+14=0
Ans: x=-1 or x=-1.75
Q2430) Solve 21x^2+43x+22=0
Ans: x=-1 or x=-1.05
Q2431) Solve 19x^2+27x+8=0
Ans: x=-0.42 or x=-1
Q2432) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q2433) Solve 9x^2+50x+41=0
Ans: x=-1 or x=-4.56
Q2434) Solve 22x^2+47x+18=0
Ans: x=-0.5 or x=-1.64
Q2435) Solve 15x^2+16x+1=0
Ans: x=-0.07 or x=-1
Q2436) Solve x^2+15x+44=0
Ans: x=-4 or x=-11
Q2437) Solve 5x^2+44x+39=0
Ans: x=-1 or x=-7.8
Q2438) Solve 12x^2+42x+18=0
Ans: x=-0.5 or x=-3
Q2439) Solve 23x^2+50x+8=0
Ans: x=-0.17 or x=-2
Q2440) Solve 28x^2+38x+12=0
Ans: x=-0.5 or x=-0.86
Q2441) Solve 26x^2+45x+16=0
Ans: x=-0.5 or x=-1.23
Q2442) Solve 28x^2+30x+8=0
Ans: x=-0.5 or x=-0.57
Q2443) Solve 2x^2+46x+44=0
Ans: x=-1 or x=-22
Q2444) Solve 22x^2+45x+23=0
Ans: x=-1 or x=-1.05
Q2445) Solve 16x^2+38x+12=0
Ans: x=-0.37 or x=-2
Q2446) Solve 39x^2+49x+12=0
Ans: x=-0.33 or x=-0.92
Q2447) Solve 46x^2+27x+2=0
Ans: x=-0.09 or x=-0.5
Q2448) Solve 10x^2+29x+10=0
Ans: x=-0.4 or x=-2.5
Q2449) Solve 16x^2+28x+12=0
Ans: x=-0.75 or x=-1
Q2450) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q2451) Solve 10x^2+35x+30=0
Ans: x=-1.5 or x=-2
Q2452) Solve x^2+14x+40=0
Ans: x=-4 or x=-10
Q2453) Solve 7x^2+15x+8=0
Ans: x=-1 or x=-1.14
Q2454) Solve 25x^2+43x+18=0
Ans: x=-0.72 or x=-1
Q2455) Solve 20x^2+36x+9=0
Ans: x=-0.3 or x=-1.5
Q2456) Solve 8x^2+36x+40=0
Ans: x=-2 or x=-2.5
Q2457) Solve 9x^2+48x+39=0
Ans: x=-1 or x=-4.33
Q2458) Solve 16x^2+40x+9=0
Ans: x=-0.25 or x=-2.25
Q2459) Solve 24x^2+46x+10=0
Ans: x=-0.25 or x=-1.67
Q2460) Solve 3x^2+17x+20=0
Ans: x=-1.67 or x=-4
Q2461) Solve 11x^2+19x+8=0
Ans: x=-0.73 or x=-1
Q2462) Solve 9x^2+40x+16=0
Ans: x=-0.44 or x=-4
Q2463) Solve 2x^2+31x+42=0
Ans: x=-1.5 or x=-14
Q2464) Solve 34x^2+42x+8=0
Ans: x=-0.24 or x=-1
Q2465) Solve 28x^2+39x+5=0
Ans: x=-0.14 or x=-1.25
Q2466) Solve 6x^2+20x+16=0
Ans: x=-1.33 or x=-2
Q2467) Solve 15x^2+17x+4=0
Ans: x=-0.33 or x=-0.8
Q2468) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q2469) Solve x^2+23x+42=0
Ans: x=-2 or x=-21
Q2470) Solve 50x^2+43x+9=0
Ans: x=-0.36 or x=-0.5
Q2471) Solve 3x^2+12x+9=0
Ans: x=-1 or x=-3
Q2472) Solve 9x^2+36x+11=0
Ans: x=-0.33 or x=-3.67
Q2473) Solve 34x^2+49x+15=0
Ans: x=-0.44 or x=-1
Q2474) Solve 5x^2+23x+24=0
Ans: x=-1.6 or x=-3
Q2475) Solve 35x^2+38x+8=0
Ans: x=-0.29 or x=-0.8
Q2476) Solve 32x^2+48x+10=0
Ans: x=-0.25 or x=-1.25
Q2477) Solve 6x^2+24x+24=0
Ans: x=-2 or x=-2
Q2478) Solve 17x^2+24x+7=0
Ans: x=-0.41 or x=-1
Q2479) Solve 35x^2+39x+10=0
Ans: x=-0.4 or x=-0.71
Q2480) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q2481) Solve 11x^2+45x+34=0
Ans: x=-1 or x=-3.09
Q2482) Solve 20x^2+27x+7=0
Ans: x=-0.35 or x=-1
Q2483) Solve 11x^2+47x+42=0
Ans: x=-1.27 or x=-3
Q2484) Solve 15x^2+19x+4=0
Ans: x=-0.27 or x=-1
Q2485) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q2486) Solve 5x^2+32x+48=0
Ans: x=-2.4 or x=-4
Q2487) Solve 4x^2+25x+39=0
Ans: x=-3 or x=-3.25
Q2488) Solve 20x^2+33x+13=0
Ans: x=-0.65 or x=-1
Q2489) Solve 9x^2+13x+4=0
Ans: x=-0.44 or x=-1
Q2490) Solve 27x^2+44x+17=0
Ans: x=-0.63 or x=-1
Q2491) Solve 14x^2+45x+34=0
Ans: x=-1.21 or x=-2
Q2492) Solve 18x^2+18x+4=0
Ans: x=-0.33 or x=-0.67
Q2493) Solve 3x^2+25x+28=0
Ans: x=-1.33 or x=-7
Q2494) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q2495) Solve 3x^2+44x+41=0
Ans: x=-1 or x=-13.67
Q2496) Solve 45x^2+23x+2=0
Ans: x=-0.11 or x=-0.4
Q2497) Solve 42x^2+47x+13=0
Ans: x=-0.5 or x=-0.62
Q2498) Solve 6x^2+21x+18=0
Ans: x=-1.5 or x=-2
Q2499) Solve 16x^2+45x+29=0
Ans: x=-1 or x=-1.81
Q2500) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q2501) Solve 7x^2+47x+40=0
Ans: x=-1 or x=-5.71
Q2502) Solve 23x^2+40x+17=0
Ans: x=-0.74 or x=-1
Q2503) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q2504) Solve 33x^2+38x+9=0
Ans: x=-0.33 or x=-0.82
Q2505) Solve 11x^2+32x+21=0
Ans: x=-1 or x=-1.91
Q2506) Solve 14x^2+50x+44=0
Ans: x=-1.57 or x=-2
Q2507) Solve 6x^2+31x+5=0
Ans: x=-0.17 or x=-5
Q2508) Solve 35x^2+12x+1=0
Ans: x=-0.14 or x=-0.2
Q2509) Solve 42x^2+34x+4=0
Ans: x=-0.14 or x=-0.67
Q2510) Solve 9x^2+39x+40=0
Ans: x=-1.67 or x=-2.67
Q2511) Solve 44x^2+23x+3=0
Ans: x=-0.25 or x=-0.27
Q2512) Solve 34x^2+49x+15=0
Ans: x=-0.44 or x=-1
Q2513) Solve x^2+48x+47=0
Ans: x=-1 or x=-47
Q2514) Solve 42x^2+45x+3=0
Ans: x=-0.07 or x=-1
Q2515) Solve 6x^2+41x+13=0
Ans: x=-0.33 or x=-6.5
Q2516) Solve 22x^2+23x+6=0
Ans: x=-0.5 or x=-0.55
Q2517) Solve 24x^2+37x+14=0
Ans: x=-0.67 or x=-0.87
Q2518) Solve 12x^2+35x+22=0
Ans: x=-0.92 or x=-2
Q2519) Solve 41x^2+42x+1=0
Ans: x=-0.02 or x=-1
Q2520) Solve 3x^2+22x+40=0
Ans: x=-3.33 or x=-4
Q2521) Solve 30x^2+49x+6=0
Ans: x=-0.13 or x=-1.5
Q2522) Solve 21x^2+23x+6=0
Ans: x=-0.43 or x=-0.67
Q2523) Solve 17x^2+38x+8=0
Ans: x=-0.24 or x=-2
Q2524) Solve 11x^2+17x+6=0
Ans: x=-0.55 or x=-1
Q2525) Solve x^2+11x+10=0
Ans: x=-1 or x=-10
Q2526) Solve 8x^2+38x+42=0
Ans: x=-1.75 or x=-3
Q2527) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q2528) Solve 6x^2+25x+11=0
Ans: x=-0.5 or x=-3.67
Q2529) Solve 34x^2+37x+10=0
Ans: x=-0.5 or x=-0.59
Q2530) Solve 45x^2+42x+8=0
Ans: x=-0.27 or x=-0.67
Q2531) Solve 3x^2+14x+8=0
Ans: x=-0.67 or x=-4
Q2532) Solve 22x^2+32x+10=0
Ans: x=-0.45 or x=-1
Q2533) Solve 7x^2+20x+12=0
Ans: x=-0.86 or x=-2
Q2534) Solve 20x^2+42x+22=0
Ans: x=-1 or x=-1.1
Q2535) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q2536) Solve 9x^2+33x+24=0
Ans: x=-1 or x=-2.67
Q2537) Solve 12x^2+12x+3=0
Ans: x=-0.5 or x=-0.5
Q2538) Solve 6x^2+43x+20=0
Ans: x=-0.5 or x=-6.67
Q2539) Solve 24x^2+26x+7=0
Ans: x=-0.5 or x=-0.58
Q2540) Solve 49x^2+42x+8=0
Ans: x=-0.29 or x=-0.57
Q2541) Solve 10x^2+49x+39=0
Ans: x=-1 or x=-3.9
Q2542) Solve 30x^2+47x+14=0
Ans: x=-0.4 or x=-1.17
Q2543) Solve 26x^2+31x+9=0
Ans: x=-0.5 or x=-0.69
Q2544) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q2545) Solve 5x^2+13x+6=0
Ans: x=-0.6 or x=-2
Q2546) Solve 6x^2+19x+15=0
Ans: x=-1.5 or x=-1.67
Q2547) Solve 15x^2+31x+2=0
Ans: x=-0.07 or x=-2
Q2548) Solve 10x^2+29x+12=0
Ans: x=-0.5 or x=-2.4
Q2549) Solve 28x^2+35x+7=0
Ans: x=-0.25 or x=-1
Q2550) Solve 10x^2+40x+40=0
Ans: x=-2 or x=-2
Q2551) Solve 4x^2+23x+15=0
Ans: x=-0.75 or x=-5
Q2552) Solve 6x^2+17x+7=0
Ans: x=-0.5 or x=-2.33
Q2553) Solve 3x^2+22x+40=0
Ans: x=-3.33 or x=-4
Q2554) Solve 46x^2+35x+6=0
Ans: x=-0.26 or x=-0.5
Q2555) Solve 28x^2+41x+15=0
Ans: x=-0.71 or x=-0.75
Q2556) Solve x^2+12x+32=0
Ans: x=-4 or x=-8
Q2557) Solve 38x^2+46x+8=0
Ans: x=-0.21 or x=-1
Q2558) Solve 19x^2+40x+4=0
Ans: x=-0.11 or x=-2
Q2559) Solve 12x^2+20x+7=0
Ans: x=-0.5 or x=-1.17
Q2560) Solve 40x^2+28x+4=0
Ans: x=-0.2 or x=-0.5
Q2561) Solve 25x^2+25x+6=0
Ans: x=-0.4 or x=-0.6
Q2562) Solve 8x^2+34x+21=0
Ans: x=-0.75 or x=-3.5
Q2563) Solve 11x^2+22x+11=0
Ans: x=-1 or x=-1
Q2564) Solve 18x^2+47x+5=0
Ans: x=-0.11 or x=-2.5
Q2565) Solve 4x^2+20x+16=0
Ans: x=-1 or x=-4
Q2566) Solve 10x^2+21x+8=0
Ans: x=-0.5 or x=-1.6
Q2567) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q2568) Solve 2x^2+21x+19=0
Ans: x=-1 or x=-9.5
Q2569) Solve 15x^2+28x+13=0
Ans: x=-0.87 or x=-1
Q2570) Solve 3x^2+19x+20=0
Ans: x=-1.33 or x=-5
Q2571) Solve 3x^2+13x+10=0
Ans: x=-1 or x=-3.33
Q2572) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q2573) Solve 18x^2+45x+22=0
Ans: x=-0.67 or x=-1.83
Q2574) Solve 31x^2+48x+17=0
Ans: x=-0.55 or x=-1
Q2575) Solve 40x^2+36x+8=0
Ans: x=-0.4 or x=-0.5
Q2576) Solve 39x^2+49x+12=0
Ans: x=-0.33 or x=-0.92
Q2577) Solve 13x^2+40x+27=0
Ans: x=-1 or x=-2.08
Q2578) Solve 12x^2+30x+18=0
Ans: x=-1 or x=-1.5
Q2579) Solve 15x^2+49x+26=0
Ans: x=-0.67 or x=-2.6
Q2580) Solve 15x^2+24x+9=0
Ans: x=-0.6 or x=-1
Q2581) Solve 22x^2+19x+4=0
Ans: x=-0.36 or x=-0.5
Q2582) Solve 8x^2+37x+39=0
Ans: x=-1.62 or x=-3
Q2583) Solve 16x^2+48x+20=0
Ans: x=-0.5 or x=-2.5
Q2584) Solve 6x^2+24x+24=0
Ans: x=-2 or x=-2
Q2585) Solve 22x^2+45x+23=0
Ans: x=-1 or x=-1.05
Q2586) Solve 9x^2+40x+31=0
Ans: x=-1 or x=-3.44
Q2587) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q2588) Solve 3x^2+49x+16=0
Ans: x=-0.33 or x=-16
Q2589) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q2590) Solve 3x^2+31x+28=0
Ans: x=-1 or x=-9.33
Q2591) Solve 9x^2+46x+40=0
Ans: x=-1.11 or x=-4
Q2592) Solve 12x^2+25x+12=0
Ans: x=-0.75 or x=-1.33
Q2593) Solve 12x^2+41x+29=0
Ans: x=-1 or x=-2.42
Q2594) Solve 27x^2+48x+5=0
Ans: x=-0.11 or x=-1.67
Q2595) Solve 3x^2+19x+26=0
Ans: x=-2 or x=-4.33
Q2596) Solve 9x^2+39x+40=0
Ans: x=-1.67 or x=-2.67
Q2597) Solve 24x^2+46x+20=0
Ans: x=-0.67 or x=-1.25
Q2598) Solve 19x^2+22x+3=0
Ans: x=-0.16 or x=-1
Q2599) Solve 24x^2+50x+26=0
Ans: x=-1 or x=-1.08
Q2600) Solve 16x^2+45x+29=0
Ans: x=-1 or x=-1.81
Q2601) Solve 12x^2+39x+30=0
Ans: x=-1.25 or x=-2
Q2602) Solve 20x^2+37x+15=0
Ans: x=-0.6 or x=-1.25
Q2603) Solve 18x^2+29x+3=0
Ans: x=-0.11 or x=-1.5
Q2604) Solve 6x^2+13x+6=0
Ans: x=-0.67 or x=-1.5
Q2605) Solve 11x^2+44x+44=0
Ans: x=-2 or x=-2
Q2606) Solve 8x^2+13x+5=0
Ans: x=-0.62 or x=-1
Q2607) Solve 8x^2+38x+42=0
Ans: x=-1.75 or x=-3
Q2608) Solve 15x^2+36x+12=0
Ans: x=-0.4 or x=-2
Q2609) Solve 3x^2+23x+30=0
Ans: x=-1.67 or x=-6
Q2610) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q2611) Solve 13x^2+34x+21=0
Ans: x=-1 or x=-1.62
Q2612) Solve 14x^2+31x+17=0
Ans: x=-1 or x=-1.21
Q2613) Solve 2x^2+23x+21=0
Ans: x=-1 or x=-10.5
Q2614) Solve 16x^2+28x+10=0
Ans: x=-0.5 or x=-1.25
Q2615) Solve 12x^2+22x+6=0
Ans: x=-0.33 or x=-1.5
Q2616) Solve 8x^2+45x+25=0
Ans: x=-0.62 or x=-5
Q2617) Solve 10x^2+14x+4=0
Ans: x=-0.4 or x=-1
Q2618) Solve 44x^2+31x+5=0
Ans: x=-0.25 or x=-0.45
Q2619) Solve 42x^2+41x+10=0
Ans: x=-0.48 or x=-0.5
Q2620) Solve 10x^2+33x+9=0
Ans: x=-0.3 or x=-3
Q2621) Solve 21x^2+37x+12=0
Ans: x=-0.43 or x=-1.33
Q2622) Solve 21x^2+46x+24=0
Ans: x=-0.86 or x=-1.33
Q2623) Solve 16x^2+46x+15=0
Ans: x=-0.37 or x=-2.5
Q2624) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q2625) Solve 34x^2+31x+7=0
Ans: x=-0.41 or x=-0.5
Q2626) Solve 12x^2+24x+9=0
Ans: x=-0.5 or x=-1.5
Q2627) Solve 49x^2+50x+1=0
Ans: x=-0.02 or x=-1
Q2628) Solve 4x^2+8x+3=0
Ans: x=-0.5 or x=-1.5
Q2629) Solve 7x^2+15x+2=0
Ans: x=-0.14 or x=-2
Q2630) Solve 27x^2+42x+8=0
Ans: x=-0.22 or x=-1.33
Q2631) Solve 25x^2+45x+14=0
Ans: x=-0.4 or x=-1.4
Q2632) Solve 18x^2+38x+4=0
Ans: x=-0.11 or x=-2
Q2633) Solve 5x^2+46x+48=0
Ans: x=-1.2 or x=-8
Q2634) Solve 16x^2+17x+1=0
Ans: x=-0.06 or x=-1
Q2635) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q2636) Solve 10x^2+47x+9=0
Ans: x=-0.2 or x=-4.5
Q2637) Solve 8x^2+8x+2=0
Ans: x=-0.5 or x=-0.5
Q2638) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q2639) Solve 20x^2+27x+7=0
Ans: x=-0.35 or x=-1
Q2640) Solve 12x^2+16x+4=0
Ans: x=-0.33 or x=-1
Q2641) Solve 10x^2+43x+28=0
Ans: x=-0.8 or x=-3.5
Q2642) Solve 48x^2+46x+11=0
Ans: x=-0.46 or x=-0.5
Q2643) Solve 24x^2+10x+1=0
Ans: x=-0.17 or x=-0.25
Q2644) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q2645) Solve 31x^2+43x+12=0
Ans: x=-0.39 or x=-1
Q2646) Solve 3x^2+23x+14=0
Ans: x=-0.67 or x=-7
Q2647) Solve 6x^2+11x+3=0
Ans: x=-0.33 or x=-1.5
Q2648) Solve 24x^2+34x+11=0
Ans: x=-0.5 or x=-0.92
Q2649) Solve x^2+13x+30=0
Ans: x=-3 or x=-10
Q2650) Solve 48x^2+41x+6=0
Ans: x=-0.19 or x=-0.67
Q2651) Solve 38x^2+43x+12=0
Ans: x=-0.5 or x=-0.63
Q2652) Solve 46x^2+33x+5=0
Ans: x=-0.22 or x=-0.5
Q2653) Solve 18x^2+28x+10=0
Ans: x=-0.56 or x=-1
Q2654) Solve 16x^2+48x+20=0
Ans: x=-0.5 or x=-2.5
Q2655) Solve 15x^2+48x+33=0
Ans: x=-1 or x=-2.2
Q2656) Solve 15x^2+16x+1=0
Ans: x=-0.07 or x=-1
Q2657) Solve 14x^2+29x+11=0
Ans: x=-0.5 or x=-1.57
Q2658) Solve 33x^2+47x+4=0
Ans: x=-0.09 or x=-1.33
Q2659) Solve 28x^2+46x+6=0
Ans: x=-0.14 or x=-1.5
Q2660) Solve 6x^2+41x+30=0
Ans: x=-0.83 or x=-6
Q2661) Solve 8x^2+35x+33=0
Ans: x=-1.37 or x=-3
Q2662) Solve 26x^2+46x+20=0
Ans: x=-0.77 or x=-1
Q2663) Solve 8x^2+10x+2=0
Ans: x=-0.25 or x=-1
Q2664) Solve 19x^2+49x+30=0
Ans: x=-1 or x=-1.58
Q2665) Solve 45x^2+39x+8=0
Ans: x=-0.33 or x=-0.53
Q2666) Solve 2x^2+44x+42=0
Ans: x=-1 or x=-21
Q2667) Solve 35x^2+37x+2=0
Ans: x=-0.06 or x=-1
Q2668) Solve 17x^2+28x+11=0
Ans: x=-0.65 or x=-1
Q2669) Solve 12x^2+29x+14=0
Ans: x=-0.67 or x=-1.75
Q2670) Solve 17x^2+49x+30=0
Ans: x=-0.88 or x=-2
Q2671) Solve 33x^2+32x+7=0
Ans: x=-0.33 or x=-0.64
Q2672) Solve 8x^2+30x+28=0
Ans: x=-1.75 or x=-2
Q2673) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q2674) Solve 3x^2+26x+35=0
Ans: x=-1.67 or x=-7
Q2675) Solve 20x^2+22x+2=0
Ans: x=-0.1 or x=-1
Q2676) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q2677) Solve 40x^2+22x+3=0
Ans: x=-0.25 or x=-0.3
Q2678) Solve 10x^2+13x+3=0
Ans: x=-0.3 or x=-1
Q2679) Solve 22x^2+41x+15=0
Ans: x=-0.5 or x=-1.36
Q2680) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q2681) Solve x^2+28x+27=0
Ans: x=-1 or x=-27
Q2682) Solve x^2+20x+19=0
Ans: x=-1 or x=-19
Q2683) Solve 3x^2+23x+40=0
Ans: x=-2.67 or x=-5
Q2684) Solve 48x^2+50x+2=0
Ans: x=-0.04 or x=-1
Q2685) Solve 30x^2+46x+12=0
Ans: x=-0.33 or x=-1.2
Q2686) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q2687) Solve 9x^2+36x+35=0
Ans: x=-1.67 or x=-2.33
Q2688) Solve 48x^2+40x+7=0
Ans: x=-0.25 or x=-0.58
Q2689) Solve 38x^2+44x+6=0
Ans: x=-0.16 or x=-1
Q2690) Solve 11x^2+37x+12=0
Ans: x=-0.36 or x=-3
Q2691) Solve 4x^2+25x+39=0
Ans: x=-3 or x=-3.25
Q2692) Solve 32x^2+50x+18=0
Ans: x=-0.56 or x=-1
Q2693) Solve 34x^2+25x+4=0
Ans: x=-0.24 or x=-0.5
Q2694) Solve 2x^2+35x+33=0
Ans: x=-1 or x=-16.5
Q2695) Solve 6x^2+36x+48=0
Ans: x=-2 or x=-4
Q2696) Solve x^2+12x+20=0
Ans: x=-2 or x=-10
Q2697) Solve 42x^2+17x+1=0
Ans: x=-0.07 or x=-0.33
Q2698) Solve 16x^2+37x+10=0
Ans: x=-0.31 or x=-2
Q2699) Solve 34x^2+35x+9=0
Ans: x=-0.5 or x=-0.53
Q2700) Solve 9x^2+42x+24=0
Ans: x=-0.67 or x=-4
Q2701) Solve 30x^2+27x+6=0
Ans: x=-0.4 or x=-0.5
Q2702) Solve 4x^2+33x+50=0
Ans: x=-2 or x=-6.25
Q2703) Solve 6x^2+29x+13=0
Ans: x=-0.5 or x=-4.33
Q2704) Solve 21x^2+43x+22=0
Ans: x=-1 or x=-1.05
Q2705) Solve 3x^2+14x+15=0
Ans: x=-1.67 or x=-3
Q2706) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q2707) Solve 26x^2+35x+11=0
Ans: x=-0.5 or x=-0.85
Q2708) Solve 4x^2+32x+15=0
Ans: x=-0.5 or x=-7.5
Q2709) Solve 16x^2+40x+21=0
Ans: x=-0.75 or x=-1.75
Q2710) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q2711) Solve 3x^2+44x+28=0
Ans: x=-0.67 or x=-14
Q2712) Solve 14x^2+37x+23=0
Ans: x=-1 or x=-1.64
Q2713) Solve 5x^2+19x+18=0
Ans: x=-1.8 or x=-2
Q2714) Solve 10x^2+42x+32=0
Ans: x=-1 or x=-3.2
Q2715) Solve 32x^2+38x+6=0
Ans: x=-0.19 or x=-1
Q2716) Solve 32x^2+46x+14=0
Ans: x=-0.44 or x=-1
Q2717) Solve 10x^2+21x+2=0
Ans: x=-0.1 or x=-2
Q2718) Solve 16x^2+49x+34=0
Ans: x=-1.06 or x=-2
Q2719) Solve 11x^2+38x+32=0
Ans: x=-1.45 or x=-2
Q2720) Solve 6x^2+49x+23=0
Ans: x=-0.5 or x=-7.67
Q2721) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q2722) Solve 4x^2+12x+9=0
Ans: x=-1.5 or x=-1.5
Q2723) Solve 11x^2+26x+8=0
Ans: x=-0.36 or x=-2
Q2724) Solve 14x^2+37x+5=0
Ans: x=-0.14 or x=-2.5
Q2725) Solve 34x^2+36x+2=0
Ans: x=-0.06 or x=-1
Q2726) Solve 30x^2+31x+7=0
Ans: x=-0.33 or x=-0.7
Q2727) Solve 4x^2+20x+25=0
Ans: x=-2.5 or x=-2.5
Q2728) Solve 9x^2+48x+15=0
Ans: x=-0.33 or x=-5
Q2729) Solve 10x^2+35x+15=0
Ans: x=-0.5 or x=-3
Q2730) Solve 2x^2+19x+9=0
Ans: x=-0.5 or x=-9
Q2731) Solve 35x^2+50x+15=0
Ans: x=-0.43 or x=-1
Q2732) Solve 2x^2+23x+21=0
Ans: x=-1 or x=-10.5
Q2733) Solve 30x^2+23x+4=0
Ans: x=-0.27 or x=-0.5
Q2734) Solve 36x^2+21x+3=0
Ans: x=-0.25 or x=-0.33
Q2735) Solve 14x^2+29x+2=0
Ans: x=-0.07 or x=-2
Q2736) Solve 36x^2+41x+8=0
Ans: x=-0.25 or x=-0.89
Q2737) Solve 36x^2+15x+1=0
Ans: x=-0.08 or x=-0.33
Q2738) Solve 12x^2+35x+8=0
Ans: x=-0.25 or x=-2.67
Q2739) Solve 27x^2+36x+9=0
Ans: x=-0.33 or x=-1
Q2740) Solve 3x^2+34x+11=0
Ans: x=-0.33 or x=-11
Q2741) Solve x^2+14x+49=0
Ans: x=-7 or x=-7
Q2742) Solve 28x^2+23x+4=0
Ans: x=-0.25 or x=-0.57
Q2743) Solve 30x^2+42x+12=0
Ans: x=-0.4 or x=-1
Q2744) Solve 12x^2+44x+24=0
Ans: x=-0.67 or x=-3
Q2745) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q2746) Solve 18x^2+39x+6=0
Ans: x=-0.17 or x=-2
Q2747) Solve 4x^2+28x+49=0
Ans: x=-3.5 or x=-3.5
Q2748) Solve 3x^2+41x+38=0
Ans: x=-1 or x=-12.67
Q2749) Solve 7x^2+30x+32=0
Ans: x=-2 or x=-2.29
Q2750) Solve 6x^2+23x+21=0
Ans: x=-1.5 or x=-2.33
Q2751) Solve 9x^2+42x+40=0
Ans: x=-1.33 or x=-3.33
Q2752) Solve 21x^2+38x+17=0
Ans: x=-0.81 or x=-1
Q2753) Solve 3x^2+32x+45=0
Ans: x=-1.67 or x=-9
Q2754) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q2755) Solve 30x^2+37x+10=0
Ans: x=-0.4 or x=-0.83
Q2756) Solve 33x^2+40x+12=0
Ans: x=-0.55 or x=-0.67
Q2757) Solve 5x^2+21x+16=0
Ans: x=-1 or x=-3.2
Q2758) Solve 7x^2+31x+30=0
Ans: x=-1.43 or x=-3
Q2759) Solve 25x^2+29x+4=0
Ans: x=-0.16 or x=-1
Q2760) Solve 6x^2+39x+45=0
Ans: x=-1.5 or x=-5
Q2761) Solve 9x^2+45x+50=0
Ans: x=-1.67 or x=-3.33
Q2762) Solve 46x^2+35x+6=0
Ans: x=-0.26 or x=-0.5
Q2763) Solve 36x^2+42x+10=0
Ans: x=-0.33 or x=-0.83
Q2764) Solve 10x^2+13x+4=0
Ans: x=-0.5 or x=-0.8
Q2765) Solve 32x^2+40x+12=0
Ans: x=-0.5 or x=-0.75
Q2766) Solve 5x^2+39x+34=0
Ans: x=-1 or x=-6.8
Q2767) Solve 36x^2+13x+1=0
Ans: x=-0.11 or x=-0.25
Q2768) Solve 33x^2+35x+2=0
Ans: x=-0.06 or x=-1
Q2769) Solve 4x^2+21x+27=0
Ans: x=-2.25 or x=-3
Q2770) Solve 3x^2+26x+48=0
Ans: x=-2.67 or x=-6
Q2771) Solve 18x^2+31x+13=0
Ans: x=-0.72 or x=-1
Q2772) Solve 34x^2+41x+12=0
Ans: x=-0.5 or x=-0.71
Q2773) Solve 8x^2+39x+46=0
Ans: x=-2 or x=-2.87
Q2774) Solve 14x^2+25x+11=0
Ans: x=-0.79 or x=-1
Q2775) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q2776) Solve 4x^2+6x+2=0
Ans: x=-0.5 or x=-1
Q2777) Solve 6x^2+23x+10=0
Ans: x=-0.5 or x=-3.33
Q2778) Solve 12x^2+46x+44=0
Ans: x=-1.83 or x=-2
Q2779) Solve 16x^2+20x+6=0
Ans: x=-0.5 or x=-0.75
Q2780) Solve 12x^2+42x+18=0
Ans: x=-0.5 or x=-3
Q2781) Solve 50x^2+50x+12=0
Ans: x=-0.4 or x=-0.6
Q2782) Solve 14x^2+30x+4=0
Ans: x=-0.14 or x=-2
Q2783) Solve 9x^2+27x+8=0
Ans: x=-0.33 or x=-2.67
Q2784) Solve x^2+25x+24=0
Ans: x=-1 or x=-24
Q2785) Solve 9x^2+17x+8=0
Ans: x=-0.89 or x=-1
Q2786) Solve 2x^2+17x+33=0
Ans: x=-3 or x=-5.5
Q2787) Solve 2x^2+12x+16=0
Ans: x=-2 or x=-4
Q2788) Solve 20x^2+45x+25=0
Ans: x=-1 or x=-1.25
Q2789) Solve 12x^2+38x+16=0
Ans: x=-0.5 or x=-2.67
Q2790) Solve 17x^2+50x+33=0
Ans: x=-1 or x=-1.94
Q2791) Solve 34x^2+37x+3=0
Ans: x=-0.09 or x=-1
Q2792) Solve 10x^2+43x+28=0
Ans: x=-0.8 or x=-3.5
Q2793) Solve x^2+46x+45=0
Ans: x=-1 or x=-45
Q2794) Solve 22x^2+35x+13=0
Ans: x=-0.59 or x=-1
Q2795) Solve 17x^2+31x+14=0
Ans: x=-0.82 or x=-1
Q2796) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q2797) Solve 16x^2+14x+3=0
Ans: x=-0.37 or x=-0.5
Q2798) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q2799) Solve 9x^2+33x+18=0
Ans: x=-0.67 or x=-3
Q2800) Solve 28x^2+36x+8=0
Ans: x=-0.29 or x=-1
Q2801) Solve 9x^2+43x+34=0
Ans: x=-1 or x=-3.78
Q2802) Solve 23x^2+26x+3=0
Ans: x=-0.13 or x=-1
Q2803) Solve 16x^2+42x+20=0
Ans: x=-0.62 or x=-2
Q2804) Solve 2x^2+25x+33=0
Ans: x=-1.5 or x=-11
Q2805) Solve x^2+17x+30=0
Ans: x=-2 or x=-15
Q2806) Solve 2x^2+35x+17=0
Ans: x=-0.5 or x=-17
Q2807) Solve 8x^2+22x+5=0
Ans: x=-0.25 or x=-2.5
Q2808) Solve 48x^2+48x+12=0
Ans: x=-0.5 or x=-0.5
Q2809) Solve 10x^2+39x+27=0
Ans: x=-0.9 or x=-3
Q2810) Solve 4x^2+38x+18=0
Ans: x=-0.5 or x=-9
Q2811) Solve 14x^2+49x+35=0
Ans: x=-1 or x=-2.5
Q2812) Solve 30x^2+41x+14=0
Ans: x=-0.67 or x=-0.7
Q2813) Solve 16x^2+44x+18=0
Ans: x=-0.5 or x=-2.25
Q2814) Solve 20x^2+45x+10=0
Ans: x=-0.25 or x=-2
Q2815) Solve 28x^2+50x+12=0
Ans: x=-0.29 or x=-1.5
Q2816) Solve 23x^2+40x+17=0
Ans: x=-0.74 or x=-1
Q2817) Solve 13x^2+40x+27=0
Ans: x=-1 or x=-2.08
Q2818) Solve 11x^2+48x+16=0
Ans: x=-0.36 or x=-4
Q2819) Solve 15x^2+46x+24=0
Ans: x=-0.67 or x=-2.4
Q2820) Solve 36x^2+42x+6=0
Ans: x=-0.17 or x=-1
Q2821) Solve 4x^2+8x+3=0
Ans: x=-0.5 or x=-1.5
Q2822) Solve 36x^2+30x+6=0
Ans: x=-0.33 or x=-0.5
Q2823) Solve 30x^2+25x+5=0
Ans: x=-0.33 or x=-0.5
Q2824) Solve 29x^2+49x+20=0
Ans: x=-0.69 or x=-1
Q2825) Solve 15x^2+22x+7=0
Ans: x=-0.47 or x=-1
Q2826) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q2827) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q2828) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q2829) Solve 33x^2+41x+10=0
Ans: x=-0.33 or x=-0.91
Q2830) Solve 4x^2+33x+29=0
Ans: x=-1 or x=-7.25
Q2831) Solve 16x^2+50x+6=0
Ans: x=-0.12 or x=-3
Q2832) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q2833) Solve 41x^2+47x+6=0
Ans: x=-0.15 or x=-1
Q2834) Solve 16x^2+50x+6=0
Ans: x=-0.12 or x=-3
Q2835) Solve 8x^2+32x+14=0
Ans: x=-0.5 or x=-3.5
Q2836) Solve 8x^2+46x+30=0
Ans: x=-0.75 or x=-5
Q2837) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q2838) Solve 14x^2+33x+10=0
Ans: x=-0.36 or x=-2
Q2839) Solve 2x^2+17x+26=0
Ans: x=-2 or x=-6.5
Q2840) Solve 6x^2+23x+17=0
Ans: x=-1 or x=-2.83
Q2841) Solve 8x^2+46x+30=0
Ans: x=-0.75 or x=-5
Q2842) Solve 6x^2+14x+4=0
Ans: x=-0.33 or x=-2
Q2843) Solve 14x^2+42x+28=0
Ans: x=-1 or x=-2
Q2844) Solve 15x^2+26x+8=0
Ans: x=-0.4 or x=-1.33
Q2845) Solve 12x^2+41x+35=0
Ans: x=-1.67 or x=-1.75
Q2846) Solve 42x^2+48x+6=0
Ans: x=-0.14 or x=-1
Q2847) Solve 13x^2+48x+44=0
Ans: x=-1.69 or x=-2
Q2848) Solve 33x^2+38x+9=0
Ans: x=-0.33 or x=-0.82
Q2849) Solve 6x^2+36x+48=0
Ans: x=-2 or x=-4
Q2850) Solve 10x^2+41x+21=0
Ans: x=-0.6 or x=-3.5
Q2851) Solve 46x^2+37x+7=0
Ans: x=-0.3 or x=-0.5
Q2852) Solve 7x^2+40x+48=0
Ans: x=-1.71 or x=-4
Q2853) Solve 28x^2+29x+1=0
Ans: x=-0.04 or x=-1
Q2854) Solve 3x^2+20x+17=0
Ans: x=-1 or x=-5.67
Q2855) Solve 26x^2+33x+7=0
Ans: x=-0.27 or x=-1
Q2856) Solve 9x^2+27x+18=0
Ans: x=-1 or x=-2
Q2857) Solve 9x^2+34x+32=0
Ans: x=-1.78 or x=-2
Q2858) Solve 17x^2+39x+10=0
Ans: x=-0.29 or x=-2
Q2859) Solve 10x^2+37x+21=0
Ans: x=-0.7 or x=-3
Q2860) Solve 19x^2+45x+14=0
Ans: x=-0.37 or x=-2
Q2861) Solve 14x^2+21x+7=0
Ans: x=-0.5 or x=-1
Q2862) Solve 13x^2+38x+24=0
Ans: x=-0.92 or x=-2
Q2863) Solve 2x^2+17x+8=0
Ans: x=-0.5 or x=-8
Q2864) Solve 22x^2+47x+21=0
Ans: x=-0.64 or x=-1.5
Q2865) Solve 3x^2+32x+29=0
Ans: x=-1 or x=-9.67
Q2866) Solve 45x^2+48x+11=0
Ans: x=-0.33 or x=-0.73
Q2867) Solve 8x^2+28x+24=0
Ans: x=-1.5 or x=-2
Q2868) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q2869) Solve 32x^2+46x+15=0
Ans: x=-0.5 or x=-0.94
Q2870) Solve 4x^2+15x+9=0
Ans: x=-0.75 or x=-3
Q2871) Solve 16x^2+25x+9=0
Ans: x=-0.56 or x=-1
Q2872) Solve 2x^2+13x+6=0
Ans: x=-0.5 or x=-6
Q2873) Solve 9x^2+39x+12=0
Ans: x=-0.33 or x=-4
Q2874) Solve 14x^2+20x+6=0
Ans: x=-0.43 or x=-1
Q2875) Solve 15x^2+24x+9=0
Ans: x=-0.6 or x=-1
Q2876) Solve 21x^2+31x+10=0
Ans: x=-0.48 or x=-1
Q2877) Solve 10x^2+34x+28=0
Ans: x=-1.4 or x=-2
Q2878) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q2879) Solve 25x^2+50x+16=0
Ans: x=-0.4 or x=-1.6
Q2880) Solve 19x^2+38x+19=0
Ans: x=-1 or x=-1
Q2881) Solve 39x^2+28x+5=0
Ans: x=-0.33 or x=-0.38
Q2882) Solve 10x^2+41x+39=0
Ans: x=-1.5 or x=-2.6
Q2883) Solve 36x^2+47x+15=0
Ans: x=-0.56 or x=-0.75
Q2884) Solve 10x^2+39x+14=0
Ans: x=-0.4 or x=-3.5
Q2885) Solve 42x^2+40x+8=0
Ans: x=-0.29 or x=-0.67
Q2886) Solve 9x^2+45x+14=0
Ans: x=-0.33 or x=-4.67
Q2887) Solve 26x^2+33x+10=0
Ans: x=-0.5 or x=-0.77
Q2888) Solve 4x^2+27x+38=0
Ans: x=-2 or x=-4.75
Q2889) Solve x^2+39x+38=0
Ans: x=-1 or x=-38
Q2890) Solve 7x^2+48x+41=0
Ans: x=-1 or x=-5.86
Q2891) Solve 42x^2+47x+5=0
Ans: x=-0.12 or x=-1
Q2892) Solve 33x^2+14x+1=0
Ans: x=-0.09 or x=-0.33
Q2893) Solve 13x^2+37x+22=0
Ans: x=-0.85 or x=-2
Q2894) Solve 36x^2+39x+9=0
Ans: x=-0.33 or x=-0.75
Q2895) Solve 32x^2+34x+9=0
Ans: x=-0.5 or x=-0.56
Q2896) Solve 9x^2+36x+35=0
Ans: x=-1.67 or x=-2.33
Q2897) Solve 27x^2+45x+18=0
Ans: x=-0.67 or x=-1
Q2898) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q2899) Solve 20x^2+41x+2=0
Ans: x=-0.05 or x=-2
Q2900) Solve 14x^2+39x+22=0
Ans: x=-0.79 or x=-2
Q2901) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q2902) Solve 14x^2+45x+36=0
Ans: x=-1.5 or x=-1.71
Q2903) Solve 13x^2+35x+18=0
Ans: x=-0.69 or x=-2
Q2904) Solve 4x^2+24x+11=0
Ans: x=-0.5 or x=-5.5
Q2905) Solve 2x^2+15x+18=0
Ans: x=-1.5 or x=-6
Q2906) Solve 27x^2+29x+2=0
Ans: x=-0.07 or x=-1
Q2907) Solve 35x^2+22x+3=0
Ans: x=-0.2 or x=-0.43
Q2908) Solve 14x^2+15x+4=0
Ans: x=-0.5 or x=-0.57
Q2909) Solve 4x^2+29x+30=0
Ans: x=-1.25 or x=-6
Q2910) Solve 2x^2+6x+4=0
Ans: x=-1 or x=-2
Q2911) Solve 13x^2+15x+2=0
Ans: x=-0.15 or x=-1
Q2912) Solve 28x^2+33x+5=0
Ans: x=-0.18 or x=-1
Q2913) Solve 5x^2+37x+14=0
Ans: x=-0.4 or x=-7
Q2914) Solve 10x^2+44x+34=0
Ans: x=-1 or x=-3.4
Q2915) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q2916) Solve 12x^2+36x+27=0
Ans: x=-1.5 or x=-1.5
Q2917) Solve 2x^2+25x+23=0
Ans: x=-1 or x=-11.5
Q2918) Solve 22x^2+25x+3=0
Ans: x=-0.14 or x=-1
Q2919) Solve 18x^2+45x+13=0
Ans: x=-0.33 or x=-2.17
Q2920) Solve 6x^2+31x+18=0
Ans: x=-0.67 or x=-4.5
Q2921) Solve 28x^2+44x+16=0
Ans: x=-0.57 or x=-1
Q2922) Solve 8x^2+30x+27=0
Ans: x=-1.5 or x=-2.25
Q2923) Solve 4x^2+20x+25=0
Ans: x=-2.5 or x=-2.5
Q2924) Solve 12x^2+47x+45=0
Ans: x=-1.67 or x=-2.25
Q2925) Solve 34x^2+50x+16=0
Ans: x=-0.47 or x=-1
Q2926) Solve 16x^2+25x+9=0
Ans: x=-0.56 or x=-1
Q2927) Solve 9x^2+30x+21=0
Ans: x=-1 or x=-2.33
Q2928) Solve 16x^2+34x+4=0
Ans: x=-0.12 or x=-2
Q2929) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q2930) Solve 6x^2+26x+20=0
Ans: x=-1 or x=-3.33
Q2931) Solve 10x^2+23x+9=0
Ans: x=-0.5 or x=-1.8
Q2932) Solve 30x^2+32x+2=0
Ans: x=-0.07 or x=-1
Q2933) Solve 8x^2+10x+3=0
Ans: x=-0.5 or x=-0.75
Q2934) Solve 15x^2+28x+12=0
Ans: x=-0.67 or x=-1.2
Q2935) Solve 19x^2+47x+28=0
Ans: x=-1 or x=-1.47
Q2936) Solve 18x^2+46x+28=0
Ans: x=-1 or x=-1.56
Q2937) Solve 15x^2+23x+8=0
Ans: x=-0.53 or x=-1
Q2938) Solve 4x^2+13x+9=0
Ans: x=-1 or x=-2.25
Q2939) Solve 12x^2+34x+10=0
Ans: x=-0.33 or x=-2.5
Q2940) Solve x^2+12x+35=0
Ans: x=-5 or x=-7
Q2941) Solve 14x^2+21x+7=0
Ans: x=-0.5 or x=-1
Q2942) Solve 13x^2+42x+9=0
Ans: x=-0.23 or x=-3
Q2943) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q2944) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q2945) Solve 4x^2+28x+48=0
Ans: x=-3 or x=-4
Q2946) Solve 4x^2+35x+31=0
Ans: x=-1 or x=-7.75
Q2947) Solve 6x^2+37x+42=0
Ans: x=-1.5 or x=-4.67
Q2948) Solve 8x^2+14x+6=0
Ans: x=-0.75 or x=-1
Q2949) Solve 39x^2+28x+5=0
Ans: x=-0.33 or x=-0.38
Q2950) Solve 30x^2+31x+8=0
Ans: x=-0.5 or x=-0.53
Q2951) Solve 9x^2+44x+35=0
Ans: x=-1 or x=-3.89
Q2952) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q2953) Solve 18x^2+33x+9=0
Ans: x=-0.33 or x=-1.5
Q2954) Solve 19x^2+45x+14=0
Ans: x=-0.37 or x=-2
Q2955) Solve 20x^2+29x+9=0
Ans: x=-0.45 or x=-1
Q2956) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q2957) Solve 2x^2+15x+18=0
Ans: x=-1.5 or x=-6
Q2958) Solve 11x^2+23x+2=0
Ans: x=-0.09 or x=-2
Q2959) Solve 28x^2+22x+4=0
Ans: x=-0.29 or x=-0.5
Q2960) Solve 37x^2+49x+12=0
Ans: x=-0.32 or x=-1
Q2961) Solve 2x^2+19x+17=0
Ans: x=-1 or x=-8.5
Q2962) Solve 45x^2+48x+11=0
Ans: x=-0.33 or x=-0.73
Q2963) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q2964) Solve 7x^2+22x+3=0
Ans: x=-0.14 or x=-3
Q2965) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q2966) Solve 2x^2+20x+48=0
Ans: x=-4 or x=-6
Q2967) Solve 4x^2+30x+14=0
Ans: x=-0.5 or x=-7
Q2968) Solve 9x^2+30x+25=0
Ans: x=-1.67 or x=-1.67
Q2969) Solve 22x^2+33x+11=0
Ans: x=-0.5 or x=-1
Q2970) Solve 20x^2+38x+14=0
Ans: x=-0.5 or x=-1.4
Q2971) Solve 34x^2+21x+2=0
Ans: x=-0.12 or x=-0.5
Q2972) Solve 15x^2+38x+16=0
Ans: x=-0.53 or x=-2
Q2973) Solve 24x^2+31x+7=0
Ans: x=-0.29 or x=-1
Q2974) Solve x^2+22x+21=0
Ans: x=-1 or x=-21
Q2975) Solve x^2+2x+1=0
Ans: x=-1 or x=-1
Q2976) Solve 23x^2+34x+11=0
Ans: x=-0.48 or x=-1
Q2977) Solve 36x^2+31x+3=0
Ans: x=-0.11 or x=-0.75
Q2978) Solve 44x^2+45x+1=0
Ans: x=-0.02 or x=-1
Q2979) Solve 27x^2+24x+5=0
Ans: x=-0.33 or x=-0.56
Q2980) Solve 37x^2+44x+7=0
Ans: x=-0.19 or x=-1
Q2981) Solve 22x^2+46x+24=0
Ans: x=-1 or x=-1.09
Q2982) Solve 2x^2+9x+9=0
Ans: x=-1.5 or x=-3
Q2983) Solve 18x^2+50x+28=0
Ans: x=-0.78 or x=-2
Q2984) Solve 7x^2+33x+20=0
Ans: x=-0.71 or x=-4
Q2985) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q2986) Solve 2x^2+17x+33=0
Ans: x=-3 or x=-5.5
Q2987) Solve 27x^2+45x+12=0
Ans: x=-0.33 or x=-1.33
Q2988) Solve 33x^2+47x+12=0
Ans: x=-0.33 or x=-1.09
Q2989) Solve 14x^2+33x+10=0
Ans: x=-0.36 or x=-2
Q2990) Solve 4x^2+13x+3=0
Ans: x=-0.25 or x=-3
Q2991) Solve 48x^2+44x+8=0
Ans: x=-0.25 or x=-0.67
Q2992) Solve 10x^2+24x+14=0
Ans: x=-1 or x=-1.4
Q2993) Solve 7x^2+37x+48=0
Ans: x=-2.29 or x=-3
Q2994) Solve 20x^2+32x+3=0
Ans: x=-0.1 or x=-1.5
Q2995) Solve 11x^2+42x+27=0
Ans: x=-0.82 or x=-3
Q2996) Solve 5x^2+21x+4=0
Ans: x=-0.2 or x=-4
Q2997) Solve 25x^2+37x+12=0
Ans: x=-0.48 or x=-1
Q2998) Solve 4x^2+26x+42=0
Ans: x=-3 or x=-3.5
Q2999) Solve 9x^2+30x+9=0
Ans: x=-0.33 or x=-3
Q3000) Solve 8x^2+24x+10=0
Ans: x=-0.5 or x=-2.5
Q3001) Solve 6x^2+13x+6=0
Ans: x=-0.67 or x=-1.5
Q3002) Solve 33x^2+41x+10=0
Ans: x=-0.33 or x=-0.91
Q3003) Solve 12x^2+28x+11=0
Ans: x=-0.5 or x=-1.83
Q3004) Solve 35x^2+12x+1=0
Ans: x=-0.14 or x=-0.2
Q3005) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q3006) Solve 47x^2+48x+1=0
Ans: x=-0.02 or x=-1
Q3007) Solve 10x^2+14x+4=0
Ans: x=-0.4 or x=-1
Q3008) Solve 8x^2+30x+13=0
Ans: x=-0.5 or x=-3.25
Q3009) Solve 6x^2+38x+40=0
Ans: x=-1.33 or x=-5
Q3010) Solve 29x^2+33x+4=0
Ans: x=-0.14 or x=-1
Q3011) Solve 42x^2+39x+9=0
Ans: x=-0.43 or x=-0.5
Q3012) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q3013) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q3014) Solve 17x^2+22x+5=0
Ans: x=-0.29 or x=-1
Q3015) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q3016) Solve 35x^2+47x+8=0
Ans: x=-0.2 or x=-1.14
Q3017) Solve 30x^2+49x+13=0
Ans: x=-0.33 or x=-1.3
Q3018) Solve 10x^2+33x+9=0
Ans: x=-0.3 or x=-3
Q3019) Solve 45x^2+48x+3=0
Ans: x=-0.07 or x=-1
Q3020) Solve 4x^2+23x+30=0
Ans: x=-2 or x=-3.75
Q3021) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q3022) Solve 12x^2+26x+10=0
Ans: x=-0.5 or x=-1.67
Q3023) Solve 3x^2+37x+44=0
Ans: x=-1.33 or x=-11
Q3024) Solve 7x^2+29x+22=0
Ans: x=-1 or x=-3.14
Q3025) Solve 16x^2+50x+39=0
Ans: x=-1.5 or x=-1.62
Q3026) Solve 40x^2+38x+7=0
Ans: x=-0.25 or x=-0.7
Q3027) Solve 13x^2+47x+34=0
Ans: x=-1 or x=-2.62
Q3028) Solve 10x^2+17x+3=0
Ans: x=-0.2 or x=-1.5
Q3029) Solve 28x^2+23x+4=0
Ans: x=-0.25 or x=-0.57
Q3030) Solve 5x^2+37x+42=0
Ans: x=-1.4 or x=-6
Q3031) Solve 14x^2+36x+16=0
Ans: x=-0.57 or x=-2
Q3032) Solve 9x^2+31x+12=0
Ans: x=-0.44 or x=-3
Q3033) Solve 16x^2+37x+21=0
Ans: x=-1 or x=-1.31
Q3034) Solve 4x^2+28x+33=0
Ans: x=-1.5 or x=-5.5
Q3035) Solve 9x^2+15x+6=0
Ans: x=-0.67 or x=-1
Q3036) Solve 19x^2+38x+19=0
Ans: x=-1 or x=-1
Q3037) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q3038) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q3039) Solve 9x^2+45x+50=0
Ans: x=-1.67 or x=-3.33
Q3040) Solve 19x^2+43x+24=0
Ans: x=-1 or x=-1.26
Q3041) Solve 12x^2+35x+25=0
Ans: x=-1.25 or x=-1.67
Q3042) Solve 50x^2+41x+8=0
Ans: x=-0.32 or x=-0.5
Q3043) Solve x^2+47x+46=0
Ans: x=-1 or x=-46
Q3044) Solve 28x^2+47x+10=0
Ans: x=-0.25 or x=-1.43
Q3045) Solve 9x^2+29x+6=0
Ans: x=-0.22 or x=-3
Q3046) Solve 18x^2+40x+8=0
Ans: x=-0.22 or x=-2
Q3047) Solve 12x^2+49x+15=0
Ans: x=-0.33 or x=-3.75
Q3048) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q3049) Solve 30x^2+31x+8=0
Ans: x=-0.5 or x=-0.53
Q3050) Solve 12x^2+38x+6=0
Ans: x=-0.17 or x=-3
Q3051) Solve 24x^2+27x+3=0
Ans: x=-0.12 or x=-1
Q3052) Solve 7x^2+33x+36=0
Ans: x=-1.71 or x=-3
Q3053) Solve 24x^2+14x+2=0
Ans: x=-0.25 or x=-0.33
Q3054) Solve 20x^2+40x+20=0
Ans: x=-1 or x=-1
Q3055) Solve 12x^2+47x+45=0
Ans: x=-1.67 or x=-2.25
Q3056) Solve 2x^2+25x+12=0
Ans: x=-0.5 or x=-12
Q3057) Solve 10x^2+38x+28=0
Ans: x=-1 or x=-2.8
Q3058) Solve 11x^2+22x+11=0
Ans: x=-1 or x=-1
Q3059) Solve 5x^2+26x+5=0
Ans: x=-0.2 or x=-5
Q3060) Solve 18x^2+32x+14=0
Ans: x=-0.78 or x=-1
Q3061) Solve 20x^2+20x+5=0
Ans: x=-0.5 or x=-0.5
Q3062) Solve 6x^2+19x+10=0
Ans: x=-0.67 or x=-2.5
Q3063) Solve 5x^2+38x+48=0
Ans: x=-1.6 or x=-6
Q3064) Solve x^2+46x+45=0
Ans: x=-1 or x=-45
Q3065) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q3066) Solve 16x^2+42x+5=0
Ans: x=-0.12 or x=-2.5
Q3067) Solve 39x^2+40x+9=0
Ans: x=-0.33 or x=-0.69
Q3068) Solve 7x^2+34x+27=0
Ans: x=-1 or x=-3.86
Q3069) Solve 43x^2+46x+3=0
Ans: x=-0.07 or x=-1
Q3070) Solve 8x^2+39x+45=0
Ans: x=-1.87 or x=-3
Q3071) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q3072) Solve 14x^2+29x+12=0
Ans: x=-0.57 or x=-1.5
Q3073) Solve 9x^2+38x+8=0
Ans: x=-0.22 or x=-4
Q3074) Solve 19x^2+39x+20=0
Ans: x=-1 or x=-1.05
Q3075) Solve 20x^2+44x+17=0
Ans: x=-0.5 or x=-1.7
Q3076) Solve 11x^2+34x+3=0
Ans: x=-0.09 or x=-3
Q3077) Solve 36x^2+24x+3=0
Ans: x=-0.17 or x=-0.5
Q3078) Solve 21x^2+22x+1=0
Ans: x=-0.05 or x=-1
Q3079) Solve 6x^2+43x+26=0
Ans: x=-0.67 or x=-6.5
Q3080) Solve 20x^2+41x+2=0
Ans: x=-0.05 or x=-2
Q3081) Solve x^2+46x+45=0
Ans: x=-1 or x=-45
Q3082) Solve 24x^2+38x+8=0
Ans: x=-0.25 or x=-1.33
Q3083) Solve 4x^2+42x+20=0
Ans: x=-0.5 or x=-10
Q3084) Solve 34x^2+42x+8=0
Ans: x=-0.24 or x=-1
Q3085) Solve 6x^2+11x+4=0
Ans: x=-0.5 or x=-1.33
Q3086) Solve 2x^2+19x+9=0
Ans: x=-0.5 or x=-9
Q3087) Solve 22x^2+31x+9=0
Ans: x=-0.41 or x=-1
Q3088) Solve 40x^2+44x+12=0
Ans: x=-0.5 or x=-0.6
Q3089) Solve 44x^2+38x+8=0
Ans: x=-0.36 or x=-0.5
Q3090) Solve 10x^2+37x+33=0
Ans: x=-1.5 or x=-2.2
Q3091) Solve 5x^2+22x+17=0
Ans: x=-1 or x=-3.4
Q3092) Solve 14x^2+49x+35=0
Ans: x=-1 or x=-2.5
Q3093) Solve 6x^2+50x+16=0
Ans: x=-0.33 or x=-8
Q3094) Solve 2x^2+26x+44=0
Ans: x=-2 or x=-11
Q3095) Solve 15x^2+44x+28=0
Ans: x=-0.93 or x=-2
Q3096) Solve 18x^2+28x+10=0
Ans: x=-0.56 or x=-1
Q3097) Solve 12x^2+47x+11=0
Ans: x=-0.25 or x=-3.67
Q3098) Solve 10x^2+21x+9=0
Ans: x=-0.6 or x=-1.5
Q3099) Solve 16x^2+39x+23=0
Ans: x=-1 or x=-1.44
Q3100) Solve 35x^2+27x+4=0
Ans: x=-0.2 or x=-0.57
Q3101) Solve 9x^2+41x+32=0
Ans: x=-1 or x=-3.56
Q3102) Solve 4x^2+32x+28=0
Ans: x=-1 or x=-7
Q3103) Solve 40x^2+42x+9=0
Ans: x=-0.3 or x=-0.75
Q3104) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q3105) Solve 15x^2+27x+12=0
Ans: x=-0.8 or x=-1
Q3106) Solve 27x^2+36x+12=0
Ans: x=-0.67 or x=-0.67
Q3107) Solve 23x^2+30x+7=0
Ans: x=-0.3 or x=-1
Q3108) Solve 38x^2+41x+3=0
Ans: x=-0.08 or x=-1
Q3109) Solve 3x^2+4x+1=0
Ans: x=-0.33 or x=-1
Q3110) Solve 22x^2+39x+9=0
Ans: x=-0.27 or x=-1.5
Q3111) Solve 12x^2+19x+4=0
Ans: x=-0.25 or x=-1.33
Q3112) Solve 35x^2+43x+12=0
Ans: x=-0.43 or x=-0.8
Q3113) Solve 6x^2+41x+19=0
Ans: x=-0.5 or x=-6.33
Q3114) Solve 24x^2+35x+11=0
Ans: x=-0.46 or x=-1
Q3115) Solve x^2+21x+38=0
Ans: x=-2 or x=-19
Q3116) Solve x^2+11x+24=0
Ans: x=-3 or x=-8
Q3117) Solve 3x^2+47x+44=0
Ans: x=-1 or x=-14.67
Q3118) Solve 8x^2+34x+26=0
Ans: x=-1 or x=-3.25
Q3119) Solve 8x^2+44x+36=0
Ans: x=-1 or x=-4.5
Q3120) Solve 39x^2+43x+10=0
Ans: x=-0.33 or x=-0.77
Q3121) Solve 10x^2+21x+11=0
Ans: x=-1 or x=-1.1
Q3122) Solve 12x^2+48x+48=0
Ans: x=-2 or x=-2
Q3123) Solve 10x^2+46x+36=0
Ans: x=-1 or x=-3.6
Q3124) Solve 12x^2+43x+36=0
Ans: x=-1.33 or x=-2.25
Q3125) Solve 31x^2+48x+17=0
Ans: x=-0.55 or x=-1
Q3126) Solve 18x^2+33x+14=0
Ans: x=-0.67 or x=-1.17
Q3127) Solve 17x^2+38x+8=0
Ans: x=-0.24 or x=-2
Q3128) Solve 4x^2+22x+24=0
Ans: x=-1.5 or x=-4
Q3129) Solve 24x^2+46x+15=0
Ans: x=-0.42 or x=-1.5
Q3130) Solve 32x^2+44x+14=0
Ans: x=-0.5 or x=-0.87
Q3131) Solve 17x^2+45x+22=0
Ans: x=-0.65 or x=-2
Q3132) Solve x^2+19x+34=0
Ans: x=-2 or x=-17
Q3133) Solve 8x^2+14x+6=0
Ans: x=-0.75 or x=-1
Q3134) Solve 16x^2+37x+10=0
Ans: x=-0.31 or x=-2
Q3135) Solve 12x^2+37x+26=0
Ans: x=-1.08 or x=-2
Q3136) Solve 11x^2+30x+19=0
Ans: x=-1 or x=-1.73
Q3137) Solve 4x^2+41x+10=0
Ans: x=-0.25 or x=-10
Q3138) Solve 4x^2+26x+40=0
Ans: x=-2.5 or x=-4
Q3139) Solve 31x^2+36x+5=0
Ans: x=-0.16 or x=-1
Q3140) Solve 16x^2+24x+8=0
Ans: x=-0.5 or x=-1
Q3141) Solve 10x^2+34x+24=0
Ans: x=-1 or x=-2.4
Q3142) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q3143) Solve 22x^2+31x+9=0
Ans: x=-0.41 or x=-1
Q3144) Solve 18x^2+15x+2=0
Ans: x=-0.17 or x=-0.67
Q3145) Solve 20x^2+41x+9=0
Ans: x=-0.25 or x=-1.8
Q3146) Solve 7x^2+27x+26=0
Ans: x=-1.86 or x=-2
Q3147) Solve 2x^2+27x+46=0
Ans: x=-2 or x=-11.5
Q3148) Solve 16x^2+50x+6=0
Ans: x=-0.12 or x=-3
Q3149) Solve 26x^2+49x+15=0
Ans: x=-0.38 or x=-1.5
Q3150) Solve 4x^2+22x+10=0
Ans: x=-0.5 or x=-5
Q3151) Solve 9x^2+9x+2=0
Ans: x=-0.33 or x=-0.67
Q3152) Solve 11x^2+45x+4=0
Ans: x=-0.09 or x=-4
Q3153) Solve 2x^2+19x+9=0
Ans: x=-0.5 or x=-9
Q3154) Solve 18x^2+46x+20=0
Ans: x=-0.56 or x=-2
Q3155) Solve 28x^2+36x+11=0
Ans: x=-0.5 or x=-0.79
Q3156) Solve 8x^2+39x+46=0
Ans: x=-2 or x=-2.87
Q3157) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Q3158) Solve 44x^2+48x+4=0
Ans: x=-0.09 or x=-1
Q3159) Solve 15x^2+40x+20=0
Ans: x=-0.67 or x=-2
Q3160) Solve 19x^2+39x+2=0
Ans: x=-0.05 or x=-2
Q3161) Solve 24x^2+14x+1=0
Ans: x=-0.08 or x=-0.5
Q3162) Solve 16x^2+47x+31=0
Ans: x=-1 or x=-1.94
Q3163) Solve 20x^2+34x+6=0
Ans: x=-0.2 or x=-1.5
Q3164) Solve 35x^2+46x+15=0
Ans: x=-0.6 or x=-0.71
Q3165) Solve 17x^2+36x+19=0
Ans: x=-1 or x=-1.12
Q3166) Solve 30x^2+28x+6=0
Ans: x=-0.33 or x=-0.6
Q3167) Solve 20x^2+25x+5=0
Ans: x=-0.25 or x=-1
Q3168) Solve 9x^2+40x+39=0
Ans: x=-1.44 or x=-3
Q3169) Solve 8x^2+28x+24=0
Ans: x=-1.5 or x=-2
Q3170) Solve 33x^2+43x+14=0
Ans: x=-0.64 or x=-0.67
Q3171) Solve 6x^2+49x+49=0
Ans: x=-1.17 or x=-7
Q3172) Solve 2x^2+29x+27=0
Ans: x=-1 or x=-13.5
Q3173) Solve 30x^2+29x+4=0
Ans: x=-0.17 or x=-0.8
Q3174) Solve 30x^2+35x+10=0
Ans: x=-0.5 or x=-0.67
Q3175) Solve 48x^2+19x+1=0
Ans: x=-0.06 or x=-0.33
Q3176) Solve 4x^2+24x+36=0
Ans: x=-3 or x=-3
Q3177) Solve 15x^2+41x+12=0
Ans: x=-0.33 or x=-2.4
Q3178) Solve 10x^2+41x+18=0
Ans: x=-0.5 or x=-3.6
Q3179) Solve 12x^2+38x+28=0
Ans: x=-1.17 or x=-2
Q3180) Solve 3x^2+27x+42=0
Ans: x=-2 or x=-7
Q3181) Solve 15x^2+37x+20=0
Ans: x=-0.8 or x=-1.67
Q3182) Solve 10x^2+11x+3=0
Ans: x=-0.5 or x=-0.6
Q3183) Solve 8x^2+42x+49=0
Ans: x=-1.75 or x=-3.5
Q3184) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q3185) Solve 7x^2+42x+35=0
Ans: x=-1 or x=-5
Q3186) Solve 14x^2+26x+12=0
Ans: x=-0.86 or x=-1
Q3187) Solve 15x^2+44x+29=0
Ans: x=-1 or x=-1.93
Q3188) Solve 20x^2+12x+1=0
Ans: x=-0.1 or x=-0.5
Q3189) Solve 45x^2+42x+9=0
Ans: x=-0.33 or x=-0.6
Q3190) Solve 34x^2+39x+5=0
Ans: x=-0.15 or x=-1
Q3191) Solve 40x^2+46x+9=0
Ans: x=-0.25 or x=-0.9
Q3192) Solve 18x^2+24x+8=0
Ans: x=-0.67 or x=-0.67
Q3193) Solve 36x^2+43x+7=0
Ans: x=-0.19 or x=-1
Q3194) Solve 12x^2+24x+12=0
Ans: x=-1 or x=-1
Q3195) Solve 6x^2+41x+35=0
Ans: x=-1 or x=-5.83
Q3196) Solve 10x^2+37x+21=0
Ans: x=-0.7 or x=-3
Q3197) Solve 19x^2+33x+14=0
Ans: x=-0.74 or x=-1
Q3198) Solve 4x^2+10x+4=0
Ans: x=-0.5 or x=-2
Q3199) Solve 11x^2+50x+24=0
Ans: x=-0.55 or x=-4
Q3200) Solve 7x^2+13x+6=0
Ans: x=-0.86 or x=-1
Q3201) Solve 32x^2+45x+13=0
Ans: x=-0.41 or x=-1
Q3202) Solve 14x^2+33x+18=0
Ans: x=-0.86 or x=-1.5
Q3203) Solve 4x^2+24x+32=0
Ans: x=-2 or x=-4
Q3204) Solve 20x^2+48x+16=0
Ans: x=-0.4 or x=-2
Q3205) Solve 16x^2+40x+21=0
Ans: x=-0.75 or x=-1.75
Q3206) Solve 3x^2+23x+20=0
Ans: x=-1 or x=-6.67
Q3207) Solve 4x^2+22x+30=0
Ans: x=-2.5 or x=-3
Q3208) Solve 32x^2+28x+6=0
Ans: x=-0.37 or x=-0.5
Q3209) Solve 12x^2+50x+22=0
Ans: x=-0.5 or x=-3.67
Q3210) Solve 10x^2+46x+48=0
Ans: x=-1.6 or x=-3
Q3211) Solve 23x^2+49x+6=0
Ans: x=-0.13 or x=-2
Q3212) Solve 21x^2+31x+4=0
Ans: x=-0.14 or x=-1.33
Q3213) Solve 20x^2+43x+23=0
Ans: x=-1 or x=-1.15
Q3214) Solve 21x^2+29x+8=0
Ans: x=-0.38 or x=-1
Q3215) Solve 32x^2+43x+11=0
Ans: x=-0.34 or x=-1
Q3216) Solve 10x^2+22x+12=0
Ans: x=-1 or x=-1.2
Q3217) Solve 36x^2+34x+8=0
Ans: x=-0.44 or x=-0.5
Q3218) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q3219) Solve 16x^2+16x+3=0
Ans: x=-0.25 or x=-0.75
Q3220) Solve 49x^2+42x+9=0
Ans: x=-0.43 or x=-0.43
Q3221) Solve 40x^2+46x+6=0
Ans: x=-0.15 or x=-1
Q3222) Solve 41x^2+43x+2=0
Ans: x=-0.05 or x=-1
Q3223) Solve 10x^2+24x+8=0
Ans: x=-0.4 or x=-2
Q3224) Solve 6x^2+29x+9=0
Ans: x=-0.33 or x=-4.5
Q3225) Solve 2x^2+19x+35=0
Ans: x=-2.5 or x=-7
Q3226) Solve 32x^2+47x+15=0
Ans: x=-0.47 or x=-1
Q3227) Solve 25x^2+31x+6=0
Ans: x=-0.24 or x=-1
Q3228) Solve 9x^2+36x+32=0
Ans: x=-1.33 or x=-2.67
Q3229) Solve 4x^2+22x+30=0
Ans: x=-2.5 or x=-3
Q3230) Solve 8x^2+20x+8=0
Ans: x=-0.5 or x=-2
Q3231) Solve 7x^2+35x+28=0
Ans: x=-1 or x=-4
Q3232) Solve 24x^2+16x+2=0
Ans: x=-0.17 or x=-0.5
Q3233) Solve 12x^2+37x+11=0
Ans: x=-0.33 or x=-2.75
Q3234) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q3235) Solve 42x^2+48x+6=0
Ans: x=-0.14 or x=-1
Q3236) Solve 37x^2+48x+11=0
Ans: x=-0.3 or x=-1
Q3237) Solve 25x^2+25x+6=0
Ans: x=-0.4 or x=-0.6
Q3238) Solve 35x^2+37x+2=0
Ans: x=-0.06 or x=-1
Q3239) Solve 37x^2+43x+6=0
Ans: x=-0.16 or x=-1
Q3240) Solve 2x^2+15x+18=0
Ans: x=-1.5 or x=-6
Q3241) Solve 40x^2+43x+3=0
Ans: x=-0.07 or x=-1
Q3242) Solve 42x^2+29x+4=0
Ans: x=-0.19 or x=-0.5
Q3243) Solve 15x^2+22x+8=0
Ans: x=-0.67 or x=-0.8
Q3244) Solve 30x^2+23x+4=0
Ans: x=-0.27 or x=-0.5
Q3245) Solve 33x^2+43x+14=0
Ans: x=-0.64 or x=-0.67
Q3246) Solve 8x^2+42x+27=0
Ans: x=-0.75 or x=-4.5
Q3247) Solve 10x^2+42x+8=0
Ans: x=-0.2 or x=-4
Q3248) Solve 18x^2+27x+4=0
Ans: x=-0.17 or x=-1.33
Q3249) Solve 3x^2+25x+38=0
Ans: x=-2 or x=-6.33
Q3250) Solve 18x^2+25x+8=0
Ans: x=-0.5 or x=-0.89
Q3251) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q3252) Solve 10x^2+20x+10=0
Ans: x=-1 or x=-1
Q3253) Solve 7x^2+19x+10=0
Ans: x=-0.71 or x=-2
Q3254) Solve 6x^2+27x+30=0
Ans: x=-2 or x=-2.5
Q3255) Solve 4x^2+19x+15=0
Ans: x=-1 or x=-3.75
Q3256) Solve 39x^2+44x+5=0
Ans: x=-0.13 or x=-1
Q3257) Solve 28x^2+32x+4=0
Ans: x=-0.14 or x=-1
Q3258) Solve 28x^2+32x+4=0
Ans: x=-0.14 or x=-1
Q3259) Solve 8x^2+18x+4=0
Ans: x=-0.25 or x=-2
Q3260) Solve 15x^2+41x+26=0
Ans: x=-1 or x=-1.73
Q3261) Solve 17x^2+32x+15=0
Ans: x=-0.88 or x=-1
Q3262) Solve 44x^2+15x+1=0
Ans: x=-0.09 or x=-0.25
Q3263) Solve 42x^2+46x+12=0
Ans: x=-0.43 or x=-0.67
Q3264) Solve 23x^2+42x+19=0
Ans: x=-0.83 or x=-1
Q3265) Solve 10x^2+35x+25=0
Ans: x=-1 or x=-2.5
Q3266) Solve 42x^2+40x+8=0
Ans: x=-0.29 or x=-0.67
Q3267) Solve 8x^2+46x+30=0
Ans: x=-0.75 or x=-5
Q3268) Solve 12x^2+44x+19=0
Ans: x=-0.5 or x=-3.17
Q3269) Solve 2x^2+19x+35=0
Ans: x=-2.5 or x=-7
Q3270) Solve 28x^2+32x+9=0
Ans: x=-0.5 or x=-0.64
Q3271) Solve 44x^2+49x+5=0
Ans: x=-0.11 or x=-1
Q3272) Solve 2x^2+25x+42=0
Ans: x=-2 or x=-10.5
Q3273) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q3274) Solve 6x^2+35x+25=0
Ans: x=-0.83 or x=-5
Q3275) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q3276) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q3277) Solve 48x^2+37x+7=0
Ans: x=-0.33 or x=-0.44
Q3278) Solve 20x^2+45x+25=0
Ans: x=-1 or x=-1.25
Q3279) Solve 30x^2+31x+1=0
Ans: x=-0.03 or x=-1
Q3280) Solve 6x^2+19x+10=0
Ans: x=-0.67 or x=-2.5
Q3281) Solve x^2+30x+29=0
Ans: x=-1 or x=-29
Q3282) Solve 8x^2+36x+28=0
Ans: x=-1 or x=-3.5
Q3283) Solve 3x^2+12x+12=0
Ans: x=-2 or x=-2
Q3284) Solve 12x^2+26x+12=0
Ans: x=-0.67 or x=-1.5
Q3285) Solve 2x^2+7x+3=0
Ans: x=-0.5 or x=-3
Q3286) Solve 9x^2+41x+32=0
Ans: x=-1 or x=-3.56
Q3287) Solve 7x^2+34x+27=0
Ans: x=-1 or x=-3.86
Q3288) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q3289) Solve 3x^2+25x+22=0
Ans: x=-1 or x=-7.33
Q3290) Solve 4x^2+13x+9=0
Ans: x=-1 or x=-2.25
Q3291) Solve 23x^2+47x+2=0
Ans: x=-0.04 or x=-2
Q3292) Solve 20x^2+17x+3=0
Ans: x=-0.25 or x=-0.6
Q3293) Solve 32x^2+40x+8=0
Ans: x=-0.25 or x=-1
Q3294) Solve 10x^2+45x+20=0
Ans: x=-0.5 or x=-4
Q3295) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q3296) Solve 8x^2+34x+8=0
Ans: x=-0.25 or x=-4
Q3297) Solve 16x^2+44x+10=0
Ans: x=-0.25 or x=-2.5
Q3298) Solve 3x^2+49x+46=0
Ans: x=-1 or x=-15.33
Q3299) Solve 9x^2+37x+30=0
Ans: x=-1.11 or x=-3
Q3300) Solve 18x^2+46x+20=0
Ans: x=-0.56 or x=-2
Q3301) Solve 9x^2+11x+2=0
Ans: x=-0.22 or x=-1
Q3302) Solve 44x^2+39x+7=0
Ans: x=-0.25 or x=-0.64
Q3303) Solve 6x^2+13x+7=0
Ans: x=-1 or x=-1.17
Q3304) Solve 11x^2+31x+20=0
Ans: x=-1 or x=-1.82
Q3305) Solve 48x^2+20x+2=0
Ans: x=-0.17 or x=-0.25
Q3306) Solve 7x^2+17x+6=0
Ans: x=-0.43 or x=-2
Q3307) Solve 45x^2+28x+4=0
Ans: x=-0.22 or x=-0.4
Q3308) Solve x^2+20x+19=0
Ans: x=-1 or x=-19
Q3309) Solve 11x^2+35x+6=0
Ans: x=-0.18 or x=-3
Q3310) Solve x^2+18x+32=0
Ans: x=-2 or x=-16
Q3311) Solve 17x^2+21x+4=0
Ans: x=-0.24 or x=-1
Q3312) Solve 10x^2+38x+36=0
Ans: x=-1.8 or x=-2
Q3313) Solve 2x^2+17x+26=0
Ans: x=-2 or x=-6.5
Q3314) Solve 18x^2+42x+24=0
Ans: x=-1 or x=-1.33
Q3315) Solve 15x^2+16x+4=0
Ans: x=-0.4 or x=-0.67
Q3316) Solve x^2+48x+47=0
Ans: x=-1 or x=-47
Q3317) Solve 30x^2+35x+5=0
Ans: x=-0.17 or x=-1
Q3318) Solve 22x^2+39x+9=0
Ans: x=-0.27 or x=-1.5
Q3319) Solve 2x^2+29x+39=0
Ans: x=-1.5 or x=-13
Q3320) Solve 28x^2+43x+9=0
Ans: x=-0.25 or x=-1.29
Q3321) Solve 8x^2+30x+22=0
Ans: x=-1 or x=-2.75
Q3322) Solve 5x^2+35x+50=0
Ans: x=-2 or x=-5
Q3323) Solve 26x^2+29x+3=0
Ans: x=-0.12 or x=-1
Q3324) Solve 8x^2+38x+9=0
Ans: x=-0.25 or x=-4.5
Q3325) Solve 24x^2+38x+8=0
Ans: x=-0.25 or x=-1.33
Q3326) Solve 6x^2+26x+28=0
Ans: x=-2 or x=-2.33
Q3327) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q3328) Solve 10x^2+35x+30=0
Ans: x=-1.5 or x=-2
Q3329) Solve 4x^2+16x+7=0
Ans: x=-0.5 or x=-3.5
Q3330) Solve 15x^2+49x+24=0
Ans: x=-0.6 or x=-2.67
Q3331) Solve 15x^2+17x+4=0
Ans: x=-0.33 or x=-0.8
Q3332) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q3333) Solve 38x^2+49x+11=0
Ans: x=-0.29 or x=-1
Q3334) Solve 48x^2+25x+3=0
Ans: x=-0.19 or x=-0.33
Q3335) Solve 12x^2+13x+1=0
Ans: x=-0.08 or x=-1
Q3336) Solve 24x^2+32x+8=0
Ans: x=-0.33 or x=-1
Q3337) Solve 6x^2+44x+14=0
Ans: x=-0.33 or x=-7
Q3338) Solve 44x^2+48x+4=0
Ans: x=-0.09 or x=-1
Q3339) Solve 25x^2+44x+19=0
Ans: x=-0.76 or x=-1
Q3340) Solve 22x^2+35x+12=0
Ans: x=-0.5 or x=-1.09
Q3341) Solve 43x^2+49x+6=0
Ans: x=-0.14 or x=-1
Q3342) Solve 18x^2+42x+20=0
Ans: x=-0.67 or x=-1.67
Q3343) Solve 21x^2+43x+2=0
Ans: x=-0.05 or x=-2
Q3344) Solve 3x^2+38x+35=0
Ans: x=-1 or x=-11.67
Q3345) Solve 18x^2+41x+16=0
Ans: x=-0.5 or x=-1.78
Q3346) Solve 36x^2+46x+14=0
Ans: x=-0.5 or x=-0.78
Q3347) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q3348) Solve 8x^2+39x+45=0
Ans: x=-1.87 or x=-3
Q3349) Solve 4x^2+39x+35=0
Ans: x=-1 or x=-8.75
Q3350) Solve 10x^2+41x+40=0
Ans: x=-1.6 or x=-2.5
Q3351) Solve 7x^2+24x+17=0
Ans: x=-1 or x=-2.43
Q3352) Solve 4x^2+14x+10=0
Ans: x=-1 or x=-2.5
Q3353) Solve 7x^2+28x+21=0
Ans: x=-1 or x=-3
Q3354) Solve x^2+14x+33=0
Ans: x=-3 or x=-11
Q3355) Solve 29x^2+35x+6=0
Ans: x=-0.21 or x=-1
Q3356) Solve 3x^2+24x+36=0
Ans: x=-2 or x=-6
Q3357) Solve 12x^2+43x+35=0
Ans: x=-1.25 or x=-2.33
Q3358) Solve 8x^2+10x+2=0
Ans: x=-0.25 or x=-1
Q3359) Solve 9x^2+44x+32=0
Ans: x=-0.89 or x=-4
Q3360) Solve 3x^2+11x+8=0
Ans: x=-1 or x=-2.67
Q3361) Solve 48x^2+24x+3=0
Ans: x=-0.25 or x=-0.25
Q3362) Solve 26x^2+32x+6=0
Ans: x=-0.23 or x=-1
Q3363) Solve 30x^2+32x+2=0
Ans: x=-0.07 or x=-1
Q3364) Solve 22x^2+41x+12=0
Ans: x=-0.36 or x=-1.5
Q3365) Solve 45x^2+39x+8=0
Ans: x=-0.33 or x=-0.53
Q3366) Solve 15x^2+43x+30=0
Ans: x=-1.2 or x=-1.67
Q3367) Solve 2x^2+13x+11=0
Ans: x=-1 or x=-5.5
Q3368) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q3369) Solve 9x^2+19x+10=0
Ans: x=-1 or x=-1.11
Q3370) Solve 9x^2+48x+28=0
Ans: x=-0.67 or x=-4.67
Q3371) Solve 15x^2+34x+8=0
Ans: x=-0.27 or x=-2
Q3372) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q3373) Solve 7x^2+20x+12=0
Ans: x=-0.86 or x=-2
Q3374) Solve 6x^2+48x+42=0
Ans: x=-1 or x=-7
Q3375) Solve 39x^2+44x+12=0
Ans: x=-0.46 or x=-0.67
Q3376) Solve 8x^2+44x+36=0
Ans: x=-1 or x=-4.5
Q3377) Solve 5x^2+29x+36=0
Ans: x=-1.8 or x=-4
Q3378) Solve 27x^2+42x+16=0
Ans: x=-0.67 or x=-0.89
Q3379) Solve 43x^2+44x+1=0
Ans: x=-0.02 or x=-1
Q3380) Solve 16x^2+44x+10=0
Ans: x=-0.25 or x=-2.5
Q3381) Solve 4x^2+28x+45=0
Ans: x=-2.5 or x=-4.5
Q3382) Solve 11x^2+42x+27=0
Ans: x=-0.82 or x=-3
Q3383) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q3384) Solve 35x^2+36x+1=0
Ans: x=-0.03 or x=-1
Q3385) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q3386) Solve 10x^2+38x+36=0
Ans: x=-1.8 or x=-2
Q3387) Solve 6x^2+36x+30=0
Ans: x=-1 or x=-5
Q3388) Solve 36x^2+41x+5=0
Ans: x=-0.14 or x=-1
Q3389) Solve 12x^2+21x+9=0
Ans: x=-0.75 or x=-1
Q3390) Solve 35x^2+36x+9=0
Ans: x=-0.43 or x=-0.6
Q3391) Solve 14x^2+40x+26=0
Ans: x=-1 or x=-1.86
Q3392) Solve 40x^2+22x+3=0
Ans: x=-0.25 or x=-0.3
Q3393) Solve 26x^2+31x+9=0
Ans: x=-0.5 or x=-0.69
Q3394) Solve 26x^2+37x+12=0
Ans: x=-0.5 or x=-0.92
Q3395) Solve 5x^2+18x+16=0
Ans: x=-1.6 or x=-2
Q3396) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q3397) Solve 10x^2+27x+17=0
Ans: x=-1 or x=-1.7
Q3398) Solve 20x^2+21x+1=0
Ans: x=-0.05 or x=-1
Q3399) Solve 24x^2+38x+3=0
Ans: x=-0.08 or x=-1.5
Q3400) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q3401) Solve 13x^2+43x+34=0
Ans: x=-1.31 or x=-2
Q3402) Solve 48x^2+20x+2=0
Ans: x=-0.17 or x=-0.25
Q3403) Solve 12x^2+25x+7=0
Ans: x=-0.33 or x=-1.75
Q3404) Solve 48x^2+30x+3=0
Ans: x=-0.12 or x=-0.5
Q3405) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q3406) Solve 12x^2+43x+21=0
Ans: x=-0.58 or x=-3
Q3407) Solve 8x^2+33x+34=0
Ans: x=-2 or x=-2.12
Q3408) Solve 12x^2+17x+6=0
Ans: x=-0.67 or x=-0.75
Q3409) Solve 10x^2+44x+42=0
Ans: x=-1.4 or x=-3
Q3410) Solve 3x^2+17x+10=0
Ans: x=-0.67 or x=-5
Q3411) Solve 12x^2+40x+32=0
Ans: x=-1.33 or x=-2
Q3412) Solve 12x^2+27x+15=0
Ans: x=-1 or x=-1.25
Q3413) Solve 28x^2+46x+18=0
Ans: x=-0.64 or x=-1
Q3414) Solve 45x^2+27x+4=0
Ans: x=-0.27 or x=-0.33
Q3415) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q3416) Solve 45x^2+47x+2=0
Ans: x=-0.04 or x=-1
Q3417) Solve 11x^2+29x+14=0
Ans: x=-0.64 or x=-2
Q3418) Solve 8x^2+46x+38=0
Ans: x=-1 or x=-4.75
Q3419) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q3420) Solve 21x^2+50x+29=0
Ans: x=-1 or x=-1.38
Q3421) Solve x^2+17x+16=0
Ans: x=-1 or x=-16
Q3422) Solve 8x^2+39x+31=0
Ans: x=-1 or x=-3.87
Q3423) Solve 19x^2+39x+2=0
Ans: x=-0.05 or x=-2
Q3424) Solve 16x^2+42x+17=0
Ans: x=-0.5 or x=-2.12
Q3425) Solve 6x^2+35x+16=0
Ans: x=-0.5 or x=-5.33
Q3426) Solve x^2+9x+18=0
Ans: x=-3 or x=-6
Q3427) Solve 2x^2+20x+32=0
Ans: x=-2 or x=-8
Q3428) Solve 3x^2+19x+26=0
Ans: x=-2 or x=-4.33
Q3429) Solve 28x^2+50x+12=0
Ans: x=-0.29 or x=-1.5
Q3430) Solve 4x^2+42x+38=0
Ans: x=-1 or x=-9.5
Q3431) Solve 35x^2+19x+2=0
Ans: x=-0.14 or x=-0.4
Q3432) Solve 15x^2+41x+26=0
Ans: x=-1 or x=-1.73
Q3433) Solve 5x^2+35x+30=0
Ans: x=-1 or x=-6
Q3434) Solve 10x^2+32x+6=0
Ans: x=-0.2 or x=-3
Q3435) Solve 48x^2+38x+4=0
Ans: x=-0.12 or x=-0.67
Q3436) Solve 9x^2+27x+14=0
Ans: x=-0.67 or x=-2.33
Q3437) Solve 8x^2+42x+19=0
Ans: x=-0.5 or x=-4.75
Q3438) Solve 36x^2+43x+12=0
Ans: x=-0.44 or x=-0.75
Q3439) Solve 30x^2+43x+11=0
Ans: x=-0.33 or x=-1.1
Q3440) Solve 16x^2+38x+15=0
Ans: x=-0.5 or x=-1.87
Q3441) Solve 12x^2+35x+22=0
Ans: x=-0.92 or x=-2
Q3442) Solve 22x^2+50x+12=0
Ans: x=-0.27 or x=-2
Q3443) Solve 5x^2+26x+32=0
Ans: x=-2 or x=-3.2
Q3444) Solve 34x^2+37x+10=0
Ans: x=-0.5 or x=-0.59
Q3445) Solve 39x^2+45x+6=0
Ans: x=-0.15 or x=-1
Q3446) Solve 24x^2+38x+8=0
Ans: x=-0.25 or x=-1.33
Q3447) Solve 30x^2+44x+14=0
Ans: x=-0.47 or x=-1
Q3448) Solve 9x^2+40x+44=0
Ans: x=-2 or x=-2.44
Q3449) Solve 4x^2+27x+45=0
Ans: x=-3 or x=-3.75
Q3450) Solve 4x^2+19x+21=0
Ans: x=-1.75 or x=-3
Q3451) Solve 8x^2+40x+48=0
Ans: x=-2 or x=-3
Q3452) Solve 26x^2+19x+3=0
Ans: x=-0.23 or x=-0.5
Q3453) Solve 49x^2+50x+1=0
Ans: x=-0.02 or x=-1
Q3454) Solve 14x^2+28x+14=0
Ans: x=-1 or x=-1
Q3455) Solve 12x^2+21x+9=0
Ans: x=-0.75 or x=-1
Q3456) Solve 6x^2+35x+49=0
Ans: x=-2.33 or x=-3.5
Q3457) Solve 16x^2+46x+15=0
Ans: x=-0.37 or x=-2.5
Q3458) Solve 5x^2+41x+42=0
Ans: x=-1.2 or x=-7
Q3459) Solve 3x^2+28x+44=0
Ans: x=-2 or x=-7.33
Q3460) Solve 9x^2+49x+20=0
Ans: x=-0.44 or x=-5
Q3461) Solve 10x^2+33x+26=0
Ans: x=-1.3 or x=-2
Q3462) Solve 4x^2+36x+45=0
Ans: x=-1.5 or x=-7.5
Q3463) Solve 5x^2+22x+24=0
Ans: x=-2 or x=-2.4
Q3464) Solve 48x^2+36x+6=0
Ans: x=-0.25 or x=-0.5
Q3465) Solve 25x^2+28x+3=0
Ans: x=-0.12 or x=-1
Q3466) Solve 7x^2+50x+7=0
Ans: x=-0.14 or x=-7
Q3467) Solve 15x^2+17x+2=0
Ans: x=-0.13 or x=-1
Q3468) Solve 13x^2+21x+8=0
Ans: x=-0.62 or x=-1
Q3469) Solve 45x^2+42x+8=0
Ans: x=-0.27 or x=-0.67
Q3470) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q3471) Solve 9x^2+11x+2=0
Ans: x=-0.22 or x=-1
Q3472) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q3473) Solve 5x^2+22x+24=0
Ans: x=-2 or x=-2.4
Q3474) Solve 6x^2+38x+12=0
Ans: x=-0.33 or x=-6
Q3475) Solve 28x^2+46x+18=0
Ans: x=-0.64 or x=-1
Q3476) Solve 27x^2+18x+3=0
Ans: x=-0.33 or x=-0.33
Q3477) Solve 39x^2+40x+1=0
Ans: x=-0.03 or x=-1
Q3478) Solve 9x^2+33x+18=0
Ans: x=-0.67 or x=-3
Q3479) Solve 18x^2+31x+13=0
Ans: x=-0.72 or x=-1
Q3480) Solve 28x^2+47x+10=0
Ans: x=-0.25 or x=-1.43
Q3481) Solve 6x^2+37x+42=0
Ans: x=-1.5 or x=-4.67
Q3482) Solve 17x^2+33x+16=0
Ans: x=-0.94 or x=-1
Q3483) Solve 7x^2+24x+17=0
Ans: x=-1 or x=-2.43
Q3484) Solve 9x^2+28x+20=0
Ans: x=-1.11 or x=-2
Q3485) Solve 6x^2+23x+7=0
Ans: x=-0.33 or x=-3.5
Q3486) Solve 4x^2+36x+45=0
Ans: x=-1.5 or x=-7.5
Q3487) Solve 20x^2+43x+23=0
Ans: x=-1 or x=-1.15
Q3488) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q3489) Solve 17x^2+36x+4=0
Ans: x=-0.12 or x=-2
Q3490) Solve 22x^2+43x+21=0
Ans: x=-0.95 or x=-1
Q3491) Solve 16x^2+42x+27=0
Ans: x=-1.12 or x=-1.5
Q3492) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q3493) Solve 18x^2+47x+19=0
Ans: x=-0.5 or x=-2.11
Q3494) Solve 48x^2+26x+3=0
Ans: x=-0.17 or x=-0.37
Q3495) Solve 39x^2+43x+10=0
Ans: x=-0.33 or x=-0.77
Q3496) Solve 6x^2+35x+16=0
Ans: x=-0.5 or x=-5.33
Q3497) Solve 16x^2+25x+9=0
Ans: x=-0.56 or x=-1
Q3498) Solve 34x^2+37x+3=0
Ans: x=-0.09 or x=-1
Q3499) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q3500) Solve 29x^2+44x+15=0
Ans: x=-0.52 or x=-1
Q3501) Solve 39x^2+28x+5=0
Ans: x=-0.33 or x=-0.38
Q3502) Solve 3x^2+9x+6=0
Ans: x=-1 or x=-2
Q3503) Solve 7x^2+15x+8=0
Ans: x=-1 or x=-1.14
Q3504) Solve 30x^2+31x+7=0
Ans: x=-0.33 or x=-0.7
Q3505) Solve 14x^2+36x+16=0
Ans: x=-0.57 or x=-2
Q3506) Solve 36x^2+44x+13=0
Ans: x=-0.5 or x=-0.72
Q3507) Solve 34x^2+44x+10=0
Ans: x=-0.29 or x=-1
Q3508) Solve 5x^2+34x+48=0
Ans: x=-2 or x=-4.8
Q3509) Solve 18x^2+21x+5=0
Ans: x=-0.33 or x=-0.83
Q3510) Solve 39x^2+49x+12=0
Ans: x=-0.33 or x=-0.92
Q3511) Solve 3x^2+29x+46=0
Ans: x=-2 or x=-7.67
Q3512) Solve 34x^2+42x+8=0
Ans: x=-0.24 or x=-1
Q3513) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q3514) Solve 44x^2+40x+9=0
Ans: x=-0.41 or x=-0.5
Q3515) Solve 14x^2+34x+12=0
Ans: x=-0.43 or x=-2
Q3516) Solve 3x^2+50x+32=0
Ans: x=-0.67 or x=-16
Q3517) Solve 7x^2+34x+27=0
Ans: x=-1 or x=-3.86
Q3518) Solve 7x^2+29x+30=0
Ans: x=-2 or x=-2.14
Q3519) Solve 10x^2+40x+40=0
Ans: x=-2 or x=-2
Q3520) Solve 25x^2+30x+9=0
Ans: x=-0.6 or x=-0.6
Q3521) Solve 10x^2+21x+9=0
Ans: x=-0.6 or x=-1.5
Q3522) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q3523) Solve 30x^2+46x+8=0
Ans: x=-0.2 or x=-1.33
Q3524) Solve 15x^2+32x+16=0
Ans: x=-0.8 or x=-1.33
Q3525) Solve 3x^2+16x+5=0
Ans: x=-0.33 or x=-5
Q3526) Solve 11x^2+14x+3=0
Ans: x=-0.27 or x=-1
Q3527) Solve 3x^2+4x+1=0
Ans: x=-0.33 or x=-1
Q3528) Solve 32x^2+36x+4=0
Ans: x=-0.12 or x=-1
Q3529) Solve 3x^2+23x+42=0
Ans: x=-3 or x=-4.67
Q3530) Solve 35x^2+22x+3=0
Ans: x=-0.2 or x=-0.43
Q3531) Solve 9x^2+47x+10=0
Ans: x=-0.22 or x=-5
Q3532) Solve 16x^2+24x+9=0
Ans: x=-0.75 or x=-0.75
Q3533) Solve 2x^2+37x+18=0
Ans: x=-0.5 or x=-18
Q3534) Solve 40x^2+38x+6=0
Ans: x=-0.2 or x=-0.75
Q3535) Solve 12x^2+47x+11=0
Ans: x=-0.25 or x=-3.67
Q3536) Solve x^2+8x+15=0
Ans: x=-3 or x=-5
Q3537) Solve 25x^2+34x+9=0
Ans: x=-0.36 or x=-1
Q3538) Solve 16x^2+46x+19=0
Ans: x=-0.5 or x=-2.37
Q3539) Solve 13x^2+38x+24=0
Ans: x=-0.92 or x=-2
Q3540) Solve 2x^2+30x+28=0
Ans: x=-1 or x=-14
Q3541) Solve 2x^2+20x+42=0
Ans: x=-3 or x=-7
Q3542) Solve 44x^2+28x+3=0
Ans: x=-0.14 or x=-0.5
Q3543) Solve 14x^2+35x+14=0
Ans: x=-0.5 or x=-2
Q3544) Solve 15x^2+37x+22=0
Ans: x=-1 or x=-1.47
Q3545) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q3546) Solve 6x^2+44x+14=0
Ans: x=-0.33 or x=-7
Q3547) Solve 8x^2+37x+39=0
Ans: x=-1.62 or x=-3
Q3548) Solve 30x^2+41x+11=0
Ans: x=-0.37 or x=-1
Q3549) Solve 35x^2+38x+8=0
Ans: x=-0.29 or x=-0.8
Q3550) Solve 6x^2+31x+40=0
Ans: x=-2.5 or x=-2.67
Q3551) Solve 32x^2+42x+10=0
Ans: x=-0.31 or x=-1
Q3552) Solve 12x^2+29x+14=0
Ans: x=-0.67 or x=-1.75
Q3553) Solve 2x^2+19x+35=0
Ans: x=-2.5 or x=-7
Q3554) Solve 42x^2+31x+4=0
Ans: x=-0.17 or x=-0.57
Q3555) Solve 9x^2+18x+9=0
Ans: x=-1 or x=-1
Q3556) Solve 36x^2+44x+13=0
Ans: x=-0.5 or x=-0.72
Q3557) Solve 40x^2+46x+6=0
Ans: x=-0.15 or x=-1
Q3558) Solve 35x^2+42x+7=0
Ans: x=-0.2 or x=-1
Q3559) Solve 14x^2+37x+18=0
Ans: x=-0.64 or x=-2
Q3560) Solve 15x^2+28x+13=0
Ans: x=-0.87 or x=-1
Q3561) Solve 36x^2+34x+8=0
Ans: x=-0.44 or x=-0.5
Q3562) Solve 15x^2+26x+7=0
Ans: x=-0.33 or x=-1.4
Q3563) Solve 27x^2+30x+8=0
Ans: x=-0.44 or x=-0.67
Q3564) Solve 3x^2+13x+14=0
Ans: x=-2 or x=-2.33
Q3565) Solve 16x^2+49x+34=0
Ans: x=-1.06 or x=-2
Q3566) Solve 3x^2+16x+21=0
Ans: x=-2.33 or x=-3
Q3567) Solve 5x^2+36x+36=0
Ans: x=-1.2 or x=-6
Q3568) Solve 48x^2+31x+5=0
Ans: x=-0.31 or x=-0.33
Q3569) Solve 8x^2+50x+33=0
Ans: x=-0.75 or x=-5.5
Q3570) Solve 20x^2+34x+6=0
Ans: x=-0.2 or x=-1.5
Q3571) Solve 7x^2+28x+21=0
Ans: x=-1 or x=-3
Q3572) Solve x^2+38x+37=0
Ans: x=-1 or x=-37
Q3573) Solve 16x^2+48x+11=0
Ans: x=-0.25 or x=-2.75
Q3574) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q3575) Solve 24x^2+30x+6=0
Ans: x=-0.25 or x=-1
Q3576) Solve 11x^2+15x+4=0
Ans: x=-0.36 or x=-1
Q3577) Solve 40x^2+34x+6=0
Ans: x=-0.25 or x=-0.6
Q3578) Solve 14x^2+41x+27=0
Ans: x=-1 or x=-1.93
Q3579) Solve 13x^2+34x+21=0
Ans: x=-1 or x=-1.62
Q3580) Solve 3x^2+28x+32=0
Ans: x=-1.33 or x=-8
Q3581) Solve 37x^2+44x+7=0
Ans: x=-0.19 or x=-1
Q3582) Solve 2x^2+33x+31=0
Ans: x=-1 or x=-15.5
Q3583) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q3584) Solve 33x^2+35x+8=0
Ans: x=-0.33 or x=-0.73
Q3585) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q3586) Solve 18x^2+31x+6=0
Ans: x=-0.22 or x=-1.5
Q3587) Solve 10x^2+48x+32=0
Ans: x=-0.8 or x=-4
Q3588) Solve 3x^2+17x+10=0
Ans: x=-0.67 or x=-5
Q3589) Solve 27x^2+24x+4=0
Ans: x=-0.22 or x=-0.67
Q3590) Solve x^2+14x+33=0
Ans: x=-3 or x=-11
Q3591) Solve 18x^2+49x+33=0
Ans: x=-1.22 or x=-1.5
Q3592) Solve 37x^2+38x+1=0
Ans: x=-0.03 or x=-1
Q3593) Solve 15x^2+50x+40=0
Ans: x=-1.33 or x=-2
Q3594) Solve 20x^2+50x+20=0
Ans: x=-0.5 or x=-2
Q3595) Solve 4x^2+30x+14=0
Ans: x=-0.5 or x=-7
Q3596) Solve 22x^2+46x+24=0
Ans: x=-1 or x=-1.09
Q3597) Solve 36x^2+20x+1=0
Ans: x=-0.06 or x=-0.5
Q3598) Solve 30x^2+42x+12=0
Ans: x=-0.4 or x=-1
Q3599) Solve 8x^2+37x+20=0
Ans: x=-0.62 or x=-4
Q3600) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q3601) Solve 16x^2+18x+5=0
Ans: x=-0.5 or x=-0.62
Q3602) Solve 14x^2+13x+3=0
Ans: x=-0.43 or x=-0.5
Q3603) Solve 10x^2+31x+22=0
Ans: x=-1.1 or x=-2
Q3604) Solve 12x^2+19x+4=0
Ans: x=-0.25 or x=-1.33
Q3605) Solve 41x^2+43x+2=0
Ans: x=-0.05 or x=-1
Q3606) Solve 34x^2+36x+2=0
Ans: x=-0.06 or x=-1
Q3607) Solve 8x^2+21x+13=0
Ans: x=-1 or x=-1.62
Q3608) Solve 14x^2+39x+27=0
Ans: x=-1.29 or x=-1.5
Q3609) Solve 16x^2+44x+18=0
Ans: x=-0.5 or x=-2.25
Q3610) Solve 2x^2+15x+13=0
Ans: x=-1 or x=-6.5
Q3611) Solve 8x^2+45x+25=0
Ans: x=-0.62 or x=-5
Q3612) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q3613) Solve 46x^2+29x+3=0
Ans: x=-0.13 or x=-0.5
Q3614) Solve 3x^2+24x+36=0
Ans: x=-2 or x=-6
Q3615) Solve 39x^2+32x+4=0
Ans: x=-0.15 or x=-0.67
Q3616) Solve 9x^2+32x+23=0
Ans: x=-1 or x=-2.56
Q3617) Solve 39x^2+48x+9=0
Ans: x=-0.23 or x=-1
Q3618) Solve 22x^2+43x+16=0
Ans: x=-0.5 or x=-1.45
Q3619) Solve 10x^2+43x+19=0
Ans: x=-0.5 or x=-3.8
Q3620) Solve 8x^2+35x+38=0
Ans: x=-2 or x=-2.37
Q3621) Solve 9x^2+35x+34=0
Ans: x=-1.89 or x=-2
Q3622) Solve 18x^2+47x+22=0
Ans: x=-0.61 or x=-2
Q3623) Solve 24x^2+30x+6=0
Ans: x=-0.25 or x=-1
Q3624) Solve 45x^2+42x+8=0
Ans: x=-0.27 or x=-0.67
Q3625) Solve 2x^2+43x+41=0
Ans: x=-1 or x=-20.5
Q3626) Solve 36x^2+45x+14=0
Ans: x=-0.58 or x=-0.67
Q3627) Solve 50x^2+45x+9=0
Ans: x=-0.3 or x=-0.6
Q3628) Solve 48x^2+26x+3=0
Ans: x=-0.17 or x=-0.37
Q3629) Solve 24x^2+27x+3=0
Ans: x=-0.12 or x=-1
Q3630) Solve 2x^2+19x+45=0
Ans: x=-4.5 or x=-5
Q3631) Solve 24x^2+35x+4=0
Ans: x=-0.12 or x=-1.33
Q3632) Solve 10x^2+47x+21=0
Ans: x=-0.5 or x=-4.2
Q3633) Solve 12x^2+32x+16=0
Ans: x=-0.67 or x=-2
Q3634) Solve 2x^2+22x+48=0
Ans: x=-3 or x=-8
Q3635) Solve 9x^2+12x+3=0
Ans: x=-0.33 or x=-1
Q3636) Solve 10x^2+47x+28=0
Ans: x=-0.7 or x=-4
Q3637) Solve 14x^2+39x+10=0
Ans: x=-0.29 or x=-2.5
Q3638) Solve 17x^2+45x+28=0
Ans: x=-1 or x=-1.65
Q3639) Solve 38x^2+44x+6=0
Ans: x=-0.16 or x=-1
Q3640) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q3641) Solve 3x^2+50x+47=0
Ans: x=-1 or x=-15.67
Q3642) Solve 8x^2+44x+20=0
Ans: x=-0.5 or x=-5
Q3643) Solve 10x^2+38x+24=0
Ans: x=-0.8 or x=-3
Q3644) Solve 15x^2+39x+24=0
Ans: x=-1 or x=-1.6
Q3645) Solve 20x^2+41x+21=0
Ans: x=-1 or x=-1.05
Q3646) Solve 15x^2+37x+14=0
Ans: x=-0.47 or x=-2
Q3647) Solve 26x^2+47x+12=0
Ans: x=-0.31 or x=-1.5
Q3648) Solve 3x^2+19x+16=0
Ans: x=-1 or x=-5.33
Q3649) Solve 11x^2+45x+46=0
Ans: x=-2 or x=-2.09
Q3650) Solve 33x^2+17x+2=0
Ans: x=-0.18 or x=-0.33
Q3651) Solve 24x^2+26x+2=0
Ans: x=-0.08 or x=-1
Q3652) Solve 21x^2+50x+25=0
Ans: x=-0.71 or x=-1.67
Q3653) Solve 20x^2+22x+2=0
Ans: x=-0.1 or x=-1
Q3654) Solve 15x^2+45x+30=0
Ans: x=-1 or x=-2
Q3655) Solve 22x^2+50x+28=0
Ans: x=-1 or x=-1.27
Q3656) Solve 14x^2+44x+6=0
Ans: x=-0.14 or x=-3
Q3657) Solve 30x^2+19x+3=0
Ans: x=-0.3 or x=-0.33
Q3658) Solve 18x^2+34x+16=0
Ans: x=-0.89 or x=-1
Q3659) Solve 9x^2+33x+10=0
Ans: x=-0.33 or x=-3.33
Q3660) Solve 19x^2+41x+22=0
Ans: x=-1 or x=-1.16
Q3661) Solve 37x^2+38x+1=0
Ans: x=-0.03 or x=-1
Q3662) Solve 12x^2+13x+1=0
Ans: x=-0.08 or x=-1
Q3663) Solve 15x^2+50x+35=0
Ans: x=-1 or x=-2.33
Q3664) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q3665) Solve 7x^2+8x+1=0
Ans: x=-0.14 or x=-1
Q3666) Solve 8x^2+21x+13=0
Ans: x=-1 or x=-1.62
Q3667) Solve 18x^2+29x+11=0
Ans: x=-0.61 or x=-1
Q3668) Solve 8x^2+24x+16=0
Ans: x=-1 or x=-2
Q3669) Solve 40x^2+47x+12=0
Ans: x=-0.37 or x=-0.8
Q3670) Solve 13x^2+22x+9=0
Ans: x=-0.69 or x=-1
Q3671) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q3672) Solve 2x^2+21x+40=0
Ans: x=-2.5 or x=-8
Q3673) Solve 4x^2+6x+2=0
Ans: x=-0.5 or x=-1
Q3674) Solve 12x^2+44x+24=0
Ans: x=-0.67 or x=-3
Q3675) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q3676) Solve 13x^2+29x+6=0
Ans: x=-0.23 or x=-2
Q3677) Solve 2x^2+18x+40=0
Ans: x=-4 or x=-5
Q3678) Solve 16x^2+44x+30=0
Ans: x=-1.25 or x=-1.5
Q3679) Solve 5x^2+43x+38=0
Ans: x=-1 or x=-7.6
Q3680) Solve 24x^2+50x+14=0
Ans: x=-0.33 or x=-1.75
Q3681) Solve 20x^2+34x+14=0
Ans: x=-0.7 or x=-1
Q3682) Solve 9x^2+42x+48=0
Ans: x=-2 or x=-2.67
Q3683) Solve 3x^2+23x+20=0
Ans: x=-1 or x=-6.67
Q3684) Solve 42x^2+40x+8=0
Ans: x=-0.29 or x=-0.67
Q3685) Solve 42x^2+31x+5=0
Ans: x=-0.24 or x=-0.5
Q3686) Solve 13x^2+27x+2=0
Ans: x=-0.08 or x=-2
Q3687) Solve 44x^2+49x+12=0
Ans: x=-0.36 or x=-0.75
Q3688) Solve 10x^2+39x+38=0
Ans: x=-1.9 or x=-2
Q3689) Solve 30x^2+26x+4=0
Ans: x=-0.2 or x=-0.67
Q3690) Solve x^2+14x+48=0
Ans: x=-6 or x=-8
Q3691) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q3692) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q3693) Solve 35x^2+26x+3=0
Ans: x=-0.14 or x=-0.6
Q3694) Solve 26x^2+25x+6=0
Ans: x=-0.46 or x=-0.5
Q3695) Solve 18x^2+20x+2=0
Ans: x=-0.11 or x=-1
Q3696) Solve 39x^2+46x+7=0
Ans: x=-0.18 or x=-1
Q3697) Solve 18x^2+39x+11=0
Ans: x=-0.33 or x=-1.83
Q3698) Solve 24x^2+50x+11=0
Ans: x=-0.25 or x=-1.83
Q3699) Solve 7x^2+38x+48=0
Ans: x=-2 or x=-3.43
Q3700) Solve 6x^2+5x+1=0
Ans: x=-0.33 or x=-0.5
Q3701) Solve 9x^2+43x+28=0
Ans: x=-0.78 or x=-4
Q3702) Solve 29x^2+33x+4=0
Ans: x=-0.14 or x=-1
Q3703) Solve 2x^2+20x+42=0
Ans: x=-3 or x=-7
Q3704) Solve 50x^2+45x+4=0
Ans: x=-0.1 or x=-0.8
Q3705) Solve 8x^2+40x+18=0
Ans: x=-0.5 or x=-4.5
Q3706) Solve 34x^2+41x+12=0
Ans: x=-0.5 or x=-0.71
Q3707) Solve 6x^2+34x+40=0
Ans: x=-1.67 or x=-4
Q3708) Solve 23x^2+33x+10=0
Ans: x=-0.43 or x=-1
Q3709) Solve 11x^2+30x+19=0
Ans: x=-1 or x=-1.73
Q3710) Solve 6x^2+19x+15=0
Ans: x=-1.5 or x=-1.67
Q3711) Solve 24x^2+26x+6=0
Ans: x=-0.33 or x=-0.75
Q3712) Solve 3x^2+8x+5=0
Ans: x=-1 or x=-1.67
Q3713) Solve 12x^2+47x+33=0
Ans: x=-0.92 or x=-3
Q3714) Solve 10x^2+37x+21=0
Ans: x=-0.7 or x=-3
Q3715) Solve 9x^2+46x+5=0
Ans: x=-0.11 or x=-5
Q3716) Solve 14x^2+41x+15=0
Ans: x=-0.43 or x=-2.5
Q3717) Solve 22x^2+44x+22=0
Ans: x=-1 or x=-1
Q3718) Solve 14x^2+32x+8=0
Ans: x=-0.29 or x=-2
Q3719) Solve 8x^2+37x+42=0
Ans: x=-2 or x=-2.62
Q3720) Solve 5x^2+31x+44=0
Ans: x=-2.2 or x=-4
Q3721) Solve 32x^2+26x+5=0
Ans: x=-0.31 or x=-0.5
Q3722) Solve 9x^2+33x+30=0
Ans: x=-1.67 or x=-2
Q3723) Solve 5x^2+33x+40=0
Ans: x=-1.6 or x=-5
Q3724) Solve 18x^2+35x+12=0
Ans: x=-0.44 or x=-1.5
Q3725) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q3726) Solve 9x^2+24x+12=0
Ans: x=-0.67 or x=-2
Q3727) Solve 42x^2+37x+5=0
Ans: x=-0.17 or x=-0.71
Q3728) Solve 2x^2+11x+5=0
Ans: x=-0.5 or x=-5
Q3729) Solve 48x^2+40x+7=0
Ans: x=-0.25 or x=-0.58
Q3730) Solve 10x^2+33x+23=0
Ans: x=-1 or x=-2.3
Q3731) Solve 32x^2+36x+9=0
Ans: x=-0.37 or x=-0.75
Q3732) Solve 5x^2+42x+49=0
Ans: x=-1.4 or x=-7
Q3733) Solve 6x^2+24x+18=0
Ans: x=-1 or x=-3
Q3734) Solve 2x^2+5x+3=0
Ans: x=-1 or x=-1.5
Q3735) Solve 8x^2+20x+12=0
Ans: x=-1 or x=-1.5
Q3736) Solve 32x^2+26x+5=0
Ans: x=-0.31 or x=-0.5
Q3737) Solve 28x^2+24x+5=0
Ans: x=-0.36 or x=-0.5
Q3738) Solve 4x^2+22x+28=0
Ans: x=-2 or x=-3.5
Q3739) Solve 4x^2+6x+2=0
Ans: x=-0.5 or x=-1
Q3740) Solve 39x^2+41x+2=0
Ans: x=-0.05 or x=-1
Q3741) Solve 21x^2+40x+11=0
Ans: x=-0.33 or x=-1.57
Q3742) Solve 16x^2+18x+2=0
Ans: x=-0.12 or x=-1
Q3743) Solve 26x^2+39x+13=0
Ans: x=-0.5 or x=-1
Q3744) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q3745) Solve 3x^2+35x+32=0
Ans: x=-1 or x=-10.67
Q3746) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q3747) Solve 30x^2+35x+5=0
Ans: x=-0.17 or x=-1
Q3748) Solve 20x^2+45x+10=0
Ans: x=-0.25 or x=-2
Q3749) Solve 2x^2+15x+27=0
Ans: x=-3 or x=-4.5
Q3750) Solve 30x^2+49x+12=0
Ans: x=-0.3 or x=-1.33
Q3751) Solve 15x^2+16x+4=0
Ans: x=-0.4 or x=-0.67
Q3752) Solve 45x^2+48x+11=0
Ans: x=-0.33 or x=-0.73
Q3753) Solve 3x^2+11x+6=0
Ans: x=-0.67 or x=-3
Q3754) Solve 7x^2+22x+15=0
Ans: x=-1 or x=-2.14
Q3755) Solve 48x^2+19x+1=0
Ans: x=-0.06 or x=-0.33
Q3756) Solve 50x^2+45x+9=0
Ans: x=-0.3 or x=-0.6
Q3757) Solve 8x^2+40x+48=0
Ans: x=-2 or x=-3
Q3758) Solve 15x^2+34x+19=0
Ans: x=-1 or x=-1.27
Q3759) Solve 16x^2+27x+11=0
Ans: x=-0.69 or x=-1
Q3760) Solve 13x^2+44x+36=0
Ans: x=-1.38 or x=-2
Q3761) Solve 11x^2+43x+30=0
Ans: x=-0.91 or x=-3
Q3762) Solve 20x^2+40x+20=0
Ans: x=-1 or x=-1
Q3763) Solve 2x^2+17x+33=0
Ans: x=-3 or x=-5.5
Q3764) Solve 18x^2+39x+21=0
Ans: x=-1 or x=-1.17
Q3765) Solve 44x^2+48x+4=0
Ans: x=-0.09 or x=-1
Q3766) Solve 5x^2+18x+13=0
Ans: x=-1 or x=-2.6
Q3767) Solve 30x^2+41x+6=0
Ans: x=-0.17 or x=-1.2
Q3768) Solve 23x^2+33x+10=0
Ans: x=-0.43 or x=-1
Q3769) Solve 20x^2+29x+6=0
Ans: x=-0.25 or x=-1.2
Q3770) Solve 40x^2+14x+1=0
Ans: x=-0.1 or x=-0.25
Q3771) Solve 9x^2+40x+31=0
Ans: x=-1 or x=-3.44
Q3772) Solve 10x^2+13x+3=0
Ans: x=-0.3 or x=-1
Q3773) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q3774) Solve 2x^2+34x+32=0
Ans: x=-1 or x=-16
Q3775) Solve 12x^2+18x+6=0
Ans: x=-0.5 or x=-1
Q3776) Solve 8x^2+50x+33=0
Ans: x=-0.75 or x=-5.5
Q3777) Solve 49x^2+35x+6=0
Ans: x=-0.29 or x=-0.43
Q3778) Solve 30x^2+41x+11=0
Ans: x=-0.37 or x=-1
Q3779) Solve 12x^2+20x+7=0
Ans: x=-0.5 or x=-1.17
Q3780) Solve 9x^2+49x+40=0
Ans: x=-1 or x=-4.44
Q3781) Solve 6x^2+31x+5=0
Ans: x=-0.17 or x=-5
Q3782) Solve 21x^2+40x+11=0
Ans: x=-0.33 or x=-1.57
Q3783) Solve 26x^2+45x+19=0
Ans: x=-0.73 or x=-1
Q3784) Solve 8x^2+32x+14=0
Ans: x=-0.5 or x=-3.5
Q3785) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q3786) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q3787) Solve 8x^2+21x+10=0
Ans: x=-0.62 or x=-2
Q3788) Solve 42x^2+35x+7=0
Ans: x=-0.33 or x=-0.5
Q3789) Solve 16x^2+20x+6=0
Ans: x=-0.5 or x=-0.75
Q3790) Solve 3x^2+31x+50=0
Ans: x=-2 or x=-8.33
Q3791) Solve 42x^2+20x+2=0
Ans: x=-0.14 or x=-0.33
Q3792) Solve 9x^2+38x+33=0
Ans: x=-1.22 or x=-3
Q3793) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q3794) Solve 15x^2+31x+10=0
Ans: x=-0.4 or x=-1.67
Q3795) Solve 18x^2+45x+28=0
Ans: x=-1.17 or x=-1.33
Q3796) Solve 5x^2+36x+31=0
Ans: x=-1 or x=-6.2
Q3797) Solve 40x^2+46x+13=0
Ans: x=-0.5 or x=-0.65
Q3798) Solve 24x^2+22x+5=0
Ans: x=-0.42 or x=-0.5
Q3799) Solve x^2+24x+44=0
Ans: x=-2 or x=-22
Q3800) Solve 33x^2+40x+12=0
Ans: x=-0.55 or x=-0.67
Q3801) Solve 28x^2+24x+5=0
Ans: x=-0.36 or x=-0.5
Q3802) Solve 8x^2+37x+42=0
Ans: x=-2 or x=-2.62
Q3803) Solve 15x^2+35x+20=0
Ans: x=-1 or x=-1.33
Q3804) Solve 15x^2+49x+12=0
Ans: x=-0.27 or x=-3
Q3805) Solve 8x^2+27x+22=0
Ans: x=-1.37 or x=-2
Q3806) Solve 9x^2+29x+22=0
Ans: x=-1.22 or x=-2
Q3807) Solve 10x^2+47x+37=0
Ans: x=-1 or x=-3.7
Q3808) Solve 12x^2+34x+22=0
Ans: x=-1 or x=-1.83
Q3809) Solve 35x^2+36x+1=0
Ans: x=-0.03 or x=-1
Q3810) Solve 2x^2+16x+32=0
Ans: x=-4 or x=-4
Q3811) Solve 5x^2+21x+18=0
Ans: x=-1.2 or x=-3
Q3812) Solve 3x^2+27x+24=0
Ans: x=-1 or x=-8
Q3813) Solve 18x^2+43x+25=0
Ans: x=-1 or x=-1.39
Q3814) Solve 44x^2+37x+3=0
Ans: x=-0.09 or x=-0.75
Q3815) Solve 40x^2+26x+3=0
Ans: x=-0.15 or x=-0.5
Q3816) Solve 41x^2+49x+8=0
Ans: x=-0.2 or x=-1
Q3817) Solve 24x^2+36x+12=0
Ans: x=-0.5 or x=-1
Q3818) Solve 42x^2+33x+6=0
Ans: x=-0.29 or x=-0.5
Q3819) Solve 15x^2+44x+13=0
Ans: x=-0.33 or x=-2.6
Q3820) Solve 4x^2+16x+12=0
Ans: x=-1 or x=-3
Q3821) Solve 12x^2+14x+4=0
Ans: x=-0.5 or x=-0.67
Q3822) Solve 10x^2+27x+18=0
Ans: x=-1.2 or x=-1.5
Q3823) Solve 2x^2+20x+32=0
Ans: x=-2 or x=-8
Q3824) Solve 28x^2+28x+7=0
Ans: x=-0.5 or x=-0.5
Q3825) Solve 6x^2+20x+14=0
Ans: x=-1 or x=-2.33
Q3826) Solve 30x^2+49x+12=0
Ans: x=-0.3 or x=-1.33
Q3827) Solve x^2+14x+45=0
Ans: x=-5 or x=-9
Q3828) Solve 14x^2+20x+6=0
Ans: x=-0.43 or x=-1
Q3829) Solve 32x^2+26x+5=0
Ans: x=-0.31 or x=-0.5
Q3830) Solve 3x^2+19x+30=0
Ans: x=-3 or x=-3.33
Q3831) Solve 32x^2+34x+2=0
Ans: x=-0.06 or x=-1
Q3832) Solve 22x^2+41x+19=0
Ans: x=-0.86 or x=-1
Q3833) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q3834) Solve 3x^2+35x+32=0
Ans: x=-1 or x=-10.67
Q3835) Solve 21x^2+31x+8=0
Ans: x=-0.33 or x=-1.14
Q3836) Solve 6x^2+30x+36=0
Ans: x=-2 or x=-3
Q3837) Solve 21x^2+24x+3=0
Ans: x=-0.14 or x=-1
Q3838) Solve 3x^2+22x+35=0
Ans: x=-2.33 or x=-5
Q3839) Solve 7x^2+32x+33=0
Ans: x=-1.57 or x=-3
Q3840) Solve 24x^2+41x+12=0
Ans: x=-0.37 or x=-1.33
Q3841) Solve 34x^2+23x+3=0
Ans: x=-0.18 or x=-0.5
Q3842) Solve 4x^2+26x+12=0
Ans: x=-0.5 or x=-6
Q3843) Solve 14x^2+47x+39=0
Ans: x=-1.5 or x=-1.86
Q3844) Solve 22x^2+26x+4=0
Ans: x=-0.18 or x=-1
Q3845) Solve 45x^2+49x+8=0
Ans: x=-0.2 or x=-0.89
Q3846) Solve 30x^2+37x+11=0
Ans: x=-0.5 or x=-0.73
Q3847) Solve 26x^2+49x+18=0
Ans: x=-0.5 or x=-1.38
Q3848) Solve 2x^2+20x+42=0
Ans: x=-3 or x=-7
Q3849) Solve 21x^2+37x+10=0
Ans: x=-0.33 or x=-1.43
Q3850) Solve 37x^2+39x+2=0
Ans: x=-0.05 or x=-1
Q3851) Solve 9x^2+29x+6=0
Ans: x=-0.22 or x=-3
Q3852) Solve 24x^2+43x+5=0
Ans: x=-0.12 or x=-1.67
Q3853) Solve 7x^2+17x+10=0
Ans: x=-1 or x=-1.43
Q3854) Solve 34x^2+50x+16=0
Ans: x=-0.47 or x=-1
Q3855) Solve 20x^2+44x+17=0
Ans: x=-0.5 or x=-1.7
Q3856) Solve 30x^2+31x+5=0
Ans: x=-0.2 or x=-0.83
Q3857) Solve 13x^2+50x+48=0
Ans: x=-1.85 or x=-2
Q3858) Solve 7x^2+44x+45=0
Ans: x=-1.29 or x=-5
Q3859) Solve 35x^2+49x+14=0
Ans: x=-0.4 or x=-1
Q3860) Solve 9x^2+35x+34=0
Ans: x=-1.89 or x=-2
Q3861) Solve 2x^2+47x+45=0
Ans: x=-1 or x=-22.5
Q3862) Solve 28x^2+48x+20=0
Ans: x=-0.71 or x=-1
Q3863) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q3864) Solve 5x^2+49x+36=0
Ans: x=-0.8 or x=-9
Q3865) Solve 14x^2+36x+16=0
Ans: x=-0.57 or x=-2
Q3866) Solve 2x^2+47x+45=0
Ans: x=-1 or x=-22.5
Q3867) Solve 28x^2+16x+1=0
Ans: x=-0.07 or x=-0.5
Q3868) Solve 5x^2+31x+44=0
Ans: x=-2.2 or x=-4
Q3869) Solve 2x^2+27x+46=0
Ans: x=-2 or x=-11.5
Q3870) Solve 9x^2+24x+15=0
Ans: x=-1 or x=-1.67
Q3871) Solve 14x^2+30x+16=0
Ans: x=-1 or x=-1.14
Q3872) Solve 18x^2+31x+6=0
Ans: x=-0.22 or x=-1.5
Q3873) Solve 7x^2+16x+4=0
Ans: x=-0.29 or x=-2
Q3874) Solve 12x^2+50x+38=0
Ans: x=-1 or x=-3.17
Q3875) Solve 4x^2+22x+28=0
Ans: x=-2 or x=-3.5
Q3876) Solve 24x^2+36x+12=0
Ans: x=-0.5 or x=-1
Q3877) Solve 16x^2+30x+9=0
Ans: x=-0.37 or x=-1.5
Q3878) Solve 45x^2+48x+11=0
Ans: x=-0.33 or x=-0.73
Q3879) Solve 36x^2+17x+2=0
Ans: x=-0.22 or x=-0.25
Q3880) Solve 8x^2+10x+3=0
Ans: x=-0.5 or x=-0.75
Q3881) Solve 10x^2+17x+3=0
Ans: x=-0.2 or x=-1.5
Q3882) Solve 6x^2+15x+6=0
Ans: x=-0.5 or x=-2
Q3883) Solve 20x^2+46x+12=0
Ans: x=-0.3 or x=-2
Q3884) Solve 2x^2+28x+48=0
Ans: x=-2 or x=-12
Q3885) Solve 22x^2+37x+13=0
Ans: x=-0.5 or x=-1.18
Q3886) Solve 4x^2+31x+42=0
Ans: x=-1.75 or x=-6
Q3887) Solve 35x^2+24x+4=0
Ans: x=-0.29 or x=-0.4
Q3888) Solve 2x^2+28x+26=0
Ans: x=-1 or x=-13
Q3889) Solve 16x^2+38x+22=0
Ans: x=-1 or x=-1.37
Q3890) Solve 12x^2+14x+4=0
Ans: x=-0.5 or x=-0.67
Q3891) Solve 40x^2+39x+9=0
Ans: x=-0.37 or x=-0.6
Q3892) Solve 8x^2+11x+3=0
Ans: x=-0.37 or x=-1
Q3893) Solve 4x^2+18x+8=0
Ans: x=-0.5 or x=-4
Q3894) Solve 40x^2+42x+9=0
Ans: x=-0.3 or x=-0.75
Q3895) Solve 35x^2+33x+4=0
Ans: x=-0.14 or x=-0.8
Q3896) Solve 15x^2+34x+16=0
Ans: x=-0.67 or x=-1.6
Q3897) Solve 8x^2+38x+9=0
Ans: x=-0.25 or x=-4.5
Q3898) Solve 5x^2+44x+39=0
Ans: x=-1 or x=-7.8
Q3899) Solve 24x^2+42x+15=0
Ans: x=-0.5 or x=-1.25
Q3900) Solve 9x^2+31x+26=0
Ans: x=-1.44 or x=-2
Q3901) Solve 17x^2+42x+16=0
Ans: x=-0.47 or x=-2
Q3902) Solve x^2+13x+42=0
Ans: x=-6 or x=-7
Q3903) Solve 30x^2+37x+11=0
Ans: x=-0.5 or x=-0.73
Q3904) Solve 18x^2+45x+22=0
Ans: x=-0.67 or x=-1.83
Q3905) Solve 18x^2+24x+8=0
Ans: x=-0.67 or x=-0.67
Q3906) Solve 44x^2+42x+10=0
Ans: x=-0.45 or x=-0.5
Q3907) Solve 16x^2+28x+12=0
Ans: x=-0.75 or x=-1
Q3908) Solve 45x^2+46x+8=0
Ans: x=-0.22 or x=-0.8
Q3909) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q3910) Solve x^2+25x+46=0
Ans: x=-2 or x=-23
Q3911) Solve 4x^2+36x+17=0
Ans: x=-0.5 or x=-8.5
Q3912) Solve 6x^2+25x+4=0
Ans: x=-0.17 or x=-4
Q3913) Solve 37x^2+48x+11=0
Ans: x=-0.3 or x=-1
Q3914) Solve 14x^2+37x+15=0
Ans: x=-0.5 or x=-2.14
Q3915) Solve 39x^2+29x+2=0
Ans: x=-0.08 or x=-0.67
Q3916) Solve 11x^2+38x+15=0
Ans: x=-0.45 or x=-3
Q3917) Solve 28x^2+46x+6=0
Ans: x=-0.14 or x=-1.5
Q3918) Solve 10x^2+26x+12=0
Ans: x=-0.6 or x=-2
Q3919) Solve 30x^2+38x+8=0
Ans: x=-0.27 or x=-1
Q3920) Solve 16x^2+29x+13=0
Ans: x=-0.81 or x=-1
Q3921) Solve 28x^2+41x+15=0
Ans: x=-0.71 or x=-0.75
Q3922) Solve 19x^2+23x+4=0
Ans: x=-0.21 or x=-1
Q3923) Solve 40x^2+34x+6=0
Ans: x=-0.25 or x=-0.6
Q3924) Solve x^2+9x+8=0
Ans: x=-1 or x=-8
Q3925) Solve 3x^2+35x+32=0
Ans: x=-1 or x=-10.67
Q3926) Solve 24x^2+49x+2=0
Ans: x=-0.04 or x=-2
Q3927) Solve 7x^2+17x+6=0
Ans: x=-0.43 or x=-2
Q3928) Solve 45x^2+39x+6=0
Ans: x=-0.2 or x=-0.67
Q3929) Solve 33x^2+46x+16=0
Ans: x=-0.67 or x=-0.73
Q3930) Solve 16x^2+30x+9=0
Ans: x=-0.37 or x=-1.5
Q3931) Solve 4x^2+25x+21=0
Ans: x=-1 or x=-5.25
Q3932) Solve 6x^2+35x+46=0
Ans: x=-2 or x=-3.83
Q3933) Solve 8x^2+16x+8=0
Ans: x=-1 or x=-1
Q3934) Solve 38x^2+44x+6=0
Ans: x=-0.16 or x=-1
Q3935) Solve 4x^2+11x+7=0
Ans: x=-1 or x=-1.75
Q3936) Solve 16x^2+45x+26=0
Ans: x=-0.81 or x=-2
Q3937) Solve 14x^2+39x+22=0
Ans: x=-0.79 or x=-2
Q3938) Solve 6x^2+34x+28=0
Ans: x=-1 or x=-4.67
Q3939) Solve 15x^2+25x+10=0
Ans: x=-0.67 or x=-1
Q3940) Solve 12x^2+43x+38=0
Ans: x=-1.58 or x=-2
Q3941) Solve 24x^2+29x+5=0
Ans: x=-0.21 or x=-1
Q3942) Solve 26x^2+31x+9=0
Ans: x=-0.5 or x=-0.69
Q3943) Solve 4x^2+27x+44=0
Ans: x=-2.75 or x=-4
Q3944) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q3945) Solve 6x^2+10x+4=0
Ans: x=-0.67 or x=-1
Q3946) Solve 12x^2+25x+7=0
Ans: x=-0.33 or x=-1.75
Q3947) Solve 2x^2+16x+32=0
Ans: x=-4 or x=-4
Q3948) Solve 11x^2+50x+39=0
Ans: x=-1 or x=-3.55
Q3949) Solve 12x^2+50x+38=0
Ans: x=-1 or x=-3.17
Q3950) Solve 18x^2+49x+20=0
Ans: x=-0.5 or x=-2.22
Q3951) Solve 28x^2+36x+8=0
Ans: x=-0.29 or x=-1
Q3952) Solve 26x^2+39x+13=0
Ans: x=-0.5 or x=-1
Q3953) Solve 11x^2+30x+19=0
Ans: x=-1 or x=-1.73
Q3954) Solve 42x^2+47x+13=0
Ans: x=-0.5 or x=-0.62
Q3955) Solve 5x^2+6x+1=0
Ans: x=-0.2 or x=-1
Q3956) Solve 16x^2+48x+35=0
Ans: x=-1.25 or x=-1.75
Q3957) Solve 22x^2+39x+14=0
Ans: x=-0.5 or x=-1.27
Q3958) Solve 31x^2+43x+12=0
Ans: x=-0.39 or x=-1
Q3959) Solve 2x^2+8x+8=0
Ans: x=-2 or x=-2
Q3960) Solve 4x^2+24x+36=0
Ans: x=-3 or x=-3
Q3961) Solve 20x^2+36x+9=0
Ans: x=-0.3 or x=-1.5
Q3962) Solve 22x^2+26x+4=0
Ans: x=-0.18 or x=-1
Q3963) Solve 46x^2+49x+13=0
Ans: x=-0.5 or x=-0.57
Q3964) Solve 12x^2+15x+3=0
Ans: x=-0.25 or x=-1
Q3965) Solve 21x^2+43x+2=0
Ans: x=-0.05 or x=-2
Q3966) Solve 27x^2+31x+4=0
Ans: x=-0.15 or x=-1
Q3967) Solve 13x^2+39x+26=0
Ans: x=-1 or x=-2
Q3968) Solve 21x^2+38x+16=0
Ans: x=-0.67 or x=-1.14
Q3969) Solve x^2+33x+32=0
Ans: x=-1 or x=-32
Q3970) Solve 50x^2+35x+3=0
Ans: x=-0.1 or x=-0.6
Q3971) Solve 40x^2+42x+8=0
Ans: x=-0.25 or x=-0.8
Q3972) Solve 42x^2+43x+10=0
Ans: x=-0.36 or x=-0.67
Q3973) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q3974) Solve 14x^2+16x+2=0
Ans: x=-0.14 or x=-1
Q3975) Solve 10x^2+41x+33=0
Ans: x=-1.1 or x=-3
Q3976) Solve 2x^2+11x+5=0
Ans: x=-0.5 or x=-5
Q3977) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q3978) Solve 4x^2+16x+15=0
Ans: x=-1.5 or x=-2.5
Q3979) Solve 17x^2+20x+3=0
Ans: x=-0.18 or x=-1
Q3980) Solve 40x^2+39x+9=0
Ans: x=-0.37 or x=-0.6
Q3981) Solve 18x^2+39x+21=0
Ans: x=-1 or x=-1.17
Q3982) Solve 4x^2+21x+20=0
Ans: x=-1.25 or x=-4
Q3983) Solve 36x^2+24x+4=0
Ans: x=-0.33 or x=-0.33
Q3984) Solve 38x^2+49x+11=0
Ans: x=-0.29 or x=-1
Q3985) Solve 9x^2+36x+20=0
Ans: x=-0.67 or x=-3.33
Q3986) Solve 6x^2+15x+6=0
Ans: x=-0.5 or x=-2
Q3987) Solve 27x^2+27x+6=0
Ans: x=-0.33 or x=-0.67
Q3988) Solve 18x^2+47x+22=0
Ans: x=-0.61 or x=-2
Q3989) Solve 19x^2+40x+21=0
Ans: x=-1 or x=-1.11
Q3990) Solve 6x^2+11x+4=0
Ans: x=-0.5 or x=-1.33
Q3991) Solve 28x^2+26x+6=0
Ans: x=-0.43 or x=-0.5
Q3992) Solve 30x^2+17x+2=0
Ans: x=-0.17 or x=-0.4
Q3993) Solve 12x^2+29x+14=0
Ans: x=-0.67 or x=-1.75
Q3994) Solve 18x^2+36x+10=0
Ans: x=-0.33 or x=-1.67
Q3995) Solve 8x^2+26x+18=0
Ans: x=-1 or x=-2.25
Q3996) Solve 44x^2+45x+9=0
Ans: x=-0.27 or x=-0.75
Q3997) Solve 18x^2+46x+20=0
Ans: x=-0.56 or x=-2
Q3998) Solve 2x^2+22x+36=0
Ans: x=-2 or x=-9
Q3999) Solve 40x^2+48x+14=0
Ans: x=-0.5 or x=-0.7
Q4000) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q4001) Solve 21x^2+29x+10=0
Ans: x=-0.67 or x=-0.71
Q4002) Solve 15x^2+43x+26=0
Ans: x=-0.87 or x=-2
Q4003) Solve 44x^2+49x+5=0
Ans: x=-0.11 or x=-1
Q4004) Solve 39x^2+48x+9=0
Ans: x=-0.23 or x=-1
Q4005) Solve x^2+12x+35=0
Ans: x=-5 or x=-7
Q4006) Solve 5x^2+12x+7=0
Ans: x=-1 or x=-1.4
Q4007) Solve 41x^2+42x+1=0
Ans: x=-0.02 or x=-1
Q4008) Solve 17x^2+20x+3=0
Ans: x=-0.18 or x=-1
Q4009) Solve 5x^2+18x+9=0
Ans: x=-0.6 or x=-3
Q4010) Solve 15x^2+46x+24=0
Ans: x=-0.67 or x=-2.4
Q4011) Solve 34x^2+42x+8=0
Ans: x=-0.24 or x=-1
Q4012) Solve 26x^2+34x+8=0
Ans: x=-0.31 or x=-1
Q4013) Solve 28x^2+37x+9=0
Ans: x=-0.32 or x=-1
Q4014) Solve 14x^2+39x+22=0
Ans: x=-0.79 or x=-2
Q4015) Solve 12x^2+40x+28=0
Ans: x=-1 or x=-2.33
Q4016) Solve 27x^2+48x+5=0
Ans: x=-0.11 or x=-1.67
Q4017) Solve 6x^2+12x+6=0
Ans: x=-1 or x=-1
Q4018) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q4019) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q4020) Solve 3x^2+35x+32=0
Ans: x=-1 or x=-10.67
Q4021) Solve 10x^2+43x+33=0
Ans: x=-1 or x=-3.3
Q4022) Solve 5x^2+18x+16=0
Ans: x=-1.6 or x=-2
Q4023) Solve 8x^2+14x+6=0
Ans: x=-0.75 or x=-1
Q4024) Solve 35x^2+17x+2=0
Ans: x=-0.2 or x=-0.29
Q4025) Solve 10x^2+37x+33=0
Ans: x=-1.5 or x=-2.2
Q4026) Solve 7x^2+25x+18=0
Ans: x=-1 or x=-2.57
Q4027) Solve 5x^2+22x+21=0
Ans: x=-1.4 or x=-3
Q4028) Solve 45x^2+23x+2=0
Ans: x=-0.11 or x=-0.4
Q4029) Solve 10x^2+42x+32=0
Ans: x=-1 or x=-3.2
Q4030) Solve 48x^2+38x+4=0
Ans: x=-0.12 or x=-0.67
Q4031) Solve 30x^2+31x+8=0
Ans: x=-0.5 or x=-0.53
Q4032) Solve 44x^2+50x+14=0
Ans: x=-0.5 or x=-0.64
Q4033) Solve 14x^2+43x+29=0
Ans: x=-1 or x=-2.07
Q4034) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q4035) Solve 14x^2+11x+2=0
Ans: x=-0.29 or x=-0.5
Q4036) Solve x^2+13x+12=0
Ans: x=-1 or x=-12
Q4037) Solve 19x^2+49x+22=0
Ans: x=-0.58 or x=-2
Q4038) Solve 21x^2+47x+20=0
Ans: x=-0.57 or x=-1.67
Q4039) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q4040) Solve 5x^2+35x+50=0
Ans: x=-2 or x=-5
Q4041) Solve 28x^2+29x+1=0
Ans: x=-0.04 or x=-1
Q4042) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q4043) Solve 19x^2+23x+4=0
Ans: x=-0.21 or x=-1
Q4044) Solve 26x^2+29x+8=0
Ans: x=-0.5 or x=-0.62
Q4045) Solve 8x^2+38x+24=0
Ans: x=-0.75 or x=-4
Q4046) Solve x^2+18x+32=0
Ans: x=-2 or x=-16
Q4047) Solve 12x^2+49x+50=0
Ans: x=-2 or x=-2.08
Q4048) Solve 29x^2+47x+18=0
Ans: x=-0.62 or x=-1
Q4049) Solve 3x^2+25x+28=0
Ans: x=-1.33 or x=-7
Q4050) Solve 16x^2+47x+30=0
Ans: x=-0.94 or x=-2
Q4051) Solve 10x^2+50x+40=0
Ans: x=-1 or x=-4
Q4052) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q4053) Solve 16x^2+30x+14=0
Ans: x=-0.87 or x=-1
Q4054) Solve 15x^2+46x+24=0
Ans: x=-0.67 or x=-2.4
Q4055) Solve 14x^2+44x+30=0
Ans: x=-1 or x=-2.14
Q4056) Solve 4x^2+17x+15=0
Ans: x=-1.25 or x=-3
Q4057) Solve 2x^2+45x+43=0
Ans: x=-1 or x=-21.5
Q4058) Solve 36x^2+27x+5=0
Ans: x=-0.33 or x=-0.42
Q4059) Solve 6x^2+32x+32=0
Ans: x=-1.33 or x=-4
Q4060) Solve 3x^2+12x+12=0
Ans: x=-2 or x=-2
Q4061) Solve 3x^2+22x+39=0
Ans: x=-3 or x=-4.33
Q4062) Solve 31x^2+32x+1=0
Ans: x=-0.03 or x=-1
Q4063) Solve 2x^2+38x+36=0
Ans: x=-1 or x=-18
Q4064) Solve 43x^2+50x+7=0
Ans: x=-0.16 or x=-1
Q4065) Solve 32x^2+18x+1=0
Ans: x=-0.06 or x=-0.5
Q4066) Solve 20x^2+48x+16=0
Ans: x=-0.4 or x=-2
Q4067) Solve 10x^2+24x+14=0
Ans: x=-1 or x=-1.4
Q4068) Solve 8x^2+19x+6=0
Ans: x=-0.37 or x=-2
Q4069) Solve 24x^2+35x+9=0
Ans: x=-0.33 or x=-1.12
Q4070) Solve 25x^2+36x+11=0
Ans: x=-0.44 or x=-1
Q4071) Solve 23x^2+25x+2=0
Ans: x=-0.09 or x=-1
Q4072) Solve x^2+20x+36=0
Ans: x=-2 or x=-18
Q4073) Solve 40x^2+46x+6=0
Ans: x=-0.15 or x=-1
Q4074) Solve 3x^2+18x+15=0
Ans: x=-1 or x=-5
Q4075) Solve 10x^2+42x+32=0
Ans: x=-1 or x=-3.2
Q4076) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q4077) Solve x^2+46x+45=0
Ans: x=-1 or x=-45
Q4078) Solve 8x^2+39x+28=0
Ans: x=-0.87 or x=-4
Q4079) Solve 4x^2+8x+4=0
Ans: x=-1 or x=-1
Q4080) Solve 7x^2+36x+44=0
Ans: x=-2 or x=-3.14
Q4081) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q4082) Solve 39x^2+48x+9=0
Ans: x=-0.23 or x=-1
Q4083) Solve 8x^2+34x+30=0
Ans: x=-1.25 or x=-3
Q4084) Solve 26x^2+28x+2=0
Ans: x=-0.08 or x=-1
Q4085) Solve 30x^2+17x+1=0
Ans: x=-0.07 or x=-0.5
Q4086) Solve 39x^2+46x+7=0
Ans: x=-0.18 or x=-1
Q4087) Solve 6x^2+7x+1=0
Ans: x=-0.17 or x=-1
Q4088) Solve 15x^2+19x+6=0
Ans: x=-0.6 or x=-0.67
Q4089) Solve 23x^2+40x+17=0
Ans: x=-0.74 or x=-1
Q4090) Solve 9x^2+49x+20=0
Ans: x=-0.44 or x=-5
Q4091) Solve 2x^2+31x+29=0
Ans: x=-1 or x=-14.5
Q4092) Solve 4x^2+46x+42=0
Ans: x=-1 or x=-10.5
Q4093) Solve x^2+11x+18=0
Ans: x=-2 or x=-9
Q4094) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q4095) Solve 3x^2+26x+48=0
Ans: x=-2.67 or x=-6
Q4096) Solve 27x^2+48x+13=0
Ans: x=-0.33 or x=-1.44
Q4097) Solve 15x^2+35x+10=0
Ans: x=-0.33 or x=-2
Q4098) Solve 4x^2+30x+36=0
Ans: x=-1.5 or x=-6
Q4099) Solve 17x^2+46x+29=0
Ans: x=-1 or x=-1.71
Q4100) Solve 6x^2+31x+38=0
Ans: x=-2 or x=-3.17
Q4101) Solve 50x^2+45x+10=0
Ans: x=-0.4 or x=-0.5
Q4102) Solve 11x^2+12x+1=0
Ans: x=-0.09 or x=-1
Q4103) Solve 16x^2+39x+14=0
Ans: x=-0.44 or x=-2
Q4104) Solve 24x^2+26x+5=0
Ans: x=-0.25 or x=-0.83
Q4105) Solve 12x^2+48x+36=0
Ans: x=-1 or x=-3
Q4106) Solve 12x^2+44x+24=0
Ans: x=-0.67 or x=-3
Q4107) Solve 42x^2+33x+6=0
Ans: x=-0.29 or x=-0.5
Q4108) Solve 48x^2+24x+3=0
Ans: x=-0.25 or x=-0.25
Q4109) Solve 5x^2+44x+32=0
Ans: x=-0.8 or x=-8
Q4110) Solve 40x^2+23x+3=0
Ans: x=-0.2 or x=-0.37
Q4111) Solve 3x^2+36x+33=0
Ans: x=-1 or x=-11
Q4112) Solve 4x^2+25x+6=0
Ans: x=-0.25 or x=-6
Q4113) Solve 4x^2+16x+7=0
Ans: x=-0.5 or x=-3.5
Q4114) Solve 45x^2+38x+8=0
Ans: x=-0.4 or x=-0.44
Q4115) Solve 7x^2+39x+20=0
Ans: x=-0.57 or x=-5
Q4116) Solve 10x^2+43x+46=0
Ans: x=-2 or x=-2.3
Q4117) Solve 9x^2+33x+24=0
Ans: x=-1 or x=-2.67
Q4118) Solve 28x^2+49x+21=0
Ans: x=-0.75 or x=-1
Q4119) Solve 26x^2+37x+11=0
Ans: x=-0.42 or x=-1
Q4120) Solve 14x^2+33x+10=0
Ans: x=-0.36 or x=-2
Q4121) Solve 28x^2+16x+1=0
Ans: x=-0.07 or x=-0.5
Q4122) Solve 25x^2+30x+9=0
Ans: x=-0.6 or x=-0.6
Q4123) Solve 7x^2+41x+34=0
Ans: x=-1 or x=-4.86
Q4124) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q4125) Solve 44x^2+24x+1=0
Ans: x=-0.05 or x=-0.5
Q4126) Solve 4x^2+39x+35=0
Ans: x=-1 or x=-8.75
Q4127) Solve 38x^2+40x+2=0
Ans: x=-0.05 or x=-1
Q4128) Solve x^2+15x+50=0
Ans: x=-5 or x=-10
Q4129) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q4130) Solve 8x^2+31x+30=0
Ans: x=-1.87 or x=-2
Q4131) Solve 15x^2+48x+36=0
Ans: x=-1.2 or x=-2
Q4132) Solve 12x^2+19x+5=0
Ans: x=-0.33 or x=-1.25
Q4133) Solve 28x^2+29x+6=0
Ans: x=-0.29 or x=-0.75
Q4134) Solve 12x^2+32x+5=0
Ans: x=-0.17 or x=-2.5
Q4135) Solve 13x^2+41x+28=0
Ans: x=-1 or x=-2.15
Q4136) Solve 3x^2+11x+8=0
Ans: x=-1 or x=-2.67
Q4137) Solve 10x^2+46x+24=0
Ans: x=-0.6 or x=-4
Q4138) Solve 6x^2+29x+34=0
Ans: x=-2 or x=-2.83
Q4139) Solve 25x^2+40x+16=0
Ans: x=-0.8 or x=-0.8
Q4140) Solve 20x^2+42x+22=0
Ans: x=-1 or x=-1.1
Q4141) Solve 16x^2+36x+14=0
Ans: x=-0.5 or x=-1.75
Q4142) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q4143) Solve 10x^2+13x+3=0
Ans: x=-0.3 or x=-1
Q4144) Solve 4x^2+44x+40=0
Ans: x=-1 or x=-10
Q4145) Solve 30x^2+46x+8=0
Ans: x=-0.2 or x=-1.33
Q4146) Solve 48x^2+28x+4=0
Ans: x=-0.25 or x=-0.33
Q4147) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q4148) Solve 24x^2+30x+9=0
Ans: x=-0.5 or x=-0.75
Q4149) Solve 24x^2+50x+4=0
Ans: x=-0.08 or x=-2
Q4150) Solve 44x^2+43x+8=0
Ans: x=-0.25 or x=-0.73
Q4151) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q4152) Solve 4x^2+14x+10=0
Ans: x=-1 or x=-2.5
Q4153) Solve 24x^2+31x+7=0
Ans: x=-0.29 or x=-1
Q4154) Solve x^2+15x+50=0
Ans: x=-5 or x=-10
Q4155) Solve 4x^2+12x+9=0
Ans: x=-1.5 or x=-1.5
Q4156) Solve 38x^2+31x+6=0
Ans: x=-0.32 or x=-0.5
Q4157) Solve 12x^2+46x+42=0
Ans: x=-1.5 or x=-2.33
Q4158) Solve 44x^2+48x+13=0
Ans: x=-0.5 or x=-0.59
Q4159) Solve 20x^2+22x+6=0
Ans: x=-0.5 or x=-0.6
Q4160) Solve 13x^2+49x+30=0
Ans: x=-0.77 or x=-3
Q4161) Solve 2x^2+39x+37=0
Ans: x=-1 or x=-18.5
Q4162) Solve 10x^2+33x+26=0
Ans: x=-1.3 or x=-2
Q4163) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q4164) Solve 30x^2+46x+8=0
Ans: x=-0.2 or x=-1.33
Q4165) Solve 5x^2+36x+36=0
Ans: x=-1.2 or x=-6
Q4166) Solve 5x^2+21x+4=0
Ans: x=-0.2 or x=-4
Q4167) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q4168) Solve 7x^2+37x+30=0
Ans: x=-1 or x=-4.29
Q4169) Solve 7x^2+13x+6=0
Ans: x=-0.86 or x=-1
Q4170) Solve 48x^2+28x+4=0
Ans: x=-0.25 or x=-0.33
Q4171) Solve 18x^2+49x+20=0
Ans: x=-0.5 or x=-2.22
Q4172) Solve 9x^2+31x+26=0
Ans: x=-1.44 or x=-2
Q4173) Solve 40x^2+18x+2=0
Ans: x=-0.2 or x=-0.25
Q4174) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q4175) Solve 7x^2+25x+12=0
Ans: x=-0.57 or x=-3
Q4176) Solve 12x^2+12x+3=0
Ans: x=-0.5 or x=-0.5
Q4177) Solve 40x^2+46x+9=0
Ans: x=-0.25 or x=-0.9
Q4178) Solve 20x^2+42x+22=0
Ans: x=-1 or x=-1.1
Q4179) Solve 2x^2+39x+37=0
Ans: x=-1 or x=-18.5
Q4180) Solve 21x^2+37x+12=0
Ans: x=-0.43 or x=-1.33
Q4181) Solve 33x^2+37x+4=0
Ans: x=-0.12 or x=-1
Q4182) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q4183) Solve 7x^2+50x+48=0
Ans: x=-1.14 or x=-6
Q4184) Solve x^2+42x+41=0
Ans: x=-1 or x=-41
Q4185) Solve 16x^2+30x+14=0
Ans: x=-0.87 or x=-1
Q4186) Solve 21x^2+16x+3=0
Ans: x=-0.33 or x=-0.43
Q4187) Solve 32x^2+22x+3=0
Ans: x=-0.19 or x=-0.5
Q4188) Solve 12x^2+46x+42=0
Ans: x=-1.5 or x=-2.33
Q4189) Solve 9x^2+30x+25=0
Ans: x=-1.67 or x=-1.67
Q4190) Solve 14x^2+19x+5=0
Ans: x=-0.36 or x=-1
Q4191) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q4192) Solve 18x^2+41x+23=0
Ans: x=-1 or x=-1.28
Q4193) Solve 28x^2+48x+20=0
Ans: x=-0.71 or x=-1
Q4194) Solve 36x^2+26x+4=0
Ans: x=-0.22 or x=-0.5
Q4195) Solve 4x^2+31x+42=0
Ans: x=-1.75 or x=-6
Q4196) Solve 46x^2+49x+13=0
Ans: x=-0.5 or x=-0.57
Q4197) Solve 30x^2+37x+9=0
Ans: x=-0.33 or x=-0.9
Q4198) Solve 5x^2+31x+44=0
Ans: x=-2.2 or x=-4
Q4199) Solve 9x^2+45x+14=0
Ans: x=-0.33 or x=-4.67
Q4200) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q4201) Solve 44x^2+44x+11=0
Ans: x=-0.5 or x=-0.5
Q4202) Solve 15x^2+32x+16=0
Ans: x=-0.8 or x=-1.33
Q4203) Solve 12x^2+25x+7=0
Ans: x=-0.33 or x=-1.75
Q4204) Solve 6x^2+17x+10=0
Ans: x=-0.83 or x=-2
Q4205) Solve 9x^2+38x+33=0
Ans: x=-1.22 or x=-3
Q4206) Solve 20x^2+28x+9=0
Ans: x=-0.5 or x=-0.9
Q4207) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q4208) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q4209) Solve 14x^2+37x+5=0
Ans: x=-0.14 or x=-2.5
Q4210) Solve 9x^2+30x+24=0
Ans: x=-1.33 or x=-2
Q4211) Solve 18x^2+23x+7=0
Ans: x=-0.5 or x=-0.78
Q4212) Solve 9x^2+42x+48=0
Ans: x=-2 or x=-2.67
Q4213) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q4214) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q4215) Solve 30x^2+44x+14=0
Ans: x=-0.47 or x=-1
Q4216) Solve 6x^2+40x+34=0
Ans: x=-1 or x=-5.67
Q4217) Solve 3x^2+24x+45=0
Ans: x=-3 or x=-5
Q4218) Solve 39x^2+49x+10=0
Ans: x=-0.26 or x=-1
Q4219) Solve 40x^2+42x+2=0
Ans: x=-0.05 or x=-1
Q4220) Solve 24x^2+43x+19=0
Ans: x=-0.79 or x=-1
Q4221) Solve 22x^2+45x+2=0
Ans: x=-0.05 or x=-2
Q4222) Solve 2x^2+15x+27=0
Ans: x=-3 or x=-4.5
Q4223) Solve 3x^2+31x+10=0
Ans: x=-0.33 or x=-10
Q4224) Solve 26x^2+43x+17=0
Ans: x=-0.65 or x=-1
Q4225) Solve 21x^2+35x+14=0
Ans: x=-0.67 or x=-1
Q4226) Solve 33x^2+36x+3=0
Ans: x=-0.09 or x=-1
Q4227) Solve 10x^2+34x+12=0
Ans: x=-0.4 or x=-3
Q4228) Solve 14x^2+36x+16=0
Ans: x=-0.57 or x=-2
Q4229) Solve 20x^2+29x+9=0
Ans: x=-0.45 or x=-1
Q4230) Solve 20x^2+32x+3=0
Ans: x=-0.1 or x=-1.5
Q4231) Solve 14x^2+29x+2=0
Ans: x=-0.07 or x=-2
Q4232) Solve 7x^2+22x+16=0
Ans: x=-1.14 or x=-2
Q4233) Solve 12x^2+44x+24=0
Ans: x=-0.67 or x=-3
Q4234) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q4235) Solve 9x^2+31x+22=0
Ans: x=-1 or x=-2.44
Q4236) Solve 25x^2+10x+1=0
Ans: x=-0.2 or x=-0.2
Q4237) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q4238) Solve 11x^2+40x+21=0
Ans: x=-0.64 or x=-3
Q4239) Solve 49x^2+50x+1=0
Ans: x=-0.02 or x=-1
Q4240) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q4241) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q4242) Solve 17x^2+32x+15=0
Ans: x=-0.88 or x=-1
Q4243) Solve 7x^2+29x+24=0
Ans: x=-1.14 or x=-3
Q4244) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q4245) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q4246) Solve 22x^2+38x+16=0
Ans: x=-0.73 or x=-1
Q4247) Solve 4x^2+30x+26=0
Ans: x=-1 or x=-6.5
Q4248) Solve 18x^2+21x+3=0
Ans: x=-0.17 or x=-1
Q4249) Solve 6x^2+38x+40=0
Ans: x=-1.33 or x=-5
Q4250) Solve 9x^2+50x+25=0
Ans: x=-0.56 or x=-5
Q4251) Solve 48x^2+40x+7=0
Ans: x=-0.25 or x=-0.58
Q4252) Solve 18x^2+19x+1=0
Ans: x=-0.06 or x=-1
Q4253) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q4254) Solve 2x^2+25x+42=0
Ans: x=-2 or x=-10.5
Q4255) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q4256) Solve 27x^2+50x+23=0
Ans: x=-0.85 or x=-1
Q4257) Solve 38x^2+43x+12=0
Ans: x=-0.5 or x=-0.63
Q4258) Solve 9x^2+43x+28=0
Ans: x=-0.78 or x=-4
Q4259) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q4260) Solve 23x^2+27x+4=0
Ans: x=-0.17 or x=-1
Q4261) Solve 12x^2+26x+12=0
Ans: x=-0.67 or x=-1.5
Q4262) Solve x^2+38x+37=0
Ans: x=-1 or x=-37
Q4263) Solve 48x^2+44x+6=0
Ans: x=-0.17 or x=-0.75
Q4264) Solve 15x^2+49x+24=0
Ans: x=-0.6 or x=-2.67
Q4265) Solve 27x^2+48x+21=0
Ans: x=-0.78 or x=-1
Q4266) Solve 32x^2+50x+3=0
Ans: x=-0.06 or x=-1.5
Q4267) Solve 50x^2+47x+11=0
Ans: x=-0.44 or x=-0.5
Q4268) Solve 13x^2+30x+8=0
Ans: x=-0.31 or x=-2
Q4269) Solve 12x^2+24x+12=0
Ans: x=-1 or x=-1
Q4270) Solve x^2+14x+48=0
Ans: x=-6 or x=-8
Q4271) Solve 3x^2+16x+16=0
Ans: x=-1.33 or x=-4
Q4272) Solve 20x^2+44x+8=0
Ans: x=-0.2 or x=-2
Q4273) Solve 14x^2+29x+12=0
Ans: x=-0.57 or x=-1.5
Q4274) Solve 5x^2+24x+28=0
Ans: x=-2 or x=-2.8
Q4275) Solve 15x^2+37x+14=0
Ans: x=-0.47 or x=-2
Q4276) Solve 24x^2+24x+6=0
Ans: x=-0.5 or x=-0.5
Q4277) Solve x^2+12x+36=0
Ans: x=-6 or x=-6
Q4278) Solve 11x^2+23x+2=0
Ans: x=-0.09 or x=-2
Q4279) Solve 7x^2+41x+34=0
Ans: x=-1 or x=-4.86
Q4280) Solve 2x^2+24x+22=0
Ans: x=-1 or x=-11
Q4281) Solve 3x^2+44x+41=0
Ans: x=-1 or x=-13.67
Q4282) Solve 15x^2+17x+4=0
Ans: x=-0.33 or x=-0.8
Q4283) Solve 4x^2+33x+29=0
Ans: x=-1 or x=-7.25
Q4284) Solve 8x^2+33x+25=0
Ans: x=-1 or x=-3.12
Q4285) Solve 2x^2+10x+12=0
Ans: x=-2 or x=-3
Q4286) Solve 9x^2+48x+15=0
Ans: x=-0.33 or x=-5
Q4287) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q4288) Solve 15x^2+18x+3=0
Ans: x=-0.2 or x=-1
Q4289) Solve 4x^2+26x+42=0
Ans: x=-3 or x=-3.5
Q4290) Solve 32x^2+50x+18=0
Ans: x=-0.56 or x=-1
Q4291) Solve 5x^2+27x+36=0
Ans: x=-2.4 or x=-3
Q4292) Solve 28x^2+32x+4=0
Ans: x=-0.14 or x=-1
Q4293) Solve 8x^2+33x+4=0
Ans: x=-0.12 or x=-4
Q4294) Solve 26x^2+31x+5=0
Ans: x=-0.19 or x=-1
Q4295) Solve 25x^2+29x+4=0
Ans: x=-0.16 or x=-1
Q4296) Solve 15x^2+38x+11=0
Ans: x=-0.33 or x=-2.2
Q4297) Solve 16x^2+28x+10=0
Ans: x=-0.5 or x=-1.25
Q4298) Solve 26x^2+48x+22=0
Ans: x=-0.85 or x=-1
Q4299) Solve 34x^2+49x+15=0
Ans: x=-0.44 or x=-1
Q4300) Solve 24x^2+39x+15=0
Ans: x=-0.62 or x=-1
Q4301) Solve 32x^2+24x+4=0
Ans: x=-0.25 or x=-0.5
Q4302) Solve 15x^2+46x+32=0
Ans: x=-1.07 or x=-2
Q4303) Solve 28x^2+45x+17=0
Ans: x=-0.61 or x=-1
Q4304) Solve 4x^2+39x+35=0
Ans: x=-1 or x=-8.75
Q4305) Solve 31x^2+41x+10=0
Ans: x=-0.32 or x=-1
Q4306) Solve 30x^2+47x+14=0
Ans: x=-0.4 or x=-1.17
Q4307) Solve 3x^2+25x+50=0
Ans: x=-3.33 or x=-5
Q4308) Solve 20x^2+49x+11=0
Ans: x=-0.25 or x=-2.2
Q4309) Solve 11x^2+41x+30=0
Ans: x=-1 or x=-2.73
Q4310) Solve 14x^2+34x+12=0
Ans: x=-0.43 or x=-2
Q4311) Solve 4x^2+42x+20=0
Ans: x=-0.5 or x=-10
Q4312) Solve 6x^2+31x+18=0
Ans: x=-0.67 or x=-4.5
Q4313) Solve 16x^2+46x+30=0
Ans: x=-1 or x=-1.87
Q4314) Solve 6x^2+17x+7=0
Ans: x=-0.5 or x=-2.33
Q4315) Solve 3x^2+13x+12=0
Ans: x=-1.33 or x=-3
Q4316) Solve 16x^2+43x+22=0
Ans: x=-0.69 or x=-2
Q4317) Solve x^2+48x+47=0
Ans: x=-1 or x=-47
Q4318) Solve 9x^2+42x+24=0
Ans: x=-0.67 or x=-4
Q4319) Solve 35x^2+27x+4=0
Ans: x=-0.2 or x=-0.57
Q4320) Solve 11x^2+23x+2=0
Ans: x=-0.09 or x=-2
Q4321) Solve 34x^2+36x+2=0
Ans: x=-0.06 or x=-1
Q4322) Solve 2x^2+48x+46=0
Ans: x=-1 or x=-23
Q4323) Solve 4x^2+36x+45=0
Ans: x=-1.5 or x=-7.5
Q4324) Solve 6x^2+38x+40=0
Ans: x=-1.33 or x=-5
Q4325) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q4326) Solve 2x^2+48x+46=0
Ans: x=-1 or x=-23
Q4327) Solve 3x^2+38x+35=0
Ans: x=-1 or x=-11.67
Q4328) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q4329) Solve 41x^2+45x+4=0
Ans: x=-0.1 or x=-1
Q4330) Solve 6x^2+17x+12=0
Ans: x=-1.33 or x=-1.5
Q4331) Solve 10x^2+16x+6=0
Ans: x=-0.6 or x=-1
Q4332) Solve 16x^2+20x+6=0
Ans: x=-0.5 or x=-0.75
Q4333) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q4334) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q4335) Solve 11x^2+36x+9=0
Ans: x=-0.27 or x=-3
Q4336) Solve 15x^2+38x+7=0
Ans: x=-0.2 or x=-2.33
Q4337) Solve 32x^2+35x+3=0
Ans: x=-0.09 or x=-1
Q4338) Solve 36x^2+47x+15=0
Ans: x=-0.56 or x=-0.75
Q4339) Solve 16x^2+40x+21=0
Ans: x=-0.75 or x=-1.75
Q4340) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q4341) Solve 21x^2+31x+10=0
Ans: x=-0.48 or x=-1
Q4342) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q4343) Solve 39x^2+43x+4=0
Ans: x=-0.1 or x=-1
Q4344) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q4345) Solve 22x^2+35x+13=0
Ans: x=-0.59 or x=-1
Q4346) Solve 4x^2+23x+30=0
Ans: x=-2 or x=-3.75
Q4347) Solve 39x^2+40x+9=0
Ans: x=-0.33 or x=-0.69
Q4348) Solve 7x^2+13x+6=0
Ans: x=-0.86 or x=-1
Q4349) Solve 5x^2+46x+41=0
Ans: x=-1 or x=-8.2
Q4350) Solve 12x^2+42x+30=0
Ans: x=-1 or x=-2.5
Q4351) Solve 32x^2+39x+7=0
Ans: x=-0.22 or x=-1
Q4352) Solve 18x^2+32x+14=0
Ans: x=-0.78 or x=-1
Q4353) Solve 10x^2+19x+7=0
Ans: x=-0.5 or x=-1.4
Q4354) Solve 5x^2+7x+2=0
Ans: x=-0.4 or x=-1
Q4355) Solve 12x^2+43x+31=0
Ans: x=-1 or x=-2.58
Q4356) Solve 2x^2+15x+28=0
Ans: x=-3.5 or x=-4
Q4357) Solve 20x^2+47x+21=0
Ans: x=-0.6 or x=-1.75
Q4358) Solve 45x^2+48x+3=0
Ans: x=-0.07 or x=-1
Q4359) Solve 35x^2+22x+3=0
Ans: x=-0.2 or x=-0.43
Q4360) Solve 35x^2+41x+6=0
Ans: x=-0.17 or x=-1
Q4361) Solve 2x^2+12x+18=0
Ans: x=-3 or x=-3
Q4362) Solve 3x^2+23x+34=0
Ans: x=-2 or x=-5.67
Q4363) Solve 30x^2+44x+14=0
Ans: x=-0.47 or x=-1
Q4364) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q4365) Solve 8x^2+22x+5=0
Ans: x=-0.25 or x=-2.5
Q4366) Solve 5x^2+30x+25=0
Ans: x=-1 or x=-5
Q4367) Solve 20x^2+36x+9=0
Ans: x=-0.3 or x=-1.5
Q4368) Solve 12x^2+50x+22=0
Ans: x=-0.5 or x=-3.67
Q4369) Solve 8x^2+46x+38=0
Ans: x=-1 or x=-4.75
Q4370) Solve 23x^2+50x+27=0
Ans: x=-1 or x=-1.17
Q4371) Solve 8x^2+26x+18=0
Ans: x=-1 or x=-2.25
Q4372) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q4373) Solve 6x^2+41x+35=0
Ans: x=-1 or x=-5.83
Q4374) Solve 41x^2+43x+2=0
Ans: x=-0.05 or x=-1
Q4375) Solve 3x^2+34x+31=0
Ans: x=-1 or x=-10.33
Q4376) Solve 9x^2+40x+16=0
Ans: x=-0.44 or x=-4
Q4377) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q4378) Solve 11x^2+25x+14=0
Ans: x=-1 or x=-1.27
Q4379) Solve x^2+11x+18=0
Ans: x=-2 or x=-9
Q4380) Solve 17x^2+37x+6=0
Ans: x=-0.18 or x=-2
Q4381) Solve 3x^2+20x+33=0
Ans: x=-3 or x=-3.67
Q4382) Solve 45x^2+29x+4=0
Ans: x=-0.2 or x=-0.44
Q4383) Solve 16x^2+19x+3=0
Ans: x=-0.19 or x=-1
Q4384) Solve 28x^2+46x+18=0
Ans: x=-0.64 or x=-1
Q4385) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q4386) Solve 5x^2+22x+24=0
Ans: x=-2 or x=-2.4
Q4387) Solve 9x^2+24x+15=0
Ans: x=-1 or x=-1.67
Q4388) Solve 24x^2+46x+15=0
Ans: x=-0.42 or x=-1.5
Q4389) Solve 10x^2+41x+42=0
Ans: x=-2 or x=-2.1
Q4390) Solve 8x^2+34x+30=0
Ans: x=-1.25 or x=-3
Q4391) Solve 15x^2+30x+15=0
Ans: x=-1 or x=-1
Q4392) Solve 13x^2+27x+14=0
Ans: x=-1 or x=-1.08
Q4393) Solve 8x^2+18x+10=0
Ans: x=-1 or x=-1.25
Q4394) Solve 20x^2+28x+8=0
Ans: x=-0.4 or x=-1
Q4395) Solve 4x^2+31x+46=0
Ans: x=-2 or x=-5.75
Q4396) Solve 16x^2+44x+18=0
Ans: x=-0.5 or x=-2.25
Q4397) Solve 32x^2+33x+1=0
Ans: x=-0.03 or x=-1
Q4398) Solve 16x^2+42x+27=0
Ans: x=-1.12 or x=-1.5
Q4399) Solve 16x^2+40x+9=0
Ans: x=-0.25 or x=-2.25
Q4400) Solve x^2+50x+49=0
Ans: x=-1 or x=-49
Q4401) Solve 20x^2+44x+24=0
Ans: x=-1 or x=-1.2
Q4402) Solve 16x^2+47x+31=0
Ans: x=-1 or x=-1.94
Q4403) Solve 14x^2+33x+13=0
Ans: x=-0.5 or x=-1.86
Q4404) Solve 30x^2+16x+2=0
Ans: x=-0.2 or x=-0.33
Q4405) Solve 5x^2+46x+9=0
Ans: x=-0.2 or x=-9
Q4406) Solve 13x^2+34x+16=0
Ans: x=-0.62 or x=-2
Q4407) Solve 12x^2+14x+2=0
Ans: x=-0.17 or x=-1
Q4408) Solve 3x^2+14x+11=0
Ans: x=-1 or x=-3.67
Q4409) Solve 50x^2+29x+2=0
Ans: x=-0.08 or x=-0.5
Q4410) Solve 2x^2+14x+12=0
Ans: x=-1 or x=-6
Q4411) Solve 27x^2+42x+16=0
Ans: x=-0.67 or x=-0.89
Q4412) Solve 27x^2+33x+6=0
Ans: x=-0.22 or x=-1
Q4413) Solve 13x^2+24x+11=0
Ans: x=-0.85 or x=-1
Q4414) Solve 15x^2+47x+6=0
Ans: x=-0.13 or x=-3
Q4415) Solve 33x^2+50x+17=0
Ans: x=-0.52 or x=-1
Q4416) Solve 8x^2+30x+18=0
Ans: x=-0.75 or x=-3
Q4417) Solve 28x^2+48x+9=0
Ans: x=-0.21 or x=-1.5
Q4418) Solve 49x^2+49x+12=0
Ans: x=-0.43 or x=-0.57
Q4419) Solve 26x^2+17x+2=0
Ans: x=-0.15 or x=-0.5
Q4420) Solve 7x^2+27x+26=0
Ans: x=-1.86 or x=-2
Q4421) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q4422) Solve 16x^2+14x+3=0
Ans: x=-0.37 or x=-0.5
Q4423) Solve 3x^2+49x+46=0
Ans: x=-1 or x=-15.33
Q4424) Solve 15x^2+31x+16=0
Ans: x=-1 or x=-1.07
Q4425) Solve 14x^2+35x+14=0
Ans: x=-0.5 or x=-2
Q4426) Solve 39x^2+42x+3=0
Ans: x=-0.08 or x=-1
Q4427) Solve 6x^2+37x+17=0
Ans: x=-0.5 or x=-5.67
Q4428) Solve 35x^2+12x+1=0
Ans: x=-0.14 or x=-0.2
Q4429) Solve 7x^2+48x+36=0
Ans: x=-0.86 or x=-6
Q4430) Solve 12x^2+29x+14=0
Ans: x=-0.67 or x=-1.75
Q4431) Solve 26x^2+47x+17=0
Ans: x=-0.5 or x=-1.31
Q4432) Solve 5x^2+19x+14=0
Ans: x=-1 or x=-2.8
Q4433) Solve 24x^2+43x+19=0
Ans: x=-0.79 or x=-1
Q4434) Solve 15x^2+31x+2=0
Ans: x=-0.07 or x=-2
Q4435) Solve 19x^2+50x+31=0
Ans: x=-1 or x=-1.63
Q4436) Solve 11x^2+49x+38=0
Ans: x=-1 or x=-3.45
Q4437) Solve 6x^2+27x+30=0
Ans: x=-2 or x=-2.5
Q4438) Solve 20x^2+50x+30=0
Ans: x=-1 or x=-1.5
Q4439) Solve 16x^2+44x+30=0
Ans: x=-1.25 or x=-1.5
Q4440) Solve 5x^2+29x+38=0
Ans: x=-2 or x=-3.8
Q4441) Solve 50x^2+27x+1=0
Ans: x=-0.04 or x=-0.5
Q4442) Solve 12x^2+42x+36=0
Ans: x=-1.5 or x=-2
Q4443) Solve 19x^2+26x+7=0
Ans: x=-0.37 or x=-1
Q4444) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q4445) Solve 24x^2+32x+8=0
Ans: x=-0.33 or x=-1
Q4446) Solve 4x^2+27x+44=0
Ans: x=-2.75 or x=-4
Q4447) Solve 8x^2+42x+34=0
Ans: x=-1 or x=-4.25
Q4448) Solve 14x^2+28x+14=0
Ans: x=-1 or x=-1
Q4449) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q4450) Solve 7x^2+45x+38=0
Ans: x=-1 or x=-5.43
Q4451) Solve 23x^2+30x+7=0
Ans: x=-0.3 or x=-1
Q4452) Solve 6x^2+37x+35=0
Ans: x=-1.17 or x=-5
Q4453) Solve 5x^2+42x+37=0
Ans: x=-1 or x=-7.4
Q4454) Solve 10x^2+37x+7=0
Ans: x=-0.2 or x=-3.5
Q4455) Solve 15x^2+38x+24=0
Ans: x=-1.2 or x=-1.33
Q4456) Solve 50x^2+30x+4=0
Ans: x=-0.2 or x=-0.4
Q4457) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q4458) Solve 9x^2+35x+26=0
Ans: x=-1 or x=-2.89
Q4459) Solve 18x^2+47x+5=0
Ans: x=-0.11 or x=-2.5
Q4460) Solve 40x^2+46x+13=0
Ans: x=-0.5 or x=-0.65
Q4461) Solve 13x^2+44x+31=0
Ans: x=-1 or x=-2.38
Q4462) Solve 18x^2+24x+8=0
Ans: x=-0.67 or x=-0.67
Q4463) Solve 20x^2+33x+10=0
Ans: x=-0.4 or x=-1.25
Q4464) Solve 36x^2+34x+8=0
Ans: x=-0.44 or x=-0.5
Q4465) Solve x^2+49x+48=0
Ans: x=-1 or x=-48
Q4466) Solve 11x^2+38x+15=0
Ans: x=-0.45 or x=-3
Q4467) Solve 23x^2+46x+23=0
Ans: x=-1 or x=-1
Q4468) Solve 15x^2+46x+32=0
Ans: x=-1.07 or x=-2
Q4469) Solve 19x^2+42x+23=0
Ans: x=-1 or x=-1.21
Q4470) Solve 11x^2+43x+30=0
Ans: x=-0.91 or x=-3
Q4471) Solve 6x^2+35x+39=0
Ans: x=-1.5 or x=-4.33
Q4472) Solve 28x^2+22x+4=0
Ans: x=-0.29 or x=-0.5
Q4473) Solve 5x^2+33x+40=0
Ans: x=-1.6 or x=-5
Q4474) Solve 8x^2+35x+38=0
Ans: x=-2 or x=-2.37
Q4475) Solve 6x^2+29x+20=0
Ans: x=-0.83 or x=-4
Q4476) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q4477) Solve 4x^2+18x+8=0
Ans: x=-0.5 or x=-4
Q4478) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q4479) Solve 33x^2+14x+1=0
Ans: x=-0.09 or x=-0.33
Q4480) Solve 28x^2+29x+1=0
Ans: x=-0.04 or x=-1
Q4481) Solve 10x^2+44x+16=0
Ans: x=-0.4 or x=-4
Q4482) Solve 13x^2+24x+11=0
Ans: x=-0.85 or x=-1
Q4483) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q4484) Solve 44x^2+48x+4=0
Ans: x=-0.09 or x=-1
Q4485) Solve 5x^2+31x+48=0
Ans: x=-3 or x=-3.2
Q4486) Solve x^2+11x+24=0
Ans: x=-3 or x=-8
Q4487) Solve 14x^2+33x+10=0
Ans: x=-0.36 or x=-2
Q4488) Solve 22x^2+24x+2=0
Ans: x=-0.09 or x=-1
Q4489) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q4490) Solve 7x^2+42x+35=0
Ans: x=-1 or x=-5
Q4491) Solve 49x^2+49x+6=0
Ans: x=-0.14 or x=-0.86
Q4492) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q4493) Solve 12x^2+43x+13=0
Ans: x=-0.33 or x=-3.25
Q4494) Solve 32x^2+16x+2=0
Ans: x=-0.25 or x=-0.25
Q4495) Solve 5x^2+20x+20=0
Ans: x=-2 or x=-2
Q4496) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q4497) Solve 17x^2+37x+6=0
Ans: x=-0.18 or x=-2
Q4498) Solve 44x^2+37x+3=0
Ans: x=-0.09 or x=-0.75
Q4499) Solve 4x^2+11x+6=0
Ans: x=-0.75 or x=-2
Q4500) Solve 14x^2+43x+30=0
Ans: x=-1.07 or x=-2
Q4501) Solve 18x^2+39x+20=0
Ans: x=-0.83 or x=-1.33
Q4502) Solve 12x^2+8x+1=0
Ans: x=-0.17 or x=-0.5
Q4503) Solve 9x^2+18x+9=0
Ans: x=-1 or x=-1
Q4504) Solve 36x^2+32x+7=0
Ans: x=-0.39 or x=-0.5
Q4505) Solve 4x^2+31x+46=0
Ans: x=-2 or x=-5.75
Q4506) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q4507) Solve 36x^2+39x+3=0
Ans: x=-0.08 or x=-1
Q4508) Solve 13x^2+14x+1=0
Ans: x=-0.08 or x=-1
Q4509) Solve 11x^2+48x+16=0
Ans: x=-0.36 or x=-4
Q4510) Solve 44x^2+35x+6=0
Ans: x=-0.25 or x=-0.55
Q4511) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q4512) Solve 26x^2+38x+12=0
Ans: x=-0.46 or x=-1
Q4513) Solve 18x^2+31x+11=0
Ans: x=-0.5 or x=-1.22
Q4514) Solve 35x^2+49x+14=0
Ans: x=-0.4 or x=-1
Q4515) Solve 12x^2+26x+4=0
Ans: x=-0.17 or x=-2
Q4516) Solve 33x^2+28x+4=0
Ans: x=-0.18 or x=-0.67
Q4517) Solve 35x^2+46x+15=0
Ans: x=-0.6 or x=-0.71
Q4518) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q4519) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q4520) Solve 20x^2+28x+9=0
Ans: x=-0.5 or x=-0.9
Q4521) Solve 5x^2+25x+30=0
Ans: x=-2 or x=-3
Q4522) Solve 21x^2+46x+24=0
Ans: x=-0.86 or x=-1.33
Q4523) Solve 38x^2+29x+5=0
Ans: x=-0.26 or x=-0.5
Q4524) Solve 38x^2+23x+2=0
Ans: x=-0.11 or x=-0.5
Q4525) Solve 12x^2+48x+45=0
Ans: x=-1.5 or x=-2.5
Q4526) Solve 24x^2+50x+14=0
Ans: x=-0.33 or x=-1.75
Q4527) Solve 41x^2+45x+4=0
Ans: x=-0.1 or x=-1
Q4528) Solve 16x^2+44x+18=0
Ans: x=-0.5 or x=-2.25
Q4529) Solve 42x^2+20x+2=0
Ans: x=-0.14 or x=-0.33
Q4530) Solve 7x^2+25x+18=0
Ans: x=-1 or x=-2.57
Q4531) Solve 12x^2+34x+20=0
Ans: x=-0.83 or x=-2
Q4532) Solve 26x^2+40x+14=0
Ans: x=-0.54 or x=-1
Q4533) Solve 27x^2+39x+10=0
Ans: x=-0.33 or x=-1.11
Q4534) Solve 14x^2+48x+40=0
Ans: x=-1.43 or x=-2
Q4535) Solve 6x^2+35x+11=0
Ans: x=-0.33 or x=-5.5
Q4536) Solve 18x^2+45x+27=0
Ans: x=-1 or x=-1.5
Q4537) Solve 24x^2+49x+25=0
Ans: x=-1 or x=-1.04
Q4538) Solve 24x^2+41x+17=0
Ans: x=-0.71 or x=-1
Q4539) Solve 38x^2+47x+14=0
Ans: x=-0.5 or x=-0.74
Q4540) Solve 45x^2+36x+7=0
Ans: x=-0.33 or x=-0.47
Q4541) Solve 19x^2+49x+22=0
Ans: x=-0.58 or x=-2
Q4542) Solve 32x^2+47x+15=0
Ans: x=-0.47 or x=-1
Q4543) Solve 24x^2+46x+20=0
Ans: x=-0.67 or x=-1.25
Q4544) Solve 15x^2+36x+12=0
Ans: x=-0.4 or x=-2
Q4545) Solve 6x^2+41x+19=0
Ans: x=-0.5 or x=-6.33
Q4546) Solve 6x^2+35x+11=0
Ans: x=-0.33 or x=-5.5
Q4547) Solve 36x^2+49x+13=0
Ans: x=-0.36 or x=-1
Q4548) Solve 2x^2+33x+45=0
Ans: x=-1.5 or x=-15
Q4549) Solve 28x^2+48x+17=0
Ans: x=-0.5 or x=-1.21
Q4550) Solve 21x^2+37x+10=0
Ans: x=-0.33 or x=-1.43
Q4551) Solve 21x^2+45x+24=0
Ans: x=-1 or x=-1.14
Q4552) Solve 10x^2+40x+30=0
Ans: x=-1 or x=-3
Q4553) Solve x^2+42x+41=0
Ans: x=-1 or x=-41
Q4554) Solve 7x^2+24x+20=0
Ans: x=-1.43 or x=-2
Q4555) Solve 50x^2+30x+4=0
Ans: x=-0.2 or x=-0.4
Q4556) Solve 15x^2+32x+4=0
Ans: x=-0.13 or x=-2
Q4557) Solve 6x^2+16x+8=0
Ans: x=-0.67 or x=-2
Q4558) Solve 8x^2+26x+11=0
Ans: x=-0.5 or x=-2.75
Q4559) Solve 4x^2+29x+45=0
Ans: x=-2.25 or x=-5
Q4560) Solve 21x^2+43x+12=0
Ans: x=-0.33 or x=-1.71
Q4561) Solve 2x^2+22x+48=0
Ans: x=-3 or x=-8
Q4562) Solve 27x^2+30x+7=0
Ans: x=-0.33 or x=-0.78
Q4563) Solve 10x^2+21x+8=0
Ans: x=-0.5 or x=-1.6
Q4564) Solve 10x^2+9x+2=0
Ans: x=-0.4 or x=-0.5
Q4565) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q4566) Solve 3x^2+15x+12=0
Ans: x=-1 or x=-4
Q4567) Solve 6x^2+35x+16=0
Ans: x=-0.5 or x=-5.33
Q4568) Solve 23x^2+33x+10=0
Ans: x=-0.43 or x=-1
Q4569) Solve 32x^2+34x+9=0
Ans: x=-0.5 or x=-0.56
Q4570) Solve 42x^2+27x+3=0
Ans: x=-0.14 or x=-0.5
Q4571) Solve 20x^2+28x+9=0
Ans: x=-0.5 or x=-0.9
Q4572) Solve 16x^2+38x+21=0
Ans: x=-0.87 or x=-1.5
Q4573) Solve 44x^2+36x+7=0
Ans: x=-0.32 or x=-0.5
Q4574) Solve 9x^2+23x+14=0
Ans: x=-1 or x=-1.56
Q4575) Solve 4x^2+45x+50=0
Ans: x=-1.25 or x=-10
Q4576) Solve 33x^2+47x+4=0
Ans: x=-0.09 or x=-1.33
Q4577) Solve 4x^2+18x+8=0
Ans: x=-0.5 or x=-4
Q4578) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q4579) Solve 10x^2+41x+31=0
Ans: x=-1 or x=-3.1
Q4580) Solve 5x^2+37x+42=0
Ans: x=-1.4 or x=-6
Q4581) Solve 2x^2+37x+35=0
Ans: x=-1 or x=-17.5
Q4582) Solve 8x^2+36x+28=0
Ans: x=-1 or x=-3.5
Q4583) Solve 2x^2+12x+10=0
Ans: x=-1 or x=-5
Q4584) Solve 10x^2+37x+16=0
Ans: x=-0.5 or x=-3.2
Q4585) Solve 31x^2+32x+1=0
Ans: x=-0.03 or x=-1
Q4586) Solve 48x^2+14x+1=0
Ans: x=-0.12 or x=-0.17
Q4587) Solve 29x^2+44x+15=0
Ans: x=-0.52 or x=-1
Q4588) Solve 36x^2+30x+4=0
Ans: x=-0.17 or x=-0.67
Q4589) Solve 18x^2+45x+28=0
Ans: x=-1.17 or x=-1.33
Q4590) Solve 50x^2+43x+9=0
Ans: x=-0.36 or x=-0.5
Q4591) Solve 16x^2+12x+2=0
Ans: x=-0.25 or x=-0.5
Q4592) Solve 48x^2+35x+2=0
Ans: x=-0.06 or x=-0.67
Q4593) Solve 16x^2+47x+31=0
Ans: x=-1 or x=-1.94
Q4594) Solve 4x^2+27x+38=0
Ans: x=-2 or x=-4.75
Q4595) Solve 14x^2+49x+21=0
Ans: x=-0.5 or x=-3
Q4596) Solve 9x^2+28x+3=0
Ans: x=-0.11 or x=-3
Q4597) Solve x^2+12x+27=0
Ans: x=-3 or x=-9
Q4598) Solve 10x^2+23x+9=0
Ans: x=-0.5 or x=-1.8
Q4599) Solve 12x^2+13x+3=0
Ans: x=-0.33 or x=-0.75
Q4600) Solve 22x^2+24x+2=0
Ans: x=-0.09 or x=-1
Q4601) Solve 15x^2+32x+17=0
Ans: x=-1 or x=-1.13
Q4602) Solve 15x^2+25x+10=0
Ans: x=-0.67 or x=-1
Q4603) Solve 40x^2+42x+8=0
Ans: x=-0.25 or x=-0.8
Q4604) Solve 2x^2+10x+12=0
Ans: x=-2 or x=-3
Q4605) Solve 24x^2+40x+16=0
Ans: x=-0.67 or x=-1
Q4606) Solve 13x^2+50x+48=0
Ans: x=-1.85 or x=-2
Q4607) Solve 9x^2+32x+28=0
Ans: x=-1.56 or x=-2
Q4608) Solve 43x^2+47x+4=0
Ans: x=-0.09 or x=-1
Q4609) Solve 15x^2+28x+13=0
Ans: x=-0.87 or x=-1
Q4610) Solve 3x^2+34x+31=0
Ans: x=-1 or x=-10.33
Q4611) Solve 6x^2+32x+32=0
Ans: x=-1.33 or x=-4
Q4612) Solve 4x^2+37x+9=0
Ans: x=-0.25 or x=-9
Q4613) Solve 14x^2+45x+31=0
Ans: x=-1 or x=-2.21
Q4614) Solve 38x^2+39x+10=0
Ans: x=-0.5 or x=-0.53
Q4615) Solve 3x^2+39x+36=0
Ans: x=-1 or x=-12
Q4616) Solve 5x^2+32x+44=0
Ans: x=-2 or x=-4.4
Q4617) Solve 16x^2+37x+21=0
Ans: x=-1 or x=-1.31
Q4618) Solve 15x^2+49x+34=0
Ans: x=-1 or x=-2.27
Q4619) Solve 9x^2+43x+50=0
Ans: x=-2 or x=-2.78
Q4620) Solve 24x^2+29x+7=0
Ans: x=-0.33 or x=-0.87
Q4621) Solve 48x^2+47x+10=0
Ans: x=-0.31 or x=-0.67
Q4622) Solve 2x^2+9x+10=0
Ans: x=-2 or x=-2.5
Q4623) Solve 22x^2+49x+27=0
Ans: x=-1 or x=-1.23
Q4624) Solve 3x^2+11x+6=0
Ans: x=-0.67 or x=-3
Q4625) Solve 5x^2+13x+8=0
Ans: x=-1 or x=-1.6
Q4626) Solve 3x^2+29x+18=0
Ans: x=-0.67 or x=-9
Q4627) Solve 25x^2+32x+7=0
Ans: x=-0.28 or x=-1
Q4628) Solve 12x^2+47x+40=0
Ans: x=-1.25 or x=-2.67
Q4629) Solve 16x^2+48x+27=0
Ans: x=-0.75 or x=-2.25
Q4630) Solve 30x^2+42x+12=0
Ans: x=-0.4 or x=-1
Q4631) Solve 12x^2+20x+7=0
Ans: x=-0.5 or x=-1.17
Q4632) Solve 18x^2+41x+21=0
Ans: x=-0.78 or x=-1.5
Q4633) Solve 6x^2+26x+24=0
Ans: x=-1.33 or x=-3
Q4634) Solve 2x^2+21x+19=0
Ans: x=-1 or x=-9.5
Q4635) Solve 10x^2+37x+21=0
Ans: x=-0.7 or x=-3
Q4636) Solve 9x^2+18x+8=0
Ans: x=-0.67 or x=-1.33
Q4637) Solve 6x^2+31x+38=0
Ans: x=-2 or x=-3.17
Q4638) Solve 6x^2+29x+23=0
Ans: x=-1 or x=-3.83
Q4639) Solve 19x^2+36x+17=0
Ans: x=-0.89 or x=-1
Q4640) Solve 18x^2+49x+33=0
Ans: x=-1.22 or x=-1.5
Q4641) Solve 6x^2+35x+46=0
Ans: x=-2 or x=-3.83
Q4642) Solve 21x^2+35x+14=0
Ans: x=-0.67 or x=-1
Q4643) Solve 35x^2+44x+12=0
Ans: x=-0.4 or x=-0.86
Q4644) Solve 3x^2+10x+3=0
Ans: x=-0.33 or x=-3
Q4645) Solve 36x^2+39x+9=0
Ans: x=-0.33 or x=-0.75
Q4646) Solve 46x^2+49x+3=0
Ans: x=-0.07 or x=-1
Q4647) Solve 42x^2+37x+5=0
Ans: x=-0.17 or x=-0.71
Q4648) Solve 8x^2+46x+38=0
Ans: x=-1 or x=-4.75
Q4649) Solve 9x^2+47x+10=0
Ans: x=-0.22 or x=-5
Q4650) Solve 32x^2+50x+17=0
Ans: x=-0.5 or x=-1.06
Q4651) Solve 13x^2+45x+18=0
Ans: x=-0.46 or x=-3
Q4652) Solve 44x^2+31x+5=0
Ans: x=-0.25 or x=-0.45
Q4653) Solve 8x^2+46x+21=0
Ans: x=-0.5 or x=-5.25
Q4654) Solve 2x^2+27x+36=0
Ans: x=-1.5 or x=-12
Q4655) Solve 24x^2+34x+5=0
Ans: x=-0.17 or x=-1.25
Q4656) Solve 50x^2+30x+4=0
Ans: x=-0.2 or x=-0.4
Q4657) Solve 36x^2+38x+10=0
Ans: x=-0.5 or x=-0.56
Q4658) Solve 32x^2+33x+1=0
Ans: x=-0.03 or x=-1
Q4659) Solve 42x^2+32x+6=0
Ans: x=-0.33 or x=-0.43
Q4660) Solve 18x^2+32x+14=0
Ans: x=-0.78 or x=-1
Q4661) Solve 4x^2+23x+15=0
Ans: x=-0.75 or x=-5
Q4662) Solve 41x^2+49x+8=0
Ans: x=-0.2 or x=-1
Q4663) Solve 6x^2+35x+50=0
Ans: x=-2.5 or x=-3.33
Q4664) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q4665) Solve 12x^2+47x+40=0
Ans: x=-1.25 or x=-2.67
Q4666) Solve 9x^2+37x+4=0
Ans: x=-0.11 or x=-4
Q4667) Solve 14x^2+29x+2=0
Ans: x=-0.07 or x=-2
Q4668) Solve 9x^2+46x+5=0
Ans: x=-0.11 or x=-5
Q4669) Solve 3x^2+23x+34=0
Ans: x=-2 or x=-5.67
Q4670) Solve 5x^2+32x+44=0
Ans: x=-2 or x=-4.4
Q4671) Solve 3x^2+19x+28=0
Ans: x=-2.33 or x=-4
Q4672) Solve 8x^2+25x+17=0
Ans: x=-1 or x=-2.12
Q4673) Solve x^2+9x+8=0
Ans: x=-1 or x=-8
Q4674) Solve 9x^2+25x+14=0
Ans: x=-0.78 or x=-2
Q4675) Solve 11x^2+48x+16=0
Ans: x=-0.36 or x=-4
Q4676) Solve 2x^2+15x+13=0
Ans: x=-1 or x=-6.5
Q4677) Solve 21x^2+33x+12=0
Ans: x=-0.57 or x=-1
Q4678) Solve 21x^2+40x+11=0
Ans: x=-0.33 or x=-1.57
Q4679) Solve 20x^2+46x+26=0
Ans: x=-1 or x=-1.3
Q4680) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q4681) Solve 28x^2+43x+9=0
Ans: x=-0.25 or x=-1.29
Q4682) Solve 14x^2+19x+6=0
Ans: x=-0.5 or x=-0.86
Q4683) Solve 6x^2+37x+45=0
Ans: x=-1.67 or x=-4.5
Q4684) Solve 7x^2+23x+18=0
Ans: x=-1.29 or x=-2
Q4685) Solve 6x^2+32x+10=0
Ans: x=-0.33 or x=-5
Q4686) Solve x^2+12x+27=0
Ans: x=-3 or x=-9
Q4687) Solve 27x^2+42x+11=0
Ans: x=-0.33 or x=-1.22
Q4688) Solve 6x^2+28x+32=0
Ans: x=-2 or x=-2.67
Q4689) Solve 12x^2+48x+36=0
Ans: x=-1 or x=-3
Q4690) Solve 15x^2+31x+14=0
Ans: x=-0.67 or x=-1.4
Q4691) Solve 9x^2+12x+4=0
Ans: x=-0.67 or x=-0.67
Q4692) Solve 22x^2+47x+25=0
Ans: x=-1 or x=-1.14
Q4693) Solve 40x^2+34x+7=0
Ans: x=-0.35 or x=-0.5
Q4694) Solve 7x^2+40x+33=0
Ans: x=-1 or x=-4.71
Q4695) Solve 12x^2+38x+26=0
Ans: x=-1 or x=-2.17
Q4696) Solve 39x^2+42x+3=0
Ans: x=-0.08 or x=-1
Q4697) Solve 28x^2+30x+8=0
Ans: x=-0.5 or x=-0.57
Q4698) Solve 9x^2+42x+45=0
Ans: x=-1.67 or x=-3
Q4699) Solve 24x^2+22x+3=0
Ans: x=-0.17 or x=-0.75
Q4700) Solve 50x^2+40x+6=0
Ans: x=-0.2 or x=-0.6
Q4701) Solve 24x^2+40x+14=0
Ans: x=-0.5 or x=-1.17
Q4702) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q4703) Solve 42x^2+41x+9=0
Ans: x=-0.33 or x=-0.64
Q4704) Solve 4x^2+27x+44=0
Ans: x=-2.75 or x=-4
Q4705) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q4706) Solve 5x^2+42x+37=0
Ans: x=-1 or x=-7.4
Q4707) Solve 2x^2+19x+42=0
Ans: x=-3.5 or x=-6
Q4708) Solve 50x^2+35x+5=0
Ans: x=-0.2 or x=-0.5
Q4709) Solve 24x^2+44x+16=0
Ans: x=-0.5 or x=-1.33
Q4710) Solve 6x^2+39x+45=0
Ans: x=-1.5 or x=-5
Q4711) Solve 12x^2+44x+39=0
Ans: x=-1.5 or x=-2.17
Q4712) Solve 35x^2+19x+2=0
Ans: x=-0.14 or x=-0.4
Q4713) Solve 7x^2+29x+22=0
Ans: x=-1 or x=-3.14
Q4714) Solve 32x^2+44x+5=0
Ans: x=-0.12 or x=-1.25
Q4715) Solve 17x^2+22x+5=0
Ans: x=-0.29 or x=-1
Q4716) Solve 16x^2+40x+9=0
Ans: x=-0.25 or x=-2.25
Q4717) Solve 40x^2+32x+6=0
Ans: x=-0.3 or x=-0.5
Q4718) Solve 37x^2+41x+4=0
Ans: x=-0.11 or x=-1
Q4719) Solve 42x^2+19x+2=0
Ans: x=-0.17 or x=-0.29
Q4720) Solve 18x^2+38x+20=0
Ans: x=-1 or x=-1.11
Q4721) Solve 13x^2+46x+33=0
Ans: x=-1 or x=-2.54
Q4722) Solve 46x^2+47x+12=0
Ans: x=-0.5 or x=-0.52
Q4723) Solve 30x^2+26x+4=0
Ans: x=-0.2 or x=-0.67
Q4724) Solve 31x^2+33x+2=0
Ans: x=-0.06 or x=-1
Q4725) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q4726) Solve 31x^2+37x+6=0
Ans: x=-0.19 or x=-1
Q4727) Solve 33x^2+31x+6=0
Ans: x=-0.27 or x=-0.67
Q4728) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q4729) Solve 42x^2+32x+6=0
Ans: x=-0.33 or x=-0.43
Q4730) Solve 44x^2+26x+2=0
Ans: x=-0.09 or x=-0.5
Q4731) Solve 13x^2+37x+24=0
Ans: x=-1 or x=-1.85
Q4732) Solve 32x^2+39x+7=0
Ans: x=-0.22 or x=-1
Q4733) Solve 49x^2+35x+6=0
Ans: x=-0.29 or x=-0.43
Q4734) Solve 10x^2+35x+25=0
Ans: x=-1 or x=-2.5
Q4735) Solve 30x^2+21x+3=0
Ans: x=-0.2 or x=-0.5
Q4736) Solve 10x^2+47x+9=0
Ans: x=-0.2 or x=-4.5
Q4737) Solve 50x^2+40x+6=0
Ans: x=-0.2 or x=-0.6
Q4738) Solve 21x^2+22x+1=0
Ans: x=-0.05 or x=-1
Q4739) Solve 6x^2+29x+34=0
Ans: x=-2 or x=-2.83
Q4740) Solve 25x^2+45x+20=0
Ans: x=-0.8 or x=-1
Q4741) Solve 2x^2+17x+30=0
Ans: x=-2.5 or x=-6
Q4742) Solve 22x^2+40x+18=0
Ans: x=-0.82 or x=-1
Q4743) Solve 38x^2+49x+15=0
Ans: x=-0.5 or x=-0.79
Q4744) Solve 12x^2+41x+22=0
Ans: x=-0.67 or x=-2.75
Q4745) Solve 3x^2+48x+45=0
Ans: x=-1 or x=-15
Q4746) Solve 11x^2+45x+4=0
Ans: x=-0.09 or x=-4
Q4747) Solve 16x^2+34x+4=0
Ans: x=-0.12 or x=-2
Q4748) Solve 14x^2+35x+14=0
Ans: x=-0.5 or x=-2
Q4749) Solve 18x^2+25x+8=0
Ans: x=-0.5 or x=-0.89
Q4750) Solve 8x^2+26x+15=0
Ans: x=-0.75 or x=-2.5
Q4751) Solve 7x^2+32x+25=0
Ans: x=-1 or x=-3.57
Q4752) Solve 12x^2+26x+14=0
Ans: x=-1 or x=-1.17
Q4753) Solve 22x^2+49x+19=0
Ans: x=-0.5 or x=-1.73
Q4754) Solve 3x^2+45x+42=0
Ans: x=-1 or x=-14
Q4755) Solve 5x^2+36x+31=0
Ans: x=-1 or x=-6.2
Q4756) Solve 6x^2+28x+16=0
Ans: x=-0.67 or x=-4
Q4757) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q4758) Solve 22x^2+48x+8=0
Ans: x=-0.18 or x=-2
Q4759) Solve 6x^2+23x+10=0
Ans: x=-0.5 or x=-3.33
Q4760) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q4761) Solve 8x^2+46x+45=0
Ans: x=-1.25 or x=-4.5
Q4762) Solve 25x^2+45x+14=0
Ans: x=-0.4 or x=-1.4
Q4763) Solve 25x^2+40x+16=0
Ans: x=-0.8 or x=-0.8
Q4764) Solve 12x^2+22x+10=0
Ans: x=-0.83 or x=-1
Q4765) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q4766) Solve 2x^2+17x+30=0
Ans: x=-2.5 or x=-6
Q4767) Solve x^2+15x+44=0
Ans: x=-4 or x=-11
Q4768) Solve 10x^2+37x+7=0
Ans: x=-0.2 or x=-3.5
Q4769) Solve 25x^2+40x+16=0
Ans: x=-0.8 or x=-0.8
Q4770) Solve 3x^2+28x+32=0
Ans: x=-1.33 or x=-8
Q4771) Solve 5x^2+37x+14=0
Ans: x=-0.4 or x=-7
Q4772) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q4773) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q4774) Solve 40x^2+49x+9=0
Ans: x=-0.22 or x=-1
Q4775) Solve 30x^2+49x+6=0
Ans: x=-0.13 or x=-1.5
Q4776) Solve x^2+26x+48=0
Ans: x=-2 or x=-24
Q4777) Solve 2x^2+18x+28=0
Ans: x=-2 or x=-7
Q4778) Solve x^2+8x+16=0
Ans: x=-4 or x=-4
Q4779) Solve 18x^2+23x+7=0
Ans: x=-0.5 or x=-0.78
Q4780) Solve 8x^2+23x+15=0
Ans: x=-1 or x=-1.87
Q4781) Solve 13x^2+50x+33=0
Ans: x=-0.85 or x=-3
Q4782) Solve 22x^2+38x+16=0
Ans: x=-0.73 or x=-1
Q4783) Solve 3x^2+20x+32=0
Ans: x=-2.67 or x=-4
Q4784) Solve 44x^2+28x+3=0
Ans: x=-0.14 or x=-0.5
Q4785) Solve 11x^2+32x+21=0
Ans: x=-1 or x=-1.91
Q4786) Solve 6x^2+44x+14=0
Ans: x=-0.33 or x=-7
Q4787) Solve 5x^2+28x+39=0
Ans: x=-2.6 or x=-3
Q4788) Solve 12x^2+40x+17=0
Ans: x=-0.5 or x=-2.83
Q4789) Solve 19x^2+49x+30=0
Ans: x=-1 or x=-1.58
Q4790) Solve x^2+9x+18=0
Ans: x=-3 or x=-6
Q4791) Solve 6x^2+28x+16=0
Ans: x=-0.67 or x=-4
Q4792) Solve 10x^2+47x+48=0
Ans: x=-1.5 or x=-3.2
Q4793) Solve 7x^2+23x+6=0
Ans: x=-0.29 or x=-3
Q4794) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q4795) Solve 8x^2+41x+33=0
Ans: x=-1 or x=-4.12
Q4796) Solve 8x^2+33x+4=0
Ans: x=-0.12 or x=-4
Q4797) Solve 9x^2+25x+16=0
Ans: x=-1 or x=-1.78
Q4798) Solve 27x^2+35x+8=0
Ans: x=-0.3 or x=-1
Q4799) Solve 7x^2+43x+6=0
Ans: x=-0.14 or x=-6
Q4800) Solve 14x^2+23x+9=0
Ans: x=-0.64 or x=-1
Q4801) Solve 8x^2+28x+12=0
Ans: x=-0.5 or x=-3
Q4802) Solve 5x^2+26x+5=0
Ans: x=-0.2 or x=-5
Q4803) Solve 10x^2+31x+21=0
Ans: x=-1 or x=-2.1
Q4804) Solve 2x^2+47x+23=0
Ans: x=-0.5 or x=-23
Q4805) Solve 6x^2+35x+11=0
Ans: x=-0.33 or x=-5.5
Q4806) Solve 21x^2+33x+12=0
Ans: x=-0.57 or x=-1
Q4807) Solve 5x^2+27x+34=0
Ans: x=-2 or x=-3.4
Q4808) Solve 9x^2+49x+40=0
Ans: x=-1 or x=-4.44
Q4809) Solve 29x^2+35x+6=0
Ans: x=-0.21 or x=-1
Q4810) Solve 10x^2+41x+31=0
Ans: x=-1 or x=-3.1
Q4811) Solve 7x^2+8x+1=0
Ans: x=-0.14 or x=-1
Q4812) Solve 6x^2+14x+4=0
Ans: x=-0.33 or x=-2
Q4813) Solve 7x^2+25x+22=0
Ans: x=-1.57 or x=-2
Q4814) Solve 3x^2+22x+35=0
Ans: x=-2.33 or x=-5
Q4815) Solve 4x^2+9x+2=0
Ans: x=-0.25 or x=-2
Q4816) Solve 9x^2+42x+48=0
Ans: x=-2 or x=-2.67
Q4817) Solve 14x^2+45x+34=0
Ans: x=-1.21 or x=-2
Q4818) Solve 4x^2+28x+45=0
Ans: x=-2.5 or x=-4.5
Q4819) Solve 8x^2+44x+48=0
Ans: x=-1.5 or x=-4
Q4820) Solve 11x^2+31x+18=0
Ans: x=-0.82 or x=-2
Q4821) Solve 10x^2+39x+35=0
Ans: x=-1.4 or x=-2.5
Q4822) Solve 3x^2+23x+34=0
Ans: x=-2 or x=-5.67
Q4823) Solve 48x^2+20x+2=0
Ans: x=-0.17 or x=-0.25
Q4824) Solve 14x^2+43x+20=0
Ans: x=-0.57 or x=-2.5
Q4825) Solve 3x^2+40x+37=0
Ans: x=-1 or x=-12.33
Q4826) Solve 2x^2+39x+19=0
Ans: x=-0.5 or x=-19
Q4827) Solve 17x^2+50x+32=0
Ans: x=-0.94 or x=-2
Q4828) Solve 24x^2+47x+23=0
Ans: x=-0.96 or x=-1
Q4829) Solve 32x^2+36x+10=0
Ans: x=-0.5 or x=-0.62
Q4830) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q4831) Solve 17x^2+43x+26=0
Ans: x=-1 or x=-1.53
Q4832) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q4833) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q4834) Solve 44x^2+35x+6=0
Ans: x=-0.25 or x=-0.55
Q4835) Solve 22x^2+48x+8=0
Ans: x=-0.18 or x=-2
Q4836) Solve 10x^2+35x+15=0
Ans: x=-0.5 or x=-3
Q4837) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q4838) Solve 3x^2+19x+16=0
Ans: x=-1 or x=-5.33
Q4839) Solve 10x^2+40x+30=0
Ans: x=-1 or x=-3
Q4840) Solve 34x^2+39x+5=0
Ans: x=-0.15 or x=-1
Q4841) Solve 46x^2+48x+2=0
Ans: x=-0.04 or x=-1
Q4842) Solve 48x^2+25x+3=0
Ans: x=-0.19 or x=-0.33
Q4843) Solve 3x^2+25x+8=0
Ans: x=-0.33 or x=-8
Q4844) Solve 40x^2+49x+9=0
Ans: x=-0.22 or x=-1
Q4845) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q4846) Solve 36x^2+26x+4=0
Ans: x=-0.22 or x=-0.5
Q4847) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q4848) Solve 5x^2+45x+40=0
Ans: x=-1 or x=-8
Q4849) Solve 14x^2+28x+14=0
Ans: x=-1 or x=-1
Q4850) Solve 2x^2+19x+30=0
Ans: x=-2 or x=-7.5
Q4851) Solve 5x^2+32x+35=0
Ans: x=-1.4 or x=-5
Q4852) Solve 50x^2+35x+5=0
Ans: x=-0.2 or x=-0.5
Q4853) Solve x^2+12x+36=0
Ans: x=-6 or x=-6
Q4854) Solve 10x^2+37x+33=0
Ans: x=-1.5 or x=-2.2
Q4855) Solve 4x^2+31x+21=0
Ans: x=-0.75 or x=-7
Q4856) Solve 7x^2+28x+21=0
Ans: x=-1 or x=-3
Q4857) Solve 4x^2+39x+27=0
Ans: x=-0.75 or x=-9
Q4858) Solve 12x^2+35x+22=0
Ans: x=-0.92 or x=-2
Q4859) Solve 15x^2+25x+10=0
Ans: x=-0.67 or x=-1
Q4860) Solve 10x^2+19x+9=0
Ans: x=-0.9 or x=-1
Q4861) Solve 4x^2+42x+20=0
Ans: x=-0.5 or x=-10
Q4862) Solve 46x^2+50x+4=0
Ans: x=-0.09 or x=-1
Q4863) Solve 14x^2+39x+10=0
Ans: x=-0.29 or x=-2.5
Q4864) Solve 19x^2+48x+20=0
Ans: x=-0.53 or x=-2
Q4865) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q4866) Solve 31x^2+44x+13=0
Ans: x=-0.42 or x=-1
Q4867) Solve 44x^2+49x+5=0
Ans: x=-0.11 or x=-1
Q4868) Solve 6x^2+7x+1=0
Ans: x=-0.17 or x=-1
Q4869) Solve 7x^2+50x+48=0
Ans: x=-1.14 or x=-6
Q4870) Solve 2x^2+21x+10=0
Ans: x=-0.5 or x=-10
Q4871) Solve 9x^2+42x+48=0
Ans: x=-2 or x=-2.67
Q4872) Solve 32x^2+26x+5=0
Ans: x=-0.31 or x=-0.5
Q4873) Solve 25x^2+25x+4=0
Ans: x=-0.2 or x=-0.8
Q4874) Solve 19x^2+20x+1=0
Ans: x=-0.05 or x=-1
Q4875) Solve 9x^2+27x+20=0
Ans: x=-1.33 or x=-1.67
Q4876) Solve 21x^2+26x+8=0
Ans: x=-0.57 or x=-0.67
Q4877) Solve 2x^2+25x+23=0
Ans: x=-1 or x=-11.5
Q4878) Solve 15x^2+48x+9=0
Ans: x=-0.2 or x=-3
Q4879) Solve 2x^2+45x+43=0
Ans: x=-1 or x=-21.5
Q4880) Solve 28x^2+39x+8=0
Ans: x=-0.25 or x=-1.14
Q4881) Solve 12x^2+50x+48=0
Ans: x=-1.5 or x=-2.67
Q4882) Solve 18x^2+33x+5=0
Ans: x=-0.17 or x=-1.67
Q4883) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q4884) Solve 22x^2+46x+24=0
Ans: x=-1 or x=-1.09
Q4885) Solve x^2+38x+37=0
Ans: x=-1 or x=-37
Q4886) Solve 18x^2+49x+26=0
Ans: x=-0.72 or x=-2
Q4887) Solve 4x^2+25x+34=0
Ans: x=-2 or x=-4.25
Q4888) Solve 6x^2+25x+11=0
Ans: x=-0.5 or x=-3.67
Q4889) Solve 36x^2+40x+11=0
Ans: x=-0.5 or x=-0.61
Q4890) Solve 48x^2+50x+13=0
Ans: x=-0.5 or x=-0.54
Q4891) Solve 27x^2+15x+2=0
Ans: x=-0.22 or x=-0.33
Q4892) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q4893) Solve 31x^2+44x+13=0
Ans: x=-0.42 or x=-1
Q4894) Solve 17x^2+35x+2=0
Ans: x=-0.06 or x=-2
Q4895) Solve 11x^2+26x+8=0
Ans: x=-0.36 or x=-2
Q4896) Solve 24x^2+22x+4=0
Ans: x=-0.25 or x=-0.67
Q4897) Solve 6x^2+37x+45=0
Ans: x=-1.67 or x=-4.5
Q4898) Solve 45x^2+21x+2=0
Ans: x=-0.13 or x=-0.33
Q4899) Solve 3x^2+25x+38=0
Ans: x=-2 or x=-6.33
Q4900) Solve 8x^2+22x+15=0
Ans: x=-1.25 or x=-1.5
Q4901) Solve 3x^2+16x+13=0
Ans: x=-1 or x=-4.33
Q4902) Solve 8x^2+19x+6=0
Ans: x=-0.37 or x=-2
Q4903) Solve 32x^2+44x+9=0
Ans: x=-0.25 or x=-1.12
Q4904) Solve 39x^2+49x+12=0
Ans: x=-0.33 or x=-0.92
Q4905) Solve 13x^2+27x+14=0
Ans: x=-1 or x=-1.08
Q4906) Solve 10x^2+43x+28=0
Ans: x=-0.8 or x=-3.5
Q4907) Solve 24x^2+49x+2=0
Ans: x=-0.04 or x=-2
Q4908) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q4909) Solve 27x^2+33x+8=0
Ans: x=-0.33 or x=-0.89
Q4910) Solve 5x^2+21x+4=0
Ans: x=-0.2 or x=-4
Q4911) Solve 32x^2+44x+9=0
Ans: x=-0.25 or x=-1.12
Q4912) Solve 5x^2+31x+48=0
Ans: x=-3 or x=-3.2
Q4913) Solve 9x^2+20x+4=0
Ans: x=-0.22 or x=-2
Q4914) Solve 29x^2+34x+5=0
Ans: x=-0.17 or x=-1
Q4915) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q4916) Solve 10x^2+45x+35=0
Ans: x=-1 or x=-3.5
Q4917) Solve 2x^2+10x+12=0
Ans: x=-2 or x=-3
Q4918) Solve 24x^2+38x+10=0
Ans: x=-0.33 or x=-1.25
Q4919) Solve 26x^2+31x+9=0
Ans: x=-0.5 or x=-0.69
Q4920) Solve 6x^2+43x+20=0
Ans: x=-0.5 or x=-6.67
Q4921) Solve 9x^2+27x+20=0
Ans: x=-1.33 or x=-1.67
Q4922) Solve 5x^2+27x+28=0
Ans: x=-1.4 or x=-4
Q4923) Solve 8x^2+31x+21=0
Ans: x=-0.87 or x=-3
Q4924) Solve 24x^2+46x+10=0
Ans: x=-0.25 or x=-1.67
Q4925) Solve 30x^2+49x+20=0
Ans: x=-0.8 or x=-0.83
Q4926) Solve 2x^2+12x+10=0
Ans: x=-1 or x=-5
Q4927) Solve 13x^2+38x+25=0
Ans: x=-1 or x=-1.92
Q4928) Solve 2x^2+25x+50=0
Ans: x=-2.5 or x=-10
Q4929) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q4930) Solve 2x^2+32x+30=0
Ans: x=-1 or x=-15
Q4931) Solve 49x^2+28x+3=0
Ans: x=-0.14 or x=-0.43
Q4932) Solve 9x^2+36x+35=0
Ans: x=-1.67 or x=-2.33
Q4933) Solve 26x^2+35x+9=0
Ans: x=-0.35 or x=-1
Q4934) Solve 7x^2+36x+44=0
Ans: x=-2 or x=-3.14
Q4935) Solve 8x^2+23x+14=0
Ans: x=-0.87 or x=-2
Q4936) Solve 33x^2+17x+2=0
Ans: x=-0.18 or x=-0.33
Q4937) Solve 23x^2+39x+16=0
Ans: x=-0.7 or x=-1
Q4938) Solve 4x^2+36x+17=0
Ans: x=-0.5 or x=-8.5
Q4939) Solve 19x^2+40x+21=0
Ans: x=-1 or x=-1.11
Q4940) Solve 10x^2+40x+30=0
Ans: x=-1 or x=-3
Q4941) Solve 10x^2+45x+35=0
Ans: x=-1 or x=-3.5
Q4942) Solve 16x^2+22x+7=0
Ans: x=-0.5 or x=-0.87
Q4943) Solve 5x^2+15x+10=0
Ans: x=-1 or x=-2
Q4944) Solve 6x^2+49x+8=0
Ans: x=-0.17 or x=-8
Q4945) Solve 15x^2+38x+24=0
Ans: x=-1.2 or x=-1.33
Q4946) Solve 14x^2+43x+3=0
Ans: x=-0.07 or x=-3
Q4947) Solve 26x^2+47x+17=0
Ans: x=-0.5 or x=-1.31
Q4948) Solve 23x^2+29x+6=0
Ans: x=-0.26 or x=-1
Q4949) Solve 32x^2+33x+1=0
Ans: x=-0.03 or x=-1
Q4950) Solve 14x^2+45x+19=0
Ans: x=-0.5 or x=-2.71
Q4951) Solve 20x^2+44x+21=0
Ans: x=-0.7 or x=-1.5
Q4952) Solve 9x^2+28x+19=0
Ans: x=-1 or x=-2.11
Q4953) Solve 20x^2+49x+29=0
Ans: x=-1 or x=-1.45
Q4954) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q4955) Solve 9x^2+46x+5=0
Ans: x=-0.11 or x=-5
Q4956) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q4957) Solve 44x^2+49x+12=0
Ans: x=-0.36 or x=-0.75
Q4958) Solve 20x^2+47x+14=0
Ans: x=-0.35 or x=-2
Q4959) Solve 5x^2+13x+8=0
Ans: x=-1 or x=-1.6
Q4960) Solve x^2+24x+23=0
Ans: x=-1 or x=-23
Q4961) Solve 28x^2+25x+3=0
Ans: x=-0.14 or x=-0.75
Q4962) Solve 2x^2+26x+24=0
Ans: x=-1 or x=-12
Q4963) Solve 18x^2+40x+8=0
Ans: x=-0.22 or x=-2
Q4964) Solve 20x^2+45x+10=0
Ans: x=-0.25 or x=-2
Q4965) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q4966) Solve 20x^2+43x+6=0
Ans: x=-0.15 or x=-2
Q4967) Solve 2x^2+26x+44=0
Ans: x=-2 or x=-11
Q4968) Solve 10x^2+41x+42=0
Ans: x=-2 or x=-2.1
Q4969) Solve 18x^2+27x+10=0
Ans: x=-0.67 or x=-0.83
Q4970) Solve 11x^2+25x+14=0
Ans: x=-1 or x=-1.27
Q4971) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q4972) Solve 9x^2+30x+21=0
Ans: x=-1 or x=-2.33
Q4973) Solve 6x^2+44x+14=0
Ans: x=-0.33 or x=-7
Q4974) Solve 8x^2+31x+23=0
Ans: x=-1 or x=-2.87
Q4975) Solve 2x^2+36x+34=0
Ans: x=-1 or x=-17
Q4976) Solve 8x^2+41x+5=0
Ans: x=-0.12 or x=-5
Q4977) Solve 3x^2+36x+33=0
Ans: x=-1 or x=-11
Q4978) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q4979) Solve 12x^2+25x+12=0
Ans: x=-0.75 or x=-1.33
Q4980) Solve 34x^2+23x+3=0
Ans: x=-0.18 or x=-0.5
Q4981) Solve 8x^2+28x+12=0
Ans: x=-0.5 or x=-3
Q4982) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q4983) Solve 15x^2+33x+6=0
Ans: x=-0.2 or x=-2
Q4984) Solve 2x^2+11x+5=0
Ans: x=-0.5 or x=-5
Q4985) Solve 27x^2+36x+9=0
Ans: x=-0.33 or x=-1
Q4986) Solve 17x^2+24x+7=0
Ans: x=-0.41 or x=-1
Q4987) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q4988) Solve 20x^2+45x+25=0
Ans: x=-1 or x=-1.25
Q4989) Solve 9x^2+45x+26=0
Ans: x=-0.67 or x=-4.33
Q4990) Solve 26x^2+29x+8=0
Ans: x=-0.5 or x=-0.62
Q4991) Solve 28x^2+30x+8=0
Ans: x=-0.5 or x=-0.57
Q4992) Solve 50x^2+40x+8=0
Ans: x=-0.4 or x=-0.4
Q4993) Solve 12x^2+37x+26=0
Ans: x=-1.08 or x=-2
Q4994) Solve 24x^2+49x+2=0
Ans: x=-0.04 or x=-2
Q4995) Solve 19x^2+47x+28=0
Ans: x=-1 or x=-1.47
Q4996) Solve 7x^2+39x+50=0
Ans: x=-2 or x=-3.57
Q4997) Solve 4x^2+48x+23=0
Ans: x=-0.5 or x=-11.5
Q4998) Solve 2x^2+47x+45=0
Ans: x=-1 or x=-22.5
Q4999) Solve 25x^2+40x+7=0
Ans: x=-0.2 or x=-1.4
Q5000) Solve 14x^2+29x+12=0
Ans: x=-0.57 or x=-1.5
Q5001) Solve 25x^2+35x+12=0
Ans: x=-0.6 or x=-0.8
Q5002) Solve 7x^2+43x+36=0
Ans: x=-1 or x=-5.14
Q5003) Solve 11x^2+34x+3=0
Ans: x=-0.09 or x=-3
Q5004) Solve 13x^2+44x+31=0
Ans: x=-1 or x=-2.38
Q5005) Solve 7x^2+40x+25=0
Ans: x=-0.71 or x=-5
Q5006) Solve 8x^2+44x+48=0
Ans: x=-1.5 or x=-4
Q5007) Solve 14x^2+34x+12=0
Ans: x=-0.43 or x=-2
Q5008) Solve 26x^2+31x+5=0
Ans: x=-0.19 or x=-1
Q5009) Solve 28x^2+34x+10=0
Ans: x=-0.5 or x=-0.71
Q5010) Solve 15x^2+44x+21=0
Ans: x=-0.6 or x=-2.33
Q5011) Solve x^2+26x+25=0
Ans: x=-1 or x=-25
Q5012) Solve 4x^2+17x+18=0
Ans: x=-2 or x=-2.25
Q5013) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q5014) Solve 15x^2+36x+21=0
Ans: x=-1 or x=-1.4
Q5015) Solve 42x^2+44x+10=0
Ans: x=-0.33 or x=-0.71
Q5016) Solve 36x^2+39x+9=0
Ans: x=-0.33 or x=-0.75
Q5017) Solve 2x^2+35x+33=0
Ans: x=-1 or x=-16.5
Q5018) Solve 12x^2+14x+4=0
Ans: x=-0.5 or x=-0.67
Q5019) Solve 9x^2+34x+32=0
Ans: x=-1.78 or x=-2
Q5020) Solve 7x^2+35x+42=0
Ans: x=-2 or x=-3
Q5021) Solve 4x^2+41x+37=0
Ans: x=-1 or x=-9.25
Q5022) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q5023) Solve 6x^2+7x+1=0
Ans: x=-0.17 or x=-1
Q5024) Solve 10x^2+17x+6=0
Ans: x=-0.5 or x=-1.2
Q5025) Solve 9x^2+28x+3=0
Ans: x=-0.11 or x=-3
Q5026) Solve 12x^2+32x+21=0
Ans: x=-1.17 or x=-1.5
Q5027) Solve 6x^2+29x+20=0
Ans: x=-0.83 or x=-4
Q5028) Solve 48x^2+16x+1=0
Ans: x=-0.08 or x=-0.25
Q5029) Solve 20x^2+9x+1=0
Ans: x=-0.2 or x=-0.25
Q5030) Solve 8x^2+42x+45=0
Ans: x=-1.5 or x=-3.75
Q5031) Solve 15x^2+49x+40=0
Ans: x=-1.6 or x=-1.67
Q5032) Solve 6x^2+21x+18=0
Ans: x=-1.5 or x=-2
Q5033) Solve 6x^2+34x+28=0
Ans: x=-1 or x=-4.67
Q5034) Solve 11x^2+16x+5=0
Ans: x=-0.45 or x=-1
Q5035) Solve 10x^2+36x+18=0
Ans: x=-0.6 or x=-3
Q5036) Solve 18x^2+42x+24=0
Ans: x=-1 or x=-1.33
Q5037) Solve 34x^2+19x+1=0
Ans: x=-0.06 or x=-0.5
Q5038) Solve 12x^2+48x+45=0
Ans: x=-1.5 or x=-2.5
Q5039) Solve 9x^2+15x+4=0
Ans: x=-0.33 or x=-1.33
Q5040) Solve 21x^2+47x+10=0
Ans: x=-0.24 or x=-2
Q5041) Solve 28x^2+43x+15=0
Ans: x=-0.54 or x=-1
Q5042) Solve 2x^2+15x+22=0
Ans: x=-2 or x=-5.5
Q5043) Solve 7x^2+34x+24=0
Ans: x=-0.86 or x=-4
Q5044) Solve 26x^2+35x+11=0
Ans: x=-0.5 or x=-0.85
Q5045) Solve 8x^2+34x+36=0
Ans: x=-2 or x=-2.25
Q5046) Solve 16x^2+49x+3=0
Ans: x=-0.06 or x=-3
Q5047) Solve 19x^2+49x+30=0
Ans: x=-1 or x=-1.58
Q5048) Solve 18x^2+45x+25=0
Ans: x=-0.83 or x=-1.67
Q5049) Solve 23x^2+43x+20=0
Ans: x=-0.87 or x=-1
Q5050) Solve 15x^2+50x+40=0
Ans: x=-1.33 or x=-2
Q5051) Solve 30x^2+32x+2=0
Ans: x=-0.07 or x=-1
Q5052) Solve 19x^2+29x+10=0
Ans: x=-0.53 or x=-1
Q5053) Solve 10x^2+34x+12=0
Ans: x=-0.4 or x=-3
Q5054) Solve 8x^2+37x+39=0
Ans: x=-1.62 or x=-3
Q5055) Solve 12x^2+46x+42=0
Ans: x=-1.5 or x=-2.33
Q5056) Solve 15x^2+47x+34=0
Ans: x=-1.13 or x=-2
Q5057) Solve 14x^2+22x+8=0
Ans: x=-0.57 or x=-1
Q5058) Solve 5x^2+32x+35=0
Ans: x=-1.4 or x=-5
Q5059) Solve 3x^2+42x+39=0
Ans: x=-1 or x=-13
Q5060) Solve 44x^2+28x+3=0
Ans: x=-0.14 or x=-0.5
Q5061) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q5062) Solve 12x^2+18x+6=0
Ans: x=-0.5 or x=-1
Q5063) Solve 36x^2+22x+2=0
Ans: x=-0.11 or x=-0.5
Q5064) Solve 8x^2+43x+35=0
Ans: x=-1 or x=-4.37
Q5065) Solve 8x^2+35x+12=0
Ans: x=-0.37 or x=-4
Q5066) Solve 12x^2+38x+20=0
Ans: x=-0.67 or x=-2.5
Q5067) Solve 16x^2+44x+28=0
Ans: x=-1 or x=-1.75
Q5068) Solve 3x^2+41x+38=0
Ans: x=-1 or x=-12.67
Q5069) Solve 8x^2+40x+42=0
Ans: x=-1.5 or x=-3.5
Q5070) Solve 24x^2+47x+23=0
Ans: x=-0.96 or x=-1
Q5071) Solve 3x^2+36x+33=0
Ans: x=-1 or x=-11
Q5072) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q5073) Solve x^2+4x+4=0
Ans: x=-2 or x=-2
Q5074) Solve 2x^2+20x+48=0
Ans: x=-4 or x=-6
Q5075) Solve 32x^2+45x+13=0
Ans: x=-0.41 or x=-1
Q5076) Solve 16x^2+26x+10=0
Ans: x=-0.62 or x=-1
Q5077) Solve 25x^2+43x+18=0
Ans: x=-0.72 or x=-1
Q5078) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q5079) Solve 5x^2+42x+37=0
Ans: x=-1 or x=-7.4
Q5080) Solve 12x^2+16x+5=0
Ans: x=-0.5 or x=-0.83
Q5081) Solve 8x^2+43x+44=0
Ans: x=-1.37 or x=-4
Q5082) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q5083) Solve 30x^2+46x+16=0
Ans: x=-0.53 or x=-1
Q5084) Solve 49x^2+49x+10=0
Ans: x=-0.29 or x=-0.71
Q5085) Solve x^2+14x+45=0
Ans: x=-5 or x=-9
Q5086) Solve 9x^2+42x+13=0
Ans: x=-0.33 or x=-4.33
Q5087) Solve 5x^2+34x+24=0
Ans: x=-0.8 or x=-6
Q5088) Solve 4x^2+34x+16=0
Ans: x=-0.5 or x=-8
Q5089) Solve 7x^2+28x+28=0
Ans: x=-2 or x=-2
Q5090) Solve 14x^2+43x+30=0
Ans: x=-1.07 or x=-2
Q5091) Solve 34x^2+21x+2=0
Ans: x=-0.12 or x=-0.5
Q5092) Solve 6x^2+27x+27=0
Ans: x=-1.5 or x=-3
Q5093) Solve 9x^2+25x+14=0
Ans: x=-0.78 or x=-2
Q5094) Solve 4x^2+29x+7=0
Ans: x=-0.25 or x=-7
Q5095) Solve 37x^2+48x+11=0
Ans: x=-0.3 or x=-1
Q5096) Solve 35x^2+41x+12=0
Ans: x=-0.57 or x=-0.6
Q5097) Solve 2x^2+5x+3=0
Ans: x=-1 or x=-1.5
Q5098) Solve 11x^2+30x+16=0
Ans: x=-0.73 or x=-2
Q5099) Solve x^2+30x+29=0
Ans: x=-1 or x=-29
Q5100) Solve x^2+7x+12=0
Ans: x=-3 or x=-4
Q5101) Solve 2x^2+29x+27=0
Ans: x=-1 or x=-13.5
Q5102) Solve 8x^2+40x+50=0
Ans: x=-2.5 or x=-2.5
Q5103) Solve 14x^2+23x+9=0
Ans: x=-0.64 or x=-1
Q5104) Solve 19x^2+45x+26=0
Ans: x=-1 or x=-1.37
Q5105) Solve 9x^2+36x+36=0
Ans: x=-2 or x=-2
Q5106) Solve x^2+15x+50=0
Ans: x=-5 or x=-10
Q5107) Solve 40x^2+47x+7=0
Ans: x=-0.17 or x=-1
Q5108) Solve 7x^2+37x+10=0
Ans: x=-0.29 or x=-5
Q5109) Solve 9x^2+41x+32=0
Ans: x=-1 or x=-3.56
Q5110) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q5111) Solve 12x^2+25x+2=0
Ans: x=-0.08 or x=-2
Q5112) Solve 18x^2+37x+15=0
Ans: x=-0.56 or x=-1.5
Q5113) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q5114) Solve 28x^2+26x+6=0
Ans: x=-0.43 or x=-0.5
Q5115) Solve 14x^2+24x+10=0
Ans: x=-0.71 or x=-1
Q5116) Solve 42x^2+26x+4=0
Ans: x=-0.29 or x=-0.33
Q5117) Solve 26x^2+39x+13=0
Ans: x=-0.5 or x=-1
Q5118) Solve 40x^2+50x+15=0
Ans: x=-0.5 or x=-0.75
Q5119) Solve 21x^2+40x+19=0
Ans: x=-0.9 or x=-1
Q5120) Solve 15x^2+46x+24=0
Ans: x=-0.67 or x=-2.4
Q5121) Solve 37x^2+40x+3=0
Ans: x=-0.08 or x=-1
Q5122) Solve 50x^2+50x+12=0
Ans: x=-0.4 or x=-0.6
Q5123) Solve 12x^2+42x+36=0
Ans: x=-1.5 or x=-2
Q5124) Solve 17x^2+49x+32=0
Ans: x=-1 or x=-1.88
Q5125) Solve 13x^2+23x+10=0
Ans: x=-0.77 or x=-1
Q5126) Solve 10x^2+47x+48=0
Ans: x=-1.5 or x=-3.2
Q5127) Solve 36x^2+44x+8=0
Ans: x=-0.22 or x=-1
Q5128) Solve 27x^2+36x+12=0
Ans: x=-0.67 or x=-0.67
Q5129) Solve 18x^2+36x+16=0
Ans: x=-0.67 or x=-1.33
Q5130) Solve 22x^2+17x+3=0
Ans: x=-0.27 or x=-0.5
Q5131) Solve 3x^2+21x+18=0
Ans: x=-1 or x=-6
Q5132) Solve 7x^2+32x+33=0
Ans: x=-1.57 or x=-3
Q5133) Solve 2x^2+3x+1=0
Ans: x=-0.5 or x=-1
Q5134) Solve 10x^2+27x+17=0
Ans: x=-1 or x=-1.7
Q5135) Solve 37x^2+42x+5=0
Ans: x=-0.14 or x=-1
Q5136) Solve 33x^2+36x+3=0
Ans: x=-0.09 or x=-1
Q5137) Solve 45x^2+33x+6=0
Ans: x=-0.33 or x=-0.4
Q5138) Solve 6x^2+41x+13=0
Ans: x=-0.33 or x=-6.5
Q5139) Solve 3x^2+12x+12=0
Ans: x=-2 or x=-2
Q5140) Solve 18x^2+27x+7=0
Ans: x=-0.33 or x=-1.17
Q5141) Solve 42x^2+17x+1=0
Ans: x=-0.07 or x=-0.33
Q5142) Solve 27x^2+39x+10=0
Ans: x=-0.33 or x=-1.11
Q5143) Solve 32x^2+50x+18=0
Ans: x=-0.56 or x=-1
Q5144) Solve 9x^2+39x+30=0
Ans: x=-1 or x=-3.33
Q5145) Solve 26x^2+21x+4=0
Ans: x=-0.31 or x=-0.5
Q5146) Solve 21x^2+46x+8=0
Ans: x=-0.19 or x=-2
Q5147) Solve 18x^2+47x+22=0
Ans: x=-0.61 or x=-2
Q5148) Solve 32x^2+37x+5=0
Ans: x=-0.16 or x=-1
Q5149) Solve 15x^2+41x+14=0
Ans: x=-0.4 or x=-2.33
Q5150) Solve 16x^2+26x+10=0
Ans: x=-0.62 or x=-1
Q5151) Solve 16x^2+47x+31=0
Ans: x=-1 or x=-1.94
Q5152) Solve 15x^2+34x+19=0
Ans: x=-1 or x=-1.27
Q5153) Solve 6x^2+29x+20=0
Ans: x=-0.83 or x=-4
Q5154) Solve 23x^2+48x+4=0
Ans: x=-0.09 or x=-2
Q5155) Solve 14x^2+27x+13=0
Ans: x=-0.93 or x=-1
Q5156) Solve 38x^2+23x+2=0
Ans: x=-0.11 or x=-0.5
Q5157) Solve 40x^2+43x+7=0
Ans: x=-0.2 or x=-0.87
Q5158) Solve 20x^2+47x+21=0
Ans: x=-0.6 or x=-1.75
Q5159) Solve 16x^2+45x+26=0
Ans: x=-0.81 or x=-2
Q5160) Solve 6x^2+35x+39=0
Ans: x=-1.5 or x=-4.33
Q5161) Solve 36x^2+30x+4=0
Ans: x=-0.17 or x=-0.67
Q5162) Solve 3x^2+22x+32=0
Ans: x=-2 or x=-5.33
Q5163) Solve 23x^2+47x+24=0
Ans: x=-1 or x=-1.04
Q5164) Solve 22x^2+39x+14=0
Ans: x=-0.5 or x=-1.27
Q5165) Solve 32x^2+39x+7=0
Ans: x=-0.22 or x=-1
Q5166) Solve 3x^2+25x+42=0
Ans: x=-2.33 or x=-6
Q5167) Solve 27x^2+35x+8=0
Ans: x=-0.3 or x=-1
Q5168) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q5169) Solve 34x^2+39x+5=0
Ans: x=-0.15 or x=-1
Q5170) Solve 3x^2+20x+32=0
Ans: x=-2.67 or x=-4
Q5171) Solve 9x^2+50x+25=0
Ans: x=-0.56 or x=-5
Q5172) Solve 31x^2+33x+2=0
Ans: x=-0.06 or x=-1
Q5173) Solve 8x^2+24x+18=0
Ans: x=-1.5 or x=-1.5
Q5174) Solve 20x^2+32x+3=0
Ans: x=-0.1 or x=-1.5
Q5175) Solve 34x^2+50x+16=0
Ans: x=-0.47 or x=-1
Q5176) Solve 46x^2+49x+3=0
Ans: x=-0.07 or x=-1
Q5177) Solve 30x^2+34x+8=0
Ans: x=-0.33 or x=-0.8
Q5178) Solve 6x^2+29x+28=0
Ans: x=-1.33 or x=-3.5
Q5179) Solve 30x^2+19x+3=0
Ans: x=-0.3 or x=-0.33
Q5180) Solve 5x^2+48x+43=0
Ans: x=-1 or x=-8.6
Q5181) Solve 27x^2+46x+19=0
Ans: x=-0.7 or x=-1
Q5182) Solve 30x^2+49x+20=0
Ans: x=-0.8 or x=-0.83
Q5183) Solve 48x^2+25x+3=0
Ans: x=-0.19 or x=-0.33
Q5184) Solve 35x^2+42x+7=0
Ans: x=-0.2 or x=-1
Q5185) Solve 5x^2+47x+18=0
Ans: x=-0.4 or x=-9
Q5186) Solve 28x^2+35x+7=0
Ans: x=-0.25 or x=-1
Q5187) Solve x^2+22x+21=0
Ans: x=-1 or x=-21
Q5188) Solve 36x^2+18x+2=0
Ans: x=-0.17 or x=-0.33
Q5189) Solve 21x^2+49x+28=0
Ans: x=-1 or x=-1.33
Q5190) Solve 30x^2+47x+3=0
Ans: x=-0.07 or x=-1.5
Q5191) Solve 50x^2+25x+3=0
Ans: x=-0.2 or x=-0.3
Q5192) Solve 16x^2+47x+30=0
Ans: x=-0.94 or x=-2
Q5193) Solve 27x^2+24x+4=0
Ans: x=-0.22 or x=-0.67
Q5194) Solve 41x^2+42x+1=0
Ans: x=-0.02 or x=-1
Q5195) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q5196) Solve 15x^2+48x+9=0
Ans: x=-0.2 or x=-3
Q5197) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q5198) Solve 19x^2+32x+13=0
Ans: x=-0.68 or x=-1
Q5199) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q5200) Solve 10x^2+37x+33=0
Ans: x=-1.5 or x=-2.2
Q5201) Solve 13x^2+21x+8=0
Ans: x=-0.62 or x=-1
Q5202) Solve 10x^2+28x+16=0
Ans: x=-0.8 or x=-2
Q5203) Solve 10x^2+32x+6=0
Ans: x=-0.2 or x=-3
Q5204) Solve 32x^2+38x+11=0
Ans: x=-0.5 or x=-0.69
Q5205) Solve 6x^2+31x+39=0
Ans: x=-2.17 or x=-3
Q5206) Solve 3x^2+41x+38=0
Ans: x=-1 or x=-12.67
Q5207) Solve 14x^2+28x+14=0
Ans: x=-1 or x=-1
Q5208) Solve x^2+25x+24=0
Ans: x=-1 or x=-24
Q5209) Solve 22x^2+47x+18=0
Ans: x=-0.5 or x=-1.64
Q5210) Solve 10x^2+37x+21=0
Ans: x=-0.7 or x=-3
Q5211) Solve 28x^2+50x+12=0
Ans: x=-0.29 or x=-1.5
Q5212) Solve 24x^2+19x+2=0
Ans: x=-0.12 or x=-0.67
Q5213) Solve 22x^2+28x+6=0
Ans: x=-0.27 or x=-1
Q5214) Solve 40x^2+14x+1=0
Ans: x=-0.1 or x=-0.25
Q5215) Solve 44x^2+36x+7=0
Ans: x=-0.32 or x=-0.5
Q5216) Solve 11x^2+48x+37=0
Ans: x=-1 or x=-3.36
Q5217) Solve 4x^2+26x+22=0
Ans: x=-1 or x=-5.5
Q5218) Solve 22x^2+49x+19=0
Ans: x=-0.5 or x=-1.73
Q5219) Solve 15x^2+37x+14=0
Ans: x=-0.47 or x=-2
Q5220) Solve 24x^2+28x+8=0
Ans: x=-0.5 or x=-0.67
Q5221) Solve 10x^2+47x+48=0
Ans: x=-1.5 or x=-3.2
Q5222) Solve 27x^2+48x+21=0
Ans: x=-0.78 or x=-1
Q5223) Solve 24x^2+29x+5=0
Ans: x=-0.21 or x=-1
Q5224) Solve 19x^2+40x+21=0
Ans: x=-1 or x=-1.11
Q5225) Solve 9x^2+27x+8=0
Ans: x=-0.33 or x=-2.67
Q5226) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q5227) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q5228) Solve 4x^2+17x+13=0
Ans: x=-1 or x=-3.25
Q5229) Solve 24x^2+49x+2=0
Ans: x=-0.04 or x=-2
Q5230) Solve 36x^2+49x+13=0
Ans: x=-0.36 or x=-1
Q5231) Solve 30x^2+47x+17=0
Ans: x=-0.57 or x=-1
Q5232) Solve 25x^2+39x+14=0
Ans: x=-0.56 or x=-1
Q5233) Solve 10x^2+35x+15=0
Ans: x=-0.5 or x=-3
Q5234) Solve 50x^2+50x+12=0
Ans: x=-0.4 or x=-0.6
Q5235) Solve 17x^2+19x+2=0
Ans: x=-0.12 or x=-1
Q5236) Solve 14x^2+48x+40=0
Ans: x=-1.43 or x=-2
Q5237) Solve 20x^2+30x+10=0
Ans: x=-0.5 or x=-1
Q5238) Solve 12x^2+40x+17=0
Ans: x=-0.5 or x=-2.83
Q5239) Solve 12x^2+43x+38=0
Ans: x=-1.58 or x=-2
Q5240) Solve 16x^2+25x+9=0
Ans: x=-0.56 or x=-1
Q5241) Solve 48x^2+40x+3=0
Ans: x=-0.08 or x=-0.75
Q5242) Solve 29x^2+49x+20=0
Ans: x=-0.69 or x=-1
Q5243) Solve 10x^2+29x+19=0
Ans: x=-1 or x=-1.9
Q5244) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q5245) Solve 7x^2+31x+24=0
Ans: x=-1 or x=-3.43
Q5246) Solve 8x^2+6x+1=0
Ans: x=-0.25 or x=-0.5
Q5247) Solve 15x^2+49x+34=0
Ans: x=-1 or x=-2.27
Q5248) Solve 16x^2+26x+10=0
Ans: x=-0.62 or x=-1
Q5249) Solve 15x^2+49x+12=0
Ans: x=-0.27 or x=-3
Q5250) Solve 11x^2+45x+34=0
Ans: x=-1 or x=-3.09
Q5251) Solve 9x^2+33x+10=0
Ans: x=-0.33 or x=-3.33
Q5252) Solve 8x^2+30x+18=0
Ans: x=-0.75 or x=-3
Q5253) Solve 20x^2+31x+11=0
Ans: x=-0.55 or x=-1
Q5254) Solve 5x^2+44x+39=0
Ans: x=-1 or x=-7.8
Q5255) Solve 28x^2+19x+3=0
Ans: x=-0.25 or x=-0.43
Q5256) Solve 3x^2+25x+38=0
Ans: x=-2 or x=-6.33
Q5257) Solve 3x^2+37x+34=0
Ans: x=-1 or x=-11.33
Q5258) Solve 45x^2+48x+12=0
Ans: x=-0.4 or x=-0.67
Q5259) Solve 13x^2+35x+18=0
Ans: x=-0.69 or x=-2
Q5260) Solve 6x^2+47x+22=0
Ans: x=-0.5 or x=-7.33
Q5261) Solve 13x^2+44x+31=0
Ans: x=-1 or x=-2.38
Q5262) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q5263) Solve 15x^2+26x+11=0
Ans: x=-0.73 or x=-1
Q5264) Solve 15x^2+49x+24=0
Ans: x=-0.6 or x=-2.67
Q5265) Solve 23x^2+49x+26=0
Ans: x=-1 or x=-1.13
Q5266) Solve 16x^2+22x+7=0
Ans: x=-0.5 or x=-0.87
Q5267) Solve 14x^2+19x+5=0
Ans: x=-0.36 or x=-1
Q5268) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q5269) Solve 16x^2+46x+30=0
Ans: x=-1 or x=-1.87
Q5270) Solve 30x^2+35x+10=0
Ans: x=-0.5 or x=-0.67
Q5271) Solve 19x^2+33x+14=0
Ans: x=-0.74 or x=-1
Q5272) Solve 18x^2+47x+19=0
Ans: x=-0.5 or x=-2.11
Q5273) Solve 28x^2+18x+2=0
Ans: x=-0.14 or x=-0.5
Q5274) Solve 27x^2+30x+7=0
Ans: x=-0.33 or x=-0.78
Q5275) Solve 22x^2+13x+1=0
Ans: x=-0.09 or x=-0.5
Q5276) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q5277) Solve 26x^2+49x+18=0
Ans: x=-0.5 or x=-1.38
Q5278) Solve 2x^2+25x+50=0
Ans: x=-2.5 or x=-10
Q5279) Solve 19x^2+42x+23=0
Ans: x=-1 or x=-1.21
Q5280) Solve 8x^2+37x+42=0
Ans: x=-2 or x=-2.62
Q5281) Solve 4x^2+20x+9=0
Ans: x=-0.5 or x=-4.5
Q5282) Solve 18x^2+42x+24=0
Ans: x=-1 or x=-1.33
Q5283) Solve 2x^2+34x+32=0
Ans: x=-1 or x=-16
Q5284) Solve 9x^2+28x+3=0
Ans: x=-0.11 or x=-3
Q5285) Solve 8x^2+26x+20=0
Ans: x=-1.25 or x=-2
Q5286) Solve 12x^2+21x+9=0
Ans: x=-0.75 or x=-1
Q5287) Solve x^2+15x+14=0
Ans: x=-1 or x=-14
Q5288) Solve 6x^2+35x+49=0
Ans: x=-2.33 or x=-3.5
Q5289) Solve 21x^2+19x+4=0
Ans: x=-0.33 or x=-0.57
Q5290) Solve 9x^2+29x+22=0
Ans: x=-1.22 or x=-2
Q5291) Solve 25x^2+31x+6=0
Ans: x=-0.24 or x=-1
Q5292) Solve x^2+26x+25=0
Ans: x=-1 or x=-25
Q5293) Solve 3x^2+26x+40=0
Ans: x=-2 or x=-6.67
Q5294) Solve 8x^2+38x+24=0
Ans: x=-0.75 or x=-4
Q5295) Solve 36x^2+47x+11=0
Ans: x=-0.31 or x=-1
Q5296) Solve 31x^2+38x+7=0
Ans: x=-0.23 or x=-1
Q5297) Solve 6x^2+39x+18=0
Ans: x=-0.5 or x=-6
Q5298) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q5299) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q5300) Solve 16x^2+34x+4=0
Ans: x=-0.12 or x=-2
Q5301) Solve 16x^2+43x+22=0
Ans: x=-0.69 or x=-2
Q5302) Solve 7x^2+34x+24=0
Ans: x=-0.86 or x=-4
Q5303) Solve 39x^2+40x+1=0
Ans: x=-0.03 or x=-1
Q5304) Solve 11x^2+13x+2=0
Ans: x=-0.18 or x=-1
Q5305) Solve x^2+9x+20=0
Ans: x=-4 or x=-5
Q5306) Solve 39x^2+40x+9=0
Ans: x=-0.33 or x=-0.69
Q5307) Solve 18x^2+46x+28=0
Ans: x=-1 or x=-1.56
Q5308) Solve x^2+31x+30=0
Ans: x=-1 or x=-30
Q5309) Solve 15x^2+39x+18=0
Ans: x=-0.6 or x=-2
Q5310) Solve 26x^2+35x+11=0
Ans: x=-0.5 or x=-0.85
Q5311) Solve 9x^2+31x+26=0
Ans: x=-1.44 or x=-2
Q5312) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q5313) Solve 24x^2+32x+8=0
Ans: x=-0.33 or x=-1
Q5314) Solve 3x^2+18x+24=0
Ans: x=-2 or x=-4
Q5315) Solve 5x^2+32x+12=0
Ans: x=-0.4 or x=-6
Q5316) Solve 26x^2+46x+20=0
Ans: x=-0.77 or x=-1
Q5317) Solve 18x^2+37x+2=0
Ans: x=-0.06 or x=-2
Q5318) Solve 32x^2+36x+7=0
Ans: x=-0.25 or x=-0.87
Q5319) Solve 2x^2+10x+12=0
Ans: x=-2 or x=-3
Q5320) Solve 13x^2+31x+10=0
Ans: x=-0.38 or x=-2
Q5321) Solve 15x^2+8x+1=0
Ans: x=-0.2 or x=-0.33
Q5322) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q5323) Solve 6x^2+43x+26=0
Ans: x=-0.67 or x=-6.5
Q5324) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q5325) Solve 6x^2+35x+25=0
Ans: x=-0.83 or x=-5
Q5326) Solve 5x^2+16x+11=0
Ans: x=-1 or x=-2.2
Q5327) Solve 7x^2+22x+16=0
Ans: x=-1.14 or x=-2
Q5328) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q5329) Solve 30x^2+46x+12=0
Ans: x=-0.33 or x=-1.2
Q5330) Solve 4x^2+32x+48=0
Ans: x=-2 or x=-6
Q5331) Solve 18x^2+31x+11=0
Ans: x=-0.5 or x=-1.22
Q5332) Solve 12x^2+39x+9=0
Ans: x=-0.25 or x=-3
Q5333) Solve 10x^2+12x+2=0
Ans: x=-0.2 or x=-1
Q5334) Solve 10x^2+43x+28=0
Ans: x=-0.8 or x=-3.5
Q5335) Solve 19x^2+46x+16=0
Ans: x=-0.42 or x=-2
Q5336) Solve 50x^2+39x+7=0
Ans: x=-0.28 or x=-0.5
Q5337) Solve 32x^2+22x+3=0
Ans: x=-0.19 or x=-0.5
Q5338) Solve 30x^2+28x+6=0
Ans: x=-0.33 or x=-0.6
Q5339) Solve 40x^2+36x+8=0
Ans: x=-0.4 or x=-0.5
Q5340) Solve 26x^2+43x+15=0
Ans: x=-0.5 or x=-1.15
Q5341) Solve 5x^2+46x+41=0
Ans: x=-1 or x=-8.2
Q5342) Solve 28x^2+46x+18=0
Ans: x=-0.64 or x=-1
Q5343) Solve 13x^2+36x+23=0
Ans: x=-1 or x=-1.77
Q5344) Solve 9x^2+18x+9=0
Ans: x=-1 or x=-1
Q5345) Solve 11x^2+39x+18=0
Ans: x=-0.55 or x=-3
Q5346) Solve 44x^2+39x+7=0
Ans: x=-0.25 or x=-0.64
Q5347) Solve 16x^2+24x+9=0
Ans: x=-0.75 or x=-0.75
Q5348) Solve 8x^2+41x+36=0
Ans: x=-1.12 or x=-4
Q5349) Solve 4x^2+30x+26=0
Ans: x=-1 or x=-6.5
Q5350) Solve 24x^2+44x+20=0
Ans: x=-0.83 or x=-1
Q5351) Solve 10x^2+28x+16=0
Ans: x=-0.8 or x=-2
Q5352) Solve 20x^2+46x+18=0
Ans: x=-0.5 or x=-1.8
Q5353) Solve 21x^2+19x+4=0
Ans: x=-0.33 or x=-0.57
Q5354) Solve 14x^2+46x+12=0
Ans: x=-0.29 or x=-3
Q5355) Solve 45x^2+49x+8=0
Ans: x=-0.2 or x=-0.89
Q5356) Solve 12x^2+30x+12=0
Ans: x=-0.5 or x=-2
Q5357) Solve 15x^2+38x+7=0
Ans: x=-0.2 or x=-2.33
Q5358) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q5359) Solve 24x^2+34x+10=0
Ans: x=-0.42 or x=-1
Q5360) Solve 48x^2+26x+3=0
Ans: x=-0.17 or x=-0.37
Q5361) Solve 4x^2+30x+26=0
Ans: x=-1 or x=-6.5
Q5362) Solve 9x^2+42x+24=0
Ans: x=-0.67 or x=-4
Q5363) Solve 36x^2+50x+16=0
Ans: x=-0.5 or x=-0.89
Q5364) Solve 28x^2+39x+8=0
Ans: x=-0.25 or x=-1.14
Q5365) Solve 6x^2+43x+42=0
Ans: x=-1.17 or x=-6
Q5366) Solve 6x^2+35x+16=0
Ans: x=-0.5 or x=-5.33
Q5367) Solve 48x^2+40x+7=0
Ans: x=-0.25 or x=-0.58
Q5368) Solve 8x^2+27x+19=0
Ans: x=-1 or x=-2.37
Q5369) Solve 6x^2+20x+14=0
Ans: x=-1 or x=-2.33
Q5370) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q5371) Solve 3x^2+7x+2=0
Ans: x=-0.33 or x=-2
Q5372) Solve 12x^2+24x+12=0
Ans: x=-1 or x=-1
Q5373) Solve 44x^2+27x+4=0
Ans: x=-0.25 or x=-0.36
Q5374) Solve 30x^2+19x+3=0
Ans: x=-0.3 or x=-0.33
Q5375) Solve 30x^2+36x+6=0
Ans: x=-0.2 or x=-1
Q5376) Solve 13x^2+40x+27=0
Ans: x=-1 or x=-2.08
Q5377) Solve 8x^2+11x+3=0
Ans: x=-0.37 or x=-1
Q5378) Solve 21x^2+44x+4=0
Ans: x=-0.1 or x=-2
Q5379) Solve 39x^2+44x+12=0
Ans: x=-0.46 or x=-0.67
Q5380) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q5381) Solve 3x^2+8x+4=0
Ans: x=-0.67 or x=-2
Q5382) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q5383) Solve 7x^2+20x+13=0
Ans: x=-1 or x=-1.86
Q5384) Solve 22x^2+47x+21=0
Ans: x=-0.64 or x=-1.5
Q5385) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q5386) Solve 8x^2+44x+20=0
Ans: x=-0.5 or x=-5
Q5387) Solve 48x^2+38x+7=0
Ans: x=-0.29 or x=-0.5
Q5388) Solve 19x^2+39x+20=0
Ans: x=-1 or x=-1.05
Q5389) Solve 9x^2+35x+34=0
Ans: x=-1.89 or x=-2
Q5390) Solve 30x^2+38x+12=0
Ans: x=-0.6 or x=-0.67
Q5391) Solve 6x^2+25x+21=0
Ans: x=-1.17 or x=-3
Q5392) Solve 15x^2+31x+10=0
Ans: x=-0.4 or x=-1.67
Q5393) Solve 3x^2+41x+38=0
Ans: x=-1 or x=-12.67
Q5394) Solve 39x^2+49x+12=0
Ans: x=-0.33 or x=-0.92
Q5395) Solve 6x^2+27x+21=0
Ans: x=-1 or x=-3.5
Q5396) Solve 12x^2+21x+9=0
Ans: x=-0.75 or x=-1
Q5397) Solve 16x^2+36x+8=0
Ans: x=-0.25 or x=-2
Q5398) Solve 15x^2+22x+8=0
Ans: x=-0.67 or x=-0.8
Q5399) Solve 18x^2+43x+25=0
Ans: x=-1 or x=-1.39
Q5400) Solve 15x^2+22x+7=0
Ans: x=-0.47 or x=-1
Q5401) Solve 33x^2+45x+12=0
Ans: x=-0.36 or x=-1
Q5402) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q5403) Solve 10x^2+33x+23=0
Ans: x=-1 or x=-2.3
Q5404) Solve x^2+8x+15=0
Ans: x=-3 or x=-5
Q5405) Solve 24x^2+46x+22=0
Ans: x=-0.92 or x=-1
Q5406) Solve x^2+11x+28=0
Ans: x=-4 or x=-7
Q5407) Solve 16x^2+40x+21=0
Ans: x=-0.75 or x=-1.75
Q5408) Solve 6x^2+21x+9=0
Ans: x=-0.5 or x=-3
Q5409) Solve 40x^2+30x+5=0
Ans: x=-0.25 or x=-0.5
Q5410) Solve 15x^2+50x+40=0
Ans: x=-1.33 or x=-2
Q5411) Solve 14x^2+43x+33=0
Ans: x=-1.5 or x=-1.57
Q5412) Solve 3x^2+18x+15=0
Ans: x=-1 or x=-5
Q5413) Solve 31x^2+38x+7=0
Ans: x=-0.23 or x=-1
Q5414) Solve 12x^2+15x+3=0
Ans: x=-0.25 or x=-1
Q5415) Solve 24x^2+30x+6=0
Ans: x=-0.25 or x=-1
Q5416) Solve 34x^2+39x+5=0
Ans: x=-0.15 or x=-1
Q5417) Solve 8x^2+23x+14=0
Ans: x=-0.87 or x=-2
Q5418) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q5419) Solve 11x^2+14x+3=0
Ans: x=-0.27 or x=-1
Q5420) Solve 19x^2+27x+8=0
Ans: x=-0.42 or x=-1
Q5421) Solve 11x^2+48x+16=0
Ans: x=-0.36 or x=-4
Q5422) Solve 24x^2+43x+19=0
Ans: x=-0.79 or x=-1
Q5423) Solve 8x^2+50x+50=0
Ans: x=-1.25 or x=-5
Q5424) Solve 24x^2+44x+12=0
Ans: x=-0.33 or x=-1.5
Q5425) Solve 13x^2+44x+15=0
Ans: x=-0.38 or x=-3
Q5426) Solve 40x^2+33x+5=0
Ans: x=-0.2 or x=-0.62
Q5427) Solve 17x^2+20x+3=0
Ans: x=-0.18 or x=-1
Q5428) Solve 14x^2+49x+35=0
Ans: x=-1 or x=-2.5
Q5429) Solve 33x^2+38x+9=0
Ans: x=-0.33 or x=-0.82
Q5430) Solve 8x^2+34x+30=0
Ans: x=-1.25 or x=-3
Q5431) Solve 12x^2+38x+6=0
Ans: x=-0.17 or x=-3
Q5432) Solve 47x^2+48x+1=0
Ans: x=-0.02 or x=-1
Q5433) Solve 5x^2+33x+18=0
Ans: x=-0.6 or x=-6
Q5434) Solve 4x^2+46x+22=0
Ans: x=-0.5 or x=-11
Q5435) Solve 45x^2+44x+7=0
Ans: x=-0.2 or x=-0.78
Q5436) Solve 7x^2+50x+7=0
Ans: x=-0.14 or x=-7
Q5437) Solve 36x^2+45x+11=0
Ans: x=-0.33 or x=-0.92
Q5438) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q5439) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q5440) Solve 17x^2+33x+16=0
Ans: x=-0.94 or x=-1
Q5441) Solve 14x^2+45x+19=0
Ans: x=-0.5 or x=-2.71
Q5442) Solve 24x^2+46x+22=0
Ans: x=-0.92 or x=-1
Q5443) Solve 20x^2+48x+19=0
Ans: x=-0.5 or x=-1.9
Q5444) Solve 40x^2+28x+4=0
Ans: x=-0.2 or x=-0.5
Q5445) Solve 11x^2+38x+15=0
Ans: x=-0.45 or x=-3
Q5446) Solve 9x^2+42x+48=0
Ans: x=-2 or x=-2.67
Q5447) Solve 35x^2+12x+1=0
Ans: x=-0.14 or x=-0.2
Q5448) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q5449) Solve 8x^2+32x+32=0
Ans: x=-2 or x=-2
Q5450) Solve 24x^2+32x+10=0
Ans: x=-0.5 or x=-0.83
Q5451) Solve 9x^2+48x+28=0
Ans: x=-0.67 or x=-4.67
Q5452) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q5453) Solve 24x^2+40x+14=0
Ans: x=-0.5 or x=-1.17
Q5454) Solve 10x^2+43x+46=0
Ans: x=-2 or x=-2.3
Q5455) Solve 24x^2+42x+15=0
Ans: x=-0.5 or x=-1.25
Q5456) Solve 2x^2+20x+42=0
Ans: x=-3 or x=-7
Q5457) Solve 38x^2+50x+12=0
Ans: x=-0.32 or x=-1
Q5458) Solve 10x^2+16x+6=0
Ans: x=-0.6 or x=-1
Q5459) Solve 32x^2+50x+3=0
Ans: x=-0.06 or x=-1.5
Q5460) Solve 16x^2+42x+17=0
Ans: x=-0.5 or x=-2.12
Q5461) Solve 4x^2+45x+50=0
Ans: x=-1.25 or x=-10
Q5462) Solve 7x^2+28x+21=0
Ans: x=-1 or x=-3
Q5463) Solve 6x^2+23x+15=0
Ans: x=-0.83 or x=-3
Q5464) Solve 7x^2+24x+20=0
Ans: x=-1.43 or x=-2
Q5465) Solve 32x^2+20x+2=0
Ans: x=-0.12 or x=-0.5
Q5466) Solve 9x^2+34x+25=0
Ans: x=-1 or x=-2.78
Q5467) Solve 12x^2+37x+21=0
Ans: x=-0.75 or x=-2.33
Q5468) Solve 10x^2+33x+20=0
Ans: x=-0.8 or x=-2.5
Q5469) Solve 2x^2+29x+50=0
Ans: x=-2 or x=-12.5
Q5470) Solve 35x^2+32x+5=0
Ans: x=-0.2 or x=-0.71
Q5471) Solve 15x^2+46x+24=0
Ans: x=-0.67 or x=-2.4
Q5472) Solve 14x^2+29x+2=0
Ans: x=-0.07 or x=-2
Q5473) Solve 21x^2+31x+10=0
Ans: x=-0.48 or x=-1
Q5474) Solve 29x^2+48x+19=0
Ans: x=-0.66 or x=-1
Q5475) Solve 49x^2+49x+6=0
Ans: x=-0.14 or x=-0.86
Q5476) Solve 2x^2+18x+36=0
Ans: x=-3 or x=-6
Q5477) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q5478) Solve 8x^2+22x+5=0
Ans: x=-0.25 or x=-2.5
Q5479) Solve 13x^2+45x+32=0
Ans: x=-1 or x=-2.46
Q5480) Solve 40x^2+46x+12=0
Ans: x=-0.4 or x=-0.75
Q5481) Solve 7x^2+14x+7=0
Ans: x=-1 or x=-1
Q5482) Solve 46x^2+37x+7=0
Ans: x=-0.3 or x=-0.5
Q5483) Solve 11x^2+39x+18=0
Ans: x=-0.55 or x=-3
Q5484) Solve 32x^2+44x+14=0
Ans: x=-0.5 or x=-0.87
Q5485) Solve 9x^2+14x+5=0
Ans: x=-0.56 or x=-1
Q5486) Solve 15x^2+41x+14=0
Ans: x=-0.4 or x=-2.33
Q5487) Solve 15x^2+32x+9=0
Ans: x=-0.33 or x=-1.8
Q5488) Solve 18x^2+39x+21=0
Ans: x=-1 or x=-1.17
Q5489) Solve 18x^2+45x+13=0
Ans: x=-0.33 or x=-2.17
Q5490) Solve 39x^2+44x+12=0
Ans: x=-0.46 or x=-0.67
Q5491) Solve 48x^2+48x+9=0
Ans: x=-0.25 or x=-0.75
Q5492) Solve 32x^2+39x+7=0
Ans: x=-0.22 or x=-1
Q5493) Solve 9x^2+48x+28=0
Ans: x=-0.67 or x=-4.67
Q5494) Solve 44x^2+26x+2=0
Ans: x=-0.09 or x=-0.5
Q5495) Solve 33x^2+50x+13=0
Ans: x=-0.33 or x=-1.18
Q5496) Solve 6x^2+29x+13=0
Ans: x=-0.5 or x=-4.33
Q5497) Solve 6x^2+25x+21=0
Ans: x=-1.17 or x=-3
Q5498) Solve 18x^2+21x+5=0
Ans: x=-0.33 or x=-0.83
Q5499) Solve 17x^2+39x+22=0
Ans: x=-1 or x=-1.29
Q5500) Solve 5x^2+11x+2=0
Ans: x=-0.2 or x=-2
Q5501) Solve 14x^2+31x+17=0
Ans: x=-1 or x=-1.21
Q5502) Solve 26x^2+45x+9=0
Ans: x=-0.23 or x=-1.5
Q5503) Solve 12x^2+14x+2=0
Ans: x=-0.17 or x=-1
Q5504) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q5505) Solve 10x^2+46x+24=0
Ans: x=-0.6 or x=-4
Q5506) Solve 8x^2+46x+11=0
Ans: x=-0.25 or x=-5.5
Q5507) Solve 3x^2+12x+9=0
Ans: x=-1 or x=-3
Q5508) Solve 25x^2+41x+16=0
Ans: x=-0.64 or x=-1
Q5509) Solve 4x^2+27x+23=0
Ans: x=-1 or x=-5.75
Q5510) Solve 8x^2+27x+9=0
Ans: x=-0.37 or x=-3
Q5511) Solve 24x^2+50x+25=0
Ans: x=-0.83 or x=-1.25
Q5512) Solve 28x^2+38x+10=0
Ans: x=-0.36 or x=-1
Q5513) Solve 9x^2+42x+33=0
Ans: x=-1 or x=-3.67
Q5514) Solve 4x^2+32x+48=0
Ans: x=-2 or x=-6
Q5515) Solve 12x^2+16x+4=0
Ans: x=-0.33 or x=-1
Q5516) Solve 48x^2+49x+11=0
Ans: x=-0.33 or x=-0.69
Q5517) Solve 22x^2+45x+17=0
Ans: x=-0.5 or x=-1.55
Q5518) Solve 17x^2+35x+2=0
Ans: x=-0.06 or x=-2
Q5519) Solve 18x^2+37x+15=0
Ans: x=-0.56 or x=-1.5
Q5520) Solve 36x^2+29x+5=0
Ans: x=-0.25 or x=-0.56
Q5521) Solve 10x^2+33x+9=0
Ans: x=-0.3 or x=-3
Q5522) Solve 40x^2+42x+2=0
Ans: x=-0.05 or x=-1
Q5523) Solve 22x^2+50x+12=0
Ans: x=-0.27 or x=-2
Q5524) Solve 4x^2+24x+32=0
Ans: x=-2 or x=-4
Q5525) Solve 24x^2+50x+11=0
Ans: x=-0.25 or x=-1.83
Q5526) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q5527) Solve 50x^2+45x+9=0
Ans: x=-0.3 or x=-0.6
Q5528) Solve 26x^2+41x+14=0
Ans: x=-0.5 or x=-1.08
Q5529) Solve 42x^2+46x+4=0
Ans: x=-0.1 or x=-1
Q5530) Solve 2x^2+25x+12=0
Ans: x=-0.5 or x=-12
Q5531) Solve 32x^2+38x+11=0
Ans: x=-0.5 or x=-0.69
Q5532) Solve 3x^2+10x+3=0
Ans: x=-0.33 or x=-3
Q5533) Solve 7x^2+21x+14=0
Ans: x=-1 or x=-2
Q5534) Solve 3x^2+39x+36=0
Ans: x=-1 or x=-12
Q5535) Solve 4x^2+39x+27=0
Ans: x=-0.75 or x=-9
Q5536) Solve 3x^2+23x+42=0
Ans: x=-3 or x=-4.67
Q5537) Solve x^2+7x+6=0
Ans: x=-1 or x=-6
Q5538) Solve 43x^2+46x+3=0
Ans: x=-0.07 or x=-1
Q5539) Solve 5x^2+48x+43=0
Ans: x=-1 or x=-8.6
Q5540) Solve 3x^2+14x+8=0
Ans: x=-0.67 or x=-4
Q5541) Solve 35x^2+42x+7=0
Ans: x=-0.2 or x=-1
Q5542) Solve 20x^2+33x+7=0
Ans: x=-0.25 or x=-1.4
Q5543) Solve 37x^2+41x+4=0
Ans: x=-0.11 or x=-1
Q5544) Solve 3x^2+20x+33=0
Ans: x=-3 or x=-3.67
Q5545) Solve 10x^2+38x+36=0
Ans: x=-1.8 or x=-2
Q5546) Solve 12x^2+34x+24=0
Ans: x=-1.33 or x=-1.5
Q5547) Solve 17x^2+44x+20=0
Ans: x=-0.59 or x=-2
Q5548) Solve 8x^2+39x+28=0
Ans: x=-0.87 or x=-4
Q5549) Solve 8x^2+50x+50=0
Ans: x=-1.25 or x=-5
Q5550) Solve 39x^2+47x+14=0
Ans: x=-0.54 or x=-0.67
Q5551) Solve 14x^2+19x+5=0
Ans: x=-0.36 or x=-1
Q5552) Solve 4x^2+31x+42=0
Ans: x=-1.75 or x=-6
Q5553) Solve 2x^2+23x+30=0
Ans: x=-1.5 or x=-10
Q5554) Solve 7x^2+43x+36=0
Ans: x=-1 or x=-5.14
Q5555) Solve 32x^2+47x+15=0
Ans: x=-0.47 or x=-1
Q5556) Solve 50x^2+41x+8=0
Ans: x=-0.32 or x=-0.5
Q5557) Solve 29x^2+41x+12=0
Ans: x=-0.41 or x=-1
Q5558) Solve 14x^2+33x+13=0
Ans: x=-0.5 or x=-1.86
Q5559) Solve 12x^2+19x+7=0
Ans: x=-0.58 or x=-1
Q5560) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q5561) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q5562) Solve 4x^2+34x+30=0
Ans: x=-1 or x=-7.5
Q5563) Solve 34x^2+39x+11=0
Ans: x=-0.5 or x=-0.65
Q5564) Solve 2x^2+41x+39=0
Ans: x=-1 or x=-19.5
Q5565) Solve 28x^2+32x+9=0
Ans: x=-0.5 or x=-0.64
Q5566) Solve 4x^2+13x+10=0
Ans: x=-1.25 or x=-2
Q5567) Solve 20x^2+39x+7=0
Ans: x=-0.2 or x=-1.75
Q5568) Solve 12x^2+49x+44=0
Ans: x=-1.33 or x=-2.75
Q5569) Solve 12x^2+26x+12=0
Ans: x=-0.67 or x=-1.5
Q5570) Solve 10x^2+27x+17=0
Ans: x=-1 or x=-1.7
Q5571) Solve 45x^2+46x+8=0
Ans: x=-0.22 or x=-0.8
Q5572) Solve 16x^2+50x+39=0
Ans: x=-1.5 or x=-1.62
Q5573) Solve 2x^2+41x+39=0
Ans: x=-1 or x=-19.5
Q5574) Solve 2x^2+47x+23=0
Ans: x=-0.5 or x=-23
Q5575) Solve 12x^2+25x+12=0
Ans: x=-0.75 or x=-1.33
Q5576) Solve 30x^2+17x+1=0
Ans: x=-0.07 or x=-0.5
Q5577) Solve 20x^2+43x+14=0
Ans: x=-0.4 or x=-1.75
Q5578) Solve 10x^2+48x+38=0
Ans: x=-1 or x=-3.8
Q5579) Solve 42x^2+35x+7=0
Ans: x=-0.33 or x=-0.5
Q5580) Solve 8x^2+14x+3=0
Ans: x=-0.25 or x=-1.5
Q5581) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q5582) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q5583) Solve x^2+10x+9=0
Ans: x=-1 or x=-9
Q5584) Solve 6x^2+22x+16=0
Ans: x=-1 or x=-2.67
Q5585) Solve x^2+14x+49=0
Ans: x=-7 or x=-7
Q5586) Solve 35x^2+46x+11=0
Ans: x=-0.31 or x=-1
Q5587) Solve 9x^2+30x+24=0
Ans: x=-1.33 or x=-2
Q5588) Solve 22x^2+31x+9=0
Ans: x=-0.41 or x=-1
Q5589) Solve 8x^2+8x+2=0
Ans: x=-0.5 or x=-0.5
Q5590) Solve 15x^2+33x+6=0
Ans: x=-0.2 or x=-2
Q5591) Solve 3x^2+24x+45=0
Ans: x=-3 or x=-5
Q5592) Solve 20x^2+49x+9=0
Ans: x=-0.2 or x=-2.25
Q5593) Solve x^2+40x+39=0
Ans: x=-1 or x=-39
Q5594) Solve 8x^2+10x+2=0
Ans: x=-0.25 or x=-1
Q5595) Solve 4x^2+29x+30=0
Ans: x=-1.25 or x=-6
Q5596) Solve 12x^2+45x+27=0
Ans: x=-0.75 or x=-3
Q5597) Solve 16x^2+36x+20=0
Ans: x=-1 or x=-1.25
Q5598) Solve 15x^2+33x+6=0
Ans: x=-0.2 or x=-2
Q5599) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q5600) Solve 50x^2+27x+1=0
Ans: x=-0.04 or x=-0.5
Q5601) Solve 6x^2+17x+11=0
Ans: x=-1 or x=-1.83
Q5602) Solve 50x^2+47x+11=0
Ans: x=-0.44 or x=-0.5
Q5603) Solve 36x^2+48x+15=0
Ans: x=-0.5 or x=-0.83
Q5604) Solve 12x^2+32x+20=0
Ans: x=-1 or x=-1.67
Q5605) Solve 12x^2+41x+24=0
Ans: x=-0.75 or x=-2.67
Q5606) Solve 5x^2+28x+15=0
Ans: x=-0.6 or x=-5
Q5607) Solve 10x^2+17x+6=0
Ans: x=-0.5 or x=-1.2
Q5608) Solve 16x^2+18x+5=0
Ans: x=-0.5 or x=-0.62
Q5609) Solve 9x^2+43x+28=0
Ans: x=-0.78 or x=-4
Q5610) Solve 50x^2+50x+12=0
Ans: x=-0.4 or x=-0.6
Q5611) Solve 26x^2+35x+11=0
Ans: x=-0.5 or x=-0.85
Q5612) Solve 7x^2+24x+17=0
Ans: x=-1 or x=-2.43
Q5613) Solve 18x^2+39x+21=0
Ans: x=-1 or x=-1.17
Q5614) Solve 20x^2+38x+12=0
Ans: x=-0.4 or x=-1.5
Q5615) Solve 44x^2+50x+6=0
Ans: x=-0.14 or x=-1
Q5616) Solve 6x^2+27x+30=0
Ans: x=-2 or x=-2.5
Q5617) Solve 48x^2+26x+3=0
Ans: x=-0.17 or x=-0.37
Q5618) Solve 16x^2+21x+5=0
Ans: x=-0.31 or x=-1
Q5619) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q5620) Solve 12x^2+28x+16=0
Ans: x=-1 or x=-1.33
Q5621) Solve 8x^2+47x+35=0
Ans: x=-0.87 or x=-5
Q5622) Solve 33x^2+31x+6=0
Ans: x=-0.27 or x=-0.67
Q5623) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q5624) Solve 38x^2+49x+15=0
Ans: x=-0.5 or x=-0.79
Q5625) Solve 50x^2+33x+4=0
Ans: x=-0.16 or x=-0.5
Q5626) Solve 16x^2+34x+15=0
Ans: x=-0.62 or x=-1.5
Q5627) Solve 4x^2+37x+40=0
Ans: x=-1.25 or x=-8
Q5628) Solve 3x^2+28x+49=0
Ans: x=-2.33 or x=-7
Q5629) Solve 23x^2+34x+11=0
Ans: x=-0.48 or x=-1
Q5630) Solve 12x^2+23x+11=0
Ans: x=-0.92 or x=-1
Q5631) Solve 45x^2+39x+6=0
Ans: x=-0.2 or x=-0.67
Q5632) Solve 15x^2+31x+16=0
Ans: x=-1 or x=-1.07
Q5633) Solve 12x^2+50x+42=0
Ans: x=-1.17 or x=-3
Q5634) Solve 8x^2+32x+30=0
Ans: x=-1.5 or x=-2.5
Q5635) Solve 10x^2+41x+40=0
Ans: x=-1.6 or x=-2.5
Q5636) Solve 9x^2+32x+28=0
Ans: x=-1.56 or x=-2
Q5637) Solve 15x^2+44x+13=0
Ans: x=-0.33 or x=-2.6
Q5638) Solve 26x^2+44x+18=0
Ans: x=-0.69 or x=-1
Q5639) Solve 6x^2+11x+3=0
Ans: x=-0.33 or x=-1.5
Q5640) Solve 10x^2+31x+24=0
Ans: x=-1.5 or x=-1.6
Q5641) Solve 16x^2+38x+22=0
Ans: x=-1 or x=-1.37
Q5642) Solve 11x^2+31x+18=0
Ans: x=-0.82 or x=-2
Q5643) Solve 20x^2+19x+3=0
Ans: x=-0.2 or x=-0.75
Q5644) Solve 5x^2+32x+35=0
Ans: x=-1.4 or x=-5
Q5645) Solve 42x^2+47x+11=0
Ans: x=-0.33 or x=-0.79
Q5646) Solve 13x^2+27x+2=0
Ans: x=-0.08 or x=-2
Q5647) Solve 4x^2+45x+41=0
Ans: x=-1 or x=-10.25
Q5648) Solve 22x^2+35x+13=0
Ans: x=-0.59 or x=-1
Q5649) Solve 25x^2+45x+8=0
Ans: x=-0.2 or x=-1.6
Q5650) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q5651) Solve 17x^2+22x+5=0
Ans: x=-0.29 or x=-1
Q5652) Solve 10x^2+33x+14=0
Ans: x=-0.5 or x=-2.8
Q5653) Solve 28x^2+40x+13=0
Ans: x=-0.5 or x=-0.93
Q5654) Solve 20x^2+13x+2=0
Ans: x=-0.25 or x=-0.4
Q5655) Solve 42x^2+46x+4=0
Ans: x=-0.1 or x=-1
Q5656) Solve 7x^2+29x+22=0
Ans: x=-1 or x=-3.14
Q5657) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q5658) Solve 23x^2+41x+18=0
Ans: x=-0.78 or x=-1
Q5659) Solve 20x^2+26x+6=0
Ans: x=-0.3 or x=-1
Q5660) Solve 4x^2+30x+26=0
Ans: x=-1 or x=-6.5
Q5661) Solve x^2+28x+27=0
Ans: x=-1 or x=-27
Q5662) Solve 42x^2+43x+6=0
Ans: x=-0.17 or x=-0.86
Q5663) Solve 11x^2+37x+12=0
Ans: x=-0.36 or x=-3
Q5664) Solve 22x^2+46x+4=0
Ans: x=-0.09 or x=-2
Q5665) Solve 10x^2+47x+9=0
Ans: x=-0.2 or x=-4.5
Q5666) Solve 22x^2+37x+13=0
Ans: x=-0.5 or x=-1.18
Q5667) Solve 36x^2+24x+3=0
Ans: x=-0.17 or x=-0.5
Q5668) Solve 24x^2+18x+3=0
Ans: x=-0.25 or x=-0.5
Q5669) Solve 16x^2+41x+25=0
Ans: x=-1 or x=-1.56
Q5670) Solve 5x^2+16x+3=0
Ans: x=-0.2 or x=-3
Q5671) Solve 25x^2+40x+16=0
Ans: x=-0.8 or x=-0.8
Q5672) Solve 28x^2+50x+18=0
Ans: x=-0.5 or x=-1.29
Q5673) Solve 9x^2+42x+48=0
Ans: x=-2 or x=-2.67
Q5674) Solve 29x^2+33x+4=0
Ans: x=-0.14 or x=-1
Q5675) Solve 14x^2+23x+3=0
Ans: x=-0.14 or x=-1.5
Q5676) Solve 13x^2+47x+34=0
Ans: x=-1 or x=-2.62
Q5677) Solve 15x^2+8x+1=0
Ans: x=-0.2 or x=-0.33
Q5678) Solve x^2+31x+30=0
Ans: x=-1 or x=-30
Q5679) Solve 30x^2+32x+2=0
Ans: x=-0.07 or x=-1
Q5680) Solve 4x^2+41x+45=0
Ans: x=-1.25 or x=-9
Q5681) Solve 7x^2+37x+36=0
Ans: x=-1.29 or x=-4
Q5682) Solve 22x^2+37x+6=0
Ans: x=-0.18 or x=-1.5
Q5683) Solve 12x^2+18x+6=0
Ans: x=-0.5 or x=-1
Q5684) Solve 2x^2+34x+32=0
Ans: x=-1 or x=-16
Q5685) Solve 11x^2+47x+12=0
Ans: x=-0.27 or x=-4
Q5686) Solve 24x^2+38x+8=0
Ans: x=-0.25 or x=-1.33
Q5687) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q5688) Solve 24x^2+17x+3=0
Ans: x=-0.33 or x=-0.37
Q5689) Solve 39x^2+40x+1=0
Ans: x=-0.03 or x=-1
Q5690) Solve 3x^2+42x+39=0
Ans: x=-1 or x=-13
Q5691) Solve 7x^2+39x+20=0
Ans: x=-0.57 or x=-5
Q5692) Solve 6x^2+31x+38=0
Ans: x=-2 or x=-3.17
Q5693) Solve 7x^2+43x+6=0
Ans: x=-0.14 or x=-6
Q5694) Solve 11x^2+42x+40=0
Ans: x=-1.82 or x=-2
Q5695) Solve 17x^2+45x+22=0
Ans: x=-0.65 or x=-2
Q5696) Solve 2x^2+47x+45=0
Ans: x=-1 or x=-22.5
Q5697) Solve 22x^2+50x+28=0
Ans: x=-1 or x=-1.27
Q5698) Solve 21x^2+10x+1=0
Ans: x=-0.14 or x=-0.33
Q5699) Solve 3x^2+29x+26=0
Ans: x=-1 or x=-8.67
Q5700) Solve 2x^2+37x+18=0
Ans: x=-0.5 or x=-18
Q5701) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q5702) Solve 12x^2+38x+20=0
Ans: x=-0.67 or x=-2.5
Q5703) Solve 23x^2+48x+25=0
Ans: x=-1 or x=-1.09
Q5704) Solve 2x^2+47x+23=0
Ans: x=-0.5 or x=-23
Q5705) Solve 6x^2+31x+38=0
Ans: x=-2 or x=-3.17
Q5706) Solve 5x^2+11x+6=0
Ans: x=-1 or x=-1.2
Q5707) Solve 46x^2+43x+10=0
Ans: x=-0.43 or x=-0.5
Q5708) Solve 5x^2+24x+16=0
Ans: x=-0.8 or x=-4
Q5709) Solve 6x^2+38x+12=0
Ans: x=-0.33 or x=-6
Q5710) Solve 32x^2+49x+17=0
Ans: x=-0.53 or x=-1
Q5711) Solve 48x^2+34x+5=0
Ans: x=-0.21 or x=-0.5
Q5712) Solve 24x^2+50x+19=0
Ans: x=-0.5 or x=-1.58
Q5713) Solve 22x^2+27x+8=0
Ans: x=-0.5 or x=-0.73
Q5714) Solve 12x^2+13x+1=0
Ans: x=-0.08 or x=-1
Q5715) Solve 12x^2+43x+35=0
Ans: x=-1.25 or x=-2.33
Q5716) Solve 16x^2+8x+1=0
Ans: x=-0.25 or x=-0.25
Q5717) Solve 9x^2+11x+2=0
Ans: x=-0.22 or x=-1
Q5718) Solve 3x^2+22x+40=0
Ans: x=-3.33 or x=-4
Q5719) Solve 9x^2+48x+48=0
Ans: x=-1.33 or x=-4
Q5720) Solve 22x^2+30x+8=0
Ans: x=-0.36 or x=-1
Q5721) Solve 24x^2+29x+7=0
Ans: x=-0.33 or x=-0.87
Q5722) Solve 23x^2+25x+2=0
Ans: x=-0.09 or x=-1
Q5723) Solve 4x^2+22x+18=0
Ans: x=-1 or x=-4.5
Q5724) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q5725) Solve x^2+11x+10=0
Ans: x=-1 or x=-10
Q5726) Solve 18x^2+30x+8=0
Ans: x=-0.33 or x=-1.33
Q5727) Solve 6x^2+32x+26=0
Ans: x=-1 or x=-4.33
Q5728) Solve 22x^2+24x+2=0
Ans: x=-0.09 or x=-1
Q5729) Solve 6x^2+35x+36=0
Ans: x=-1.33 or x=-4.5
Q5730) Solve 4x^2+25x+36=0
Ans: x=-2.25 or x=-4
Q5731) Solve 4x^2+25x+21=0
Ans: x=-1 or x=-5.25
Q5732) Solve 19x^2+28x+9=0
Ans: x=-0.47 or x=-1
Q5733) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q5734) Solve 12x^2+41x+22=0
Ans: x=-0.67 or x=-2.75
Q5735) Solve 7x^2+11x+4=0
Ans: x=-0.57 or x=-1
Q5736) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q5737) Solve 4x^2+23x+19=0
Ans: x=-1 or x=-4.75
Q5738) Solve 10x^2+33x+14=0
Ans: x=-0.5 or x=-2.8
Q5739) Solve 9x^2+33x+10=0
Ans: x=-0.33 or x=-3.33
Q5740) Solve 3x^2+11x+8=0
Ans: x=-1 or x=-2.67
Q5741) Solve 26x^2+46x+20=0
Ans: x=-0.77 or x=-1
Q5742) Solve 8x^2+39x+28=0
Ans: x=-0.87 or x=-4
Q5743) Solve 34x^2+48x+14=0
Ans: x=-0.41 or x=-1
Q5744) Solve 22x^2+41x+15=0
Ans: x=-0.5 or x=-1.36
Q5745) Solve 16x^2+38x+15=0
Ans: x=-0.5 or x=-1.87
Q5746) Solve 20x^2+48x+27=0
Ans: x=-0.9 or x=-1.5
Q5747) Solve 28x^2+34x+10=0
Ans: x=-0.5 or x=-0.71
Q5748) Solve 40x^2+33x+5=0
Ans: x=-0.2 or x=-0.62
Q5749) Solve 36x^2+45x+11=0
Ans: x=-0.33 or x=-0.92
Q5750) Solve 4x^2+24x+27=0
Ans: x=-1.5 or x=-4.5
Q5751) Solve 4x^2+10x+4=0
Ans: x=-0.5 or x=-2
Q5752) Solve 18x^2+17x+4=0
Ans: x=-0.44 or x=-0.5
Q5753) Solve 40x^2+47x+12=0
Ans: x=-0.37 or x=-0.8
Q5754) Solve 16x^2+32x+16=0
Ans: x=-1 or x=-1
Q5755) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q5756) Solve 28x^2+32x+9=0
Ans: x=-0.5 or x=-0.64
Q5757) Solve 18x^2+42x+12=0
Ans: x=-0.33 or x=-2
Q5758) Solve 3x^2+26x+16=0
Ans: x=-0.67 or x=-8
Q5759) Solve 20x^2+13x+2=0
Ans: x=-0.25 or x=-0.4
Q5760) Solve 6x^2+19x+15=0
Ans: x=-1.5 or x=-1.67
Q5761) Solve x^2+11x+28=0
Ans: x=-4 or x=-7
Q5762) Solve 27x^2+40x+13=0
Ans: x=-0.48 or x=-1
Q5763) Solve 13x^2+47x+24=0
Ans: x=-0.62 or x=-3
Q5764) Solve 4x^2+45x+50=0
Ans: x=-1.25 or x=-10
Q5765) Solve 23x^2+31x+8=0
Ans: x=-0.35 or x=-1
Q5766) Solve 11x^2+47x+42=0
Ans: x=-1.27 or x=-3
Q5767) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q5768) Solve 32x^2+35x+3=0
Ans: x=-0.09 or x=-1
Q5769) Solve 20x^2+21x+1=0
Ans: x=-0.05 or x=-1
Q5770) Solve 30x^2+19x+3=0
Ans: x=-0.3 or x=-0.33
Q5771) Solve 3x^2+49x+46=0
Ans: x=-1 or x=-15.33
Q5772) Solve 8x^2+38x+44=0
Ans: x=-2 or x=-2.75
Q5773) Solve 8x^2+27x+19=0
Ans: x=-1 or x=-2.37
Q5774) Solve 16x^2+27x+11=0
Ans: x=-0.69 or x=-1
Q5775) Solve 40x^2+18x+2=0
Ans: x=-0.2 or x=-0.25
Q5776) Solve 5x^2+32x+35=0
Ans: x=-1.4 or x=-5
Q5777) Solve 25x^2+41x+16=0
Ans: x=-0.64 or x=-1
Q5778) Solve 36x^2+43x+12=0
Ans: x=-0.44 or x=-0.75
Q5779) Solve 33x^2+32x+7=0
Ans: x=-0.33 or x=-0.64
Q5780) Solve 12x^2+50x+42=0
Ans: x=-1.17 or x=-3
Q5781) Solve 35x^2+37x+6=0
Ans: x=-0.2 or x=-0.86
Q5782) Solve 9x^2+37x+30=0
Ans: x=-1.11 or x=-3
Q5783) Solve 10x^2+43x+28=0
Ans: x=-0.8 or x=-3.5
Q5784) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q5785) Solve 5x^2+11x+6=0
Ans: x=-1 or x=-1.2
Q5786) Solve 13x^2+36x+23=0
Ans: x=-1 or x=-1.77
Q5787) Solve 31x^2+32x+1=0
Ans: x=-0.03 or x=-1
Q5788) Solve 6x^2+35x+49=0
Ans: x=-2.33 or x=-3.5
Q5789) Solve 2x^2+39x+37=0
Ans: x=-1 or x=-18.5
Q5790) Solve 41x^2+49x+8=0
Ans: x=-0.2 or x=-1
Q5791) Solve 9x^2+14x+5=0
Ans: x=-0.56 or x=-1
Q5792) Solve 17x^2+46x+29=0
Ans: x=-1 or x=-1.71
Q5793) Solve 10x^2+42x+8=0
Ans: x=-0.2 or x=-4
Q5794) Solve 10x^2+42x+36=0
Ans: x=-1.2 or x=-3
Q5795) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q5796) Solve 9x^2+25x+16=0
Ans: x=-1 or x=-1.78
Q5797) Solve 24x^2+34x+7=0
Ans: x=-0.25 or x=-1.17
Q5798) Solve 16x^2+44x+30=0
Ans: x=-1.25 or x=-1.5
Q5799) Solve 8x^2+27x+22=0
Ans: x=-1.37 or x=-2
Q5800) Solve 23x^2+50x+8=0
Ans: x=-0.17 or x=-2
Q5801) Solve 50x^2+20x+2=0
Ans: x=-0.2 or x=-0.2
Q5802) Solve 3x^2+24x+45=0
Ans: x=-3 or x=-5
Q5803) Solve 12x^2+37x+28=0
Ans: x=-1.33 or x=-1.75
Q5804) Solve 37x^2+43x+6=0
Ans: x=-0.16 or x=-1
Q5805) Solve 8x^2+32x+32=0
Ans: x=-2 or x=-2
Q5806) Solve 5x^2+33x+46=0
Ans: x=-2 or x=-4.6
Q5807) Solve 6x^2+49x+43=0
Ans: x=-1 or x=-7.17
Q5808) Solve 48x^2+28x+2=0
Ans: x=-0.08 or x=-0.5
Q5809) Solve 16x^2+24x+8=0
Ans: x=-0.5 or x=-1
Q5810) Solve 29x^2+39x+10=0
Ans: x=-0.34 or x=-1
Q5811) Solve 22x^2+27x+8=0
Ans: x=-0.5 or x=-0.73
Q5812) Solve 7x^2+32x+33=0
Ans: x=-1.57 or x=-3
Q5813) Solve 24x^2+28x+4=0
Ans: x=-0.17 or x=-1
Q5814) Solve 3x^2+8x+4=0
Ans: x=-0.67 or x=-2
Q5815) Solve 6x^2+31x+5=0
Ans: x=-0.17 or x=-5
Q5816) Solve 12x^2+47x+26=0
Ans: x=-0.67 or x=-3.25
Q5817) Solve 23x^2+28x+5=0
Ans: x=-0.22 or x=-1
Q5818) Solve 2x^2+31x+29=0
Ans: x=-1 or x=-14.5
Q5819) Solve 2x^2+29x+39=0
Ans: x=-1.5 or x=-13
Q5820) Solve 3x^2+27x+42=0
Ans: x=-2 or x=-7
Q5821) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q5822) Solve 6x^2+41x+19=0
Ans: x=-0.5 or x=-6.33
Q5823) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q5824) Solve 6x^2+47x+15=0
Ans: x=-0.33 or x=-7.5
Q5825) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q5826) Solve 28x^2+40x+12=0
Ans: x=-0.43 or x=-1
Q5827) Solve 27x^2+40x+13=0
Ans: x=-0.48 or x=-1
Q5828) Solve 11x^2+35x+26=0
Ans: x=-1.18 or x=-2
Q5829) Solve 8x^2+50x+33=0
Ans: x=-0.75 or x=-5.5
Q5830) Solve 4x^2+40x+19=0
Ans: x=-0.5 or x=-9.5
Q5831) Solve 2x^2+29x+50=0
Ans: x=-2 or x=-12.5
Q5832) Solve 42x^2+37x+6=0
Ans: x=-0.21 or x=-0.67
Q5833) Solve 28x^2+15x+2=0
Ans: x=-0.25 or x=-0.29
Q5834) Solve 2x^2+44x+42=0
Ans: x=-1 or x=-21
Q5835) Solve 2x^2+18x+16=0
Ans: x=-1 or x=-8
Q5836) Solve 11x^2+33x+22=0
Ans: x=-1 or x=-2
Q5837) Solve 8x^2+15x+7=0
Ans: x=-0.87 or x=-1
Q5838) Solve 28x^2+36x+11=0
Ans: x=-0.5 or x=-0.79
Q5839) Solve 12x^2+35x+25=0
Ans: x=-1.25 or x=-1.67
Q5840) Solve 10x^2+50x+40=0
Ans: x=-1 or x=-4
Q5841) Solve 3x^2+35x+22=0
Ans: x=-0.67 or x=-11
Q5842) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q5843) Solve 3x^2+28x+25=0
Ans: x=-1 or x=-8.33
Q5844) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q5845) Solve 8x^2+49x+6=0
Ans: x=-0.12 or x=-6
Q5846) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q5847) Solve 4x^2+14x+6=0
Ans: x=-0.5 or x=-3
Q5848) Solve 5x^2+38x+48=0
Ans: x=-1.6 or x=-6
Q5849) Solve 36x^2+36x+5=0
Ans: x=-0.17 or x=-0.83
Q5850) Solve 8x^2+49x+45=0
Ans: x=-1.12 or x=-5
Q5851) Solve 5x^2+23x+26=0
Ans: x=-2 or x=-2.6
Q5852) Solve 30x^2+31x+7=0
Ans: x=-0.33 or x=-0.7
Q5853) Solve 15x^2+46x+32=0
Ans: x=-1.07 or x=-2
Q5854) Solve 26x^2+29x+3=0
Ans: x=-0.12 or x=-1
Q5855) Solve 45x^2+45x+10=0
Ans: x=-0.33 or x=-0.67
Q5856) Solve x^2+2x+1=0
Ans: x=-1 or x=-1
Q5857) Solve 8x^2+40x+32=0
Ans: x=-1 or x=-4
Q5858) Solve 2x^2+23x+11=0
Ans: x=-0.5 or x=-11
Q5859) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q5860) Solve 2x^2+17x+26=0
Ans: x=-2 or x=-6.5
Q5861) Solve 32x^2+36x+10=0
Ans: x=-0.5 or x=-0.62
Q5862) Solve x^2+27x+26=0
Ans: x=-1 or x=-26
Q5863) Solve 44x^2+40x+9=0
Ans: x=-0.41 or x=-0.5
Q5864) Solve 24x^2+26x+5=0
Ans: x=-0.25 or x=-0.83
Q5865) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q5866) Solve 21x^2+49x+14=0
Ans: x=-0.33 or x=-2
Q5867) Solve 22x^2+45x+2=0
Ans: x=-0.05 or x=-2
Q5868) Solve 5x^2+16x+12=0
Ans: x=-1.2 or x=-2
Q5869) Solve 21x^2+42x+21=0
Ans: x=-1 or x=-1
Q5870) Solve 2x^2+23x+38=0
Ans: x=-2 or x=-9.5
Q5871) Solve 45x^2+33x+2=0
Ans: x=-0.07 or x=-0.67
Q5872) Solve 12x^2+28x+8=0
Ans: x=-0.33 or x=-2
Q5873) Solve 12x^2+11x+2=0
Ans: x=-0.25 or x=-0.67
Q5874) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q5875) Solve 11x^2+45x+34=0
Ans: x=-1 or x=-3.09
Q5876) Solve 10x^2+28x+16=0
Ans: x=-0.8 or x=-2
Q5877) Solve 2x^2+16x+14=0
Ans: x=-1 or x=-7
Q5878) Solve 42x^2+50x+8=0
Ans: x=-0.19 or x=-1
Q5879) Solve 3x^2+35x+50=0
Ans: x=-1.67 or x=-10
Q5880) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q5881) Solve 12x^2+34x+24=0
Ans: x=-1.33 or x=-1.5
Q5882) Solve 23x^2+40x+17=0
Ans: x=-0.74 or x=-1
Q5883) Solve 6x^2+22x+20=0
Ans: x=-1.67 or x=-2
Q5884) Solve 21x^2+44x+4=0
Ans: x=-0.1 or x=-2
Q5885) Solve 4x^2+50x+24=0
Ans: x=-0.5 or x=-12
Q5886) Solve 12x^2+31x+7=0
Ans: x=-0.25 or x=-2.33
Q5887) Solve 28x^2+34x+10=0
Ans: x=-0.5 or x=-0.71
Q5888) Solve 13x^2+33x+14=0
Ans: x=-0.54 or x=-2
Q5889) Solve 27x^2+39x+12=0
Ans: x=-0.44 or x=-1
Q5890) Solve 39x^2+50x+16=0
Ans: x=-0.62 or x=-0.67
Q5891) Solve 18x^2+9x+1=0
Ans: x=-0.17 or x=-0.33
Q5892) Solve 10x^2+44x+34=0
Ans: x=-1 or x=-3.4
Q5893) Solve 21x^2+30x+9=0
Ans: x=-0.43 or x=-1
Q5894) Solve 6x^2+23x+21=0
Ans: x=-1.5 or x=-2.33
Q5895) Solve 34x^2+41x+12=0
Ans: x=-0.5 or x=-0.71
Q5896) Solve 16x^2+32x+7=0
Ans: x=-0.25 or x=-1.75
Q5897) Solve 3x^2+37x+34=0
Ans: x=-1 or x=-11.33
Q5898) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q5899) Solve 9x^2+34x+25=0
Ans: x=-1 or x=-2.78
Q5900) Solve 21x^2+24x+3=0
Ans: x=-0.14 or x=-1
Q5901) Solve 12x^2+26x+14=0
Ans: x=-1 or x=-1.17
Q5902) Solve 27x^2+30x+7=0
Ans: x=-0.33 or x=-0.78
Q5903) Solve 2x^2+22x+20=0
Ans: x=-1 or x=-10
Q5904) Solve 20x^2+41x+21=0
Ans: x=-1 or x=-1.05
Q5905) Solve 18x^2+46x+20=0
Ans: x=-0.56 or x=-2
Q5906) Solve 2x^2+13x+15=0
Ans: x=-1.5 or x=-5
Q5907) Solve 6x^2+50x+44=0
Ans: x=-1 or x=-7.33
Q5908) Solve 14x^2+39x+27=0
Ans: x=-1.29 or x=-1.5
Q5909) Solve x^2+5x+6=0
Ans: x=-2 or x=-3
Q5910) Solve 3x^2+7x+2=0
Ans: x=-0.33 or x=-2
Q5911) Solve 18x^2+29x+3=0
Ans: x=-0.11 or x=-1.5
Q5912) Solve 16x^2+41x+25=0
Ans: x=-1 or x=-1.56
Q5913) Solve 18x^2+39x+6=0
Ans: x=-0.17 or x=-2
Q5914) Solve 30x^2+32x+2=0
Ans: x=-0.07 or x=-1
Q5915) Solve 34x^2+37x+10=0
Ans: x=-0.5 or x=-0.59
Q5916) Solve 6x^2+25x+24=0
Ans: x=-1.5 or x=-2.67
Q5917) Solve 4x^2+37x+40=0
Ans: x=-1.25 or x=-8
Q5918) Solve 6x^2+29x+30=0
Ans: x=-1.5 or x=-3.33
Q5919) Solve x^2+12x+36=0
Ans: x=-6 or x=-6
Q5920) Solve 34x^2+36x+2=0
Ans: x=-0.06 or x=-1
Q5921) Solve x^2+34x+33=0
Ans: x=-1 or x=-33
Q5922) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q5923) Solve 33x^2+38x+9=0
Ans: x=-0.33 or x=-0.82
Q5924) Solve 50x^2+45x+9=0
Ans: x=-0.3 or x=-0.6
Q5925) Solve 32x^2+42x+10=0
Ans: x=-0.31 or x=-1
Q5926) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q5927) Solve 8x^2+48x+40=0
Ans: x=-1 or x=-5
Q5928) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q5929) Solve 3x^2+14x+8=0
Ans: x=-0.67 or x=-4
Q5930) Solve 40x^2+13x+1=0
Ans: x=-0.12 or x=-0.2
Q5931) Solve 42x^2+26x+4=0
Ans: x=-0.29 or x=-0.33
Q5932) Solve 30x^2+48x+18=0
Ans: x=-0.6 or x=-1
Q5933) Solve 14x^2+39x+27=0
Ans: x=-1.29 or x=-1.5
Q5934) Solve 21x^2+50x+29=0
Ans: x=-1 or x=-1.38
Q5935) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q5936) Solve 4x^2+17x+4=0
Ans: x=-0.25 or x=-4
Q5937) Solve 13x^2+45x+38=0
Ans: x=-1.46 or x=-2
Q5938) Solve 39x^2+38x+8=0
Ans: x=-0.31 or x=-0.67
Q5939) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q5940) Solve 26x^2+35x+11=0
Ans: x=-0.5 or x=-0.85
Q5941) Solve 2x^2+31x+42=0
Ans: x=-1.5 or x=-14
Q5942) Solve 7x^2+20x+13=0
Ans: x=-1 or x=-1.86
Q5943) Solve 20x^2+24x+7=0
Ans: x=-0.5 or x=-0.7
Q5944) Solve 33x^2+29x+6=0
Ans: x=-0.33 or x=-0.55
Q5945) Solve 5x^2+43x+38=0
Ans: x=-1 or x=-7.6
Q5946) Solve 2x^2+15x+7=0
Ans: x=-0.5 or x=-7
Q5947) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q5948) Solve 6x^2+27x+21=0
Ans: x=-1 or x=-3.5
Q5949) Solve 21x^2+17x+2=0
Ans: x=-0.14 or x=-0.67
Q5950) Solve 3x^2+35x+50=0
Ans: x=-1.67 or x=-10
Q5951) Solve 4x^2+18x+8=0
Ans: x=-0.5 or x=-4
Q5952) Solve 21x^2+26x+8=0
Ans: x=-0.57 or x=-0.67
Q5953) Solve 5x^2+35x+50=0
Ans: x=-2 or x=-5
Q5954) Solve 15x^2+13x+2=0
Ans: x=-0.2 or x=-0.67
Q5955) Solve 24x^2+32x+10=0
Ans: x=-0.5 or x=-0.83
Q5956) Solve 5x^2+37x+14=0
Ans: x=-0.4 or x=-7
Q5957) Solve 34x^2+41x+12=0
Ans: x=-0.5 or x=-0.71
Q5958) Solve 2x^2+16x+24=0
Ans: x=-2 or x=-6
Q5959) Solve 10x^2+45x+35=0
Ans: x=-1 or x=-3.5
Q5960) Solve 9x^2+41x+32=0
Ans: x=-1 or x=-3.56
Q5961) Solve 8x^2+34x+21=0
Ans: x=-0.75 or x=-3.5
Q5962) Solve 10x^2+39x+29=0
Ans: x=-1 or x=-2.9
Q5963) Solve 14x^2+41x+15=0
Ans: x=-0.43 or x=-2.5
Q5964) Solve 16x^2+46x+30=0
Ans: x=-1 or x=-1.87
Q5965) Solve 23x^2+35x+12=0
Ans: x=-0.52 or x=-1
Q5966) Solve 4x^2+50x+24=0
Ans: x=-0.5 or x=-12
Q5967) Solve 14x^2+35x+14=0
Ans: x=-0.5 or x=-2
Q5968) Solve 3x^2+17x+22=0
Ans: x=-2 or x=-3.67
Q5969) Solve 4x^2+24x+35=0
Ans: x=-2.5 or x=-3.5
Q5970) Solve 18x^2+47x+29=0
Ans: x=-1 or x=-1.61
Q5971) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q5972) Solve 28x^2+50x+22=0
Ans: x=-0.79 or x=-1
Q5973) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q5974) Solve 12x^2+47x+33=0
Ans: x=-0.92 or x=-3
Q5975) Solve 33x^2+40x+12=0
Ans: x=-0.55 or x=-0.67
Q5976) Solve 4x^2+33x+29=0
Ans: x=-1 or x=-7.25
Q5977) Solve 42x^2+13x+1=0
Ans: x=-0.14 or x=-0.17
Q5978) Solve 6x^2+31x+35=0
Ans: x=-1.67 or x=-3.5
Q5979) Solve x^2+9x+20=0
Ans: x=-4 or x=-5
Q5980) Solve 3x^2+31x+10=0
Ans: x=-0.33 or x=-10
Q5981) Solve 25x^2+47x+22=0
Ans: x=-0.88 or x=-1
Q5982) Solve 10x^2+24x+14=0
Ans: x=-1 or x=-1.4
Q5983) Solve 32x^2+33x+1=0
Ans: x=-0.03 or x=-1
Q5984) Solve 33x^2+41x+8=0
Ans: x=-0.24 or x=-1
Q5985) Solve 10x^2+27x+5=0
Ans: x=-0.2 or x=-2.5
Q5986) Solve 10x^2+37x+30=0
Ans: x=-1.2 or x=-2.5
Q5987) Solve 13x^2+36x+23=0
Ans: x=-1 or x=-1.77
Q5988) Solve 45x^2+41x+4=0
Ans: x=-0.11 or x=-0.8
Q5989) Solve x^2+21x+38=0
Ans: x=-2 or x=-19
Q5990) Solve 13x^2+40x+28=0
Ans: x=-1.08 or x=-2
Q5991) Solve 28x^2+18x+2=0
Ans: x=-0.14 or x=-0.5
Q5992) Solve 42x^2+50x+12=0
Ans: x=-0.33 or x=-0.86
Q5993) Solve 4x^2+12x+8=0
Ans: x=-1 or x=-2
Q5994) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q5995) Solve 12x^2+50x+22=0
Ans: x=-0.5 or x=-3.67
Q5996) Solve 6x^2+29x+28=0
Ans: x=-1.33 or x=-3.5
Q5997) Solve 6x^2+25x+25=0
Ans: x=-1.67 or x=-2.5
Q5998) Solve 2x^2+17x+35=0
Ans: x=-3.5 or x=-5
Q5999) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q6000) Solve 6x^2+26x+20=0
Ans: x=-1 or x=-3.33
Q6001) Solve 30x^2+39x+9=0
Ans: x=-0.3 or x=-1
Q6002) Solve 6x^2+47x+35=0
Ans: x=-0.83 or x=-7
Q6003) Solve 24x^2+23x+5=0
Ans: x=-0.33 or x=-0.62
Q6004) Solve 44x^2+28x+3=0
Ans: x=-0.14 or x=-0.5
Q6005) Solve 10x^2+39x+17=0
Ans: x=-0.5 or x=-3.4
Q6006) Solve 3x^2+14x+15=0
Ans: x=-1.67 or x=-3
Q6007) Solve 7x^2+31x+34=0
Ans: x=-2 or x=-2.43
Q6008) Solve 18x^2+33x+5=0
Ans: x=-0.17 or x=-1.67
Q6009) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q6010) Solve 16x^2+35x+19=0
Ans: x=-1 or x=-1.19
Q6011) Solve 16x^2+37x+10=0
Ans: x=-0.31 or x=-2
Q6012) Solve 17x^2+45x+28=0
Ans: x=-1 or x=-1.65
Q6013) Solve 11x^2+45x+34=0
Ans: x=-1 or x=-3.09
Q6014) Solve 27x^2+42x+15=0
Ans: x=-0.56 or x=-1
Q6015) Solve 42x^2+33x+6=0
Ans: x=-0.29 or x=-0.5
Q6016) Solve 3x^2+20x+12=0
Ans: x=-0.67 or x=-6
Q6017) Solve 44x^2+48x+13=0
Ans: x=-0.5 or x=-0.59
Q6018) Solve 3x^2+32x+45=0
Ans: x=-1.67 or x=-9
Q6019) Solve 7x^2+30x+23=0
Ans: x=-1 or x=-3.29
Q6020) Solve 16x^2+26x+9=0
Ans: x=-0.5 or x=-1.12
Q6021) Solve 40x^2+46x+12=0
Ans: x=-0.4 or x=-0.75
Q6022) Solve 32x^2+16x+2=0
Ans: x=-0.25 or x=-0.25
Q6023) Solve 19x^2+42x+23=0
Ans: x=-1 or x=-1.21
Q6024) Solve 9x^2+33x+30=0
Ans: x=-1.67 or x=-2
Q6025) Solve 12x^2+31x+20=0
Ans: x=-1.25 or x=-1.33
Q6026) Solve 21x^2+41x+10=0
Ans: x=-0.29 or x=-1.67
Q6027) Solve 5x^2+30x+25=0
Ans: x=-1 or x=-5
Q6028) Solve 6x^2+49x+49=0
Ans: x=-1.17 or x=-7
Q6029) Solve 5x^2+32x+48=0
Ans: x=-2.4 or x=-4
Q6030) Solve 21x^2+43x+2=0
Ans: x=-0.05 or x=-2
Q6031) Solve 27x^2+41x+14=0
Ans: x=-0.52 or x=-1
Q6032) Solve 6x^2+29x+33=0
Ans: x=-1.83 or x=-3
Q6033) Solve 10x^2+15x+5=0
Ans: x=-0.5 or x=-1
Q6034) Solve 14x^2+31x+12=0
Ans: x=-0.5 or x=-1.71
Q6035) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q6036) Solve 10x^2+43x+39=0
Ans: x=-1.3 or x=-3
Q6037) Solve 4x^2+31x+27=0
Ans: x=-1 or x=-6.75
Q6038) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q6039) Solve 16x^2+46x+30=0
Ans: x=-1 or x=-1.87
Q6040) Solve 16x^2+22x+7=0
Ans: x=-0.5 or x=-0.87
Q6041) Solve 4x^2+18x+18=0
Ans: x=-1.5 or x=-3
Q6042) Solve 20x^2+25x+5=0
Ans: x=-0.25 or x=-1
Q6043) Solve 22x^2+42x+20=0
Ans: x=-0.91 or x=-1
Q6044) Solve 15x^2+23x+4=0
Ans: x=-0.2 or x=-1.33
Q6045) Solve 27x^2+35x+8=0
Ans: x=-0.3 or x=-1
Q6046) Solve 7x^2+45x+38=0
Ans: x=-1 or x=-5.43
Q6047) Solve 3x^2+25x+38=0
Ans: x=-2 or x=-6.33
Q6048) Solve 24x^2+24x+6=0
Ans: x=-0.5 or x=-0.5
Q6049) Solve 2x^2+21x+40=0
Ans: x=-2.5 or x=-8
Q6050) Solve x^2+11x+28=0
Ans: x=-4 or x=-7
Q6051) Solve 3x^2+19x+6=0
Ans: x=-0.33 or x=-6
Q6052) Solve 12x^2+10x+2=0
Ans: x=-0.33 or x=-0.5
Q6053) Solve 9x^2+39x+22=0
Ans: x=-0.67 or x=-3.67
Q6054) Solve 45x^2+37x+6=0
Ans: x=-0.22 or x=-0.6
Q6055) Solve 2x^2+41x+39=0
Ans: x=-1 or x=-19.5
Q6056) Solve 16x^2+48x+36=0
Ans: x=-1.5 or x=-1.5
Q6057) Solve 9x^2+35x+34=0
Ans: x=-1.89 or x=-2
Q6058) Solve 19x^2+42x+8=0
Ans: x=-0.21 or x=-2
Q6059) Solve 11x^2+18x+7=0
Ans: x=-0.64 or x=-1
Q6060) Solve 11x^2+35x+24=0
Ans: x=-1 or x=-2.18
Q6061) Solve 32x^2+37x+5=0
Ans: x=-0.16 or x=-1
Q6062) Solve 18x^2+33x+15=0
Ans: x=-0.83 or x=-1
Q6063) Solve 4x^2+29x+7=0
Ans: x=-0.25 or x=-7
Q6064) Solve 12x^2+23x+11=0
Ans: x=-0.92 or x=-1
Q6065) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q6066) Solve 6x^2+30x+36=0
Ans: x=-2 or x=-3
Q6067) Solve 4x^2+15x+14=0
Ans: x=-1.75 or x=-2
Q6068) Solve 7x^2+33x+38=0
Ans: x=-2 or x=-2.71
Q6069) Solve 30x^2+46x+16=0
Ans: x=-0.53 or x=-1
Q6070) Solve 15x^2+34x+8=0
Ans: x=-0.27 or x=-2
Q6071) Solve 3x^2+13x+14=0
Ans: x=-2 or x=-2.33
Q6072) Solve 12x^2+41x+15=0
Ans: x=-0.42 or x=-3
Q6073) Solve 8x^2+14x+3=0
Ans: x=-0.25 or x=-1.5
Q6074) Solve 49x^2+21x+2=0
Ans: x=-0.14 or x=-0.29
Q6075) Solve 35x^2+34x+8=0
Ans: x=-0.4 or x=-0.57
Q6076) Solve 18x^2+15x+2=0
Ans: x=-0.17 or x=-0.67
Q6077) Solve 5x^2+49x+36=0
Ans: x=-0.8 or x=-9
Q6078) Solve 32x^2+48x+16=0
Ans: x=-0.5 or x=-1
Q6079) Solve 15x^2+40x+20=0
Ans: x=-0.67 or x=-2
Q6080) Solve 18x^2+39x+6=0
Ans: x=-0.17 or x=-2
Q6081) Solve 5x^2+26x+33=0
Ans: x=-2.2 or x=-3
Q6082) Solve 9x^2+46x+40=0
Ans: x=-1.11 or x=-4
Q6083) Solve 18x^2+41x+10=0
Ans: x=-0.28 or x=-2
Q6084) Solve 37x^2+43x+6=0
Ans: x=-0.16 or x=-1
Q6085) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q6086) Solve 9x^2+35x+26=0
Ans: x=-1 or x=-2.89
Q6087) Solve 22x^2+35x+3=0
Ans: x=-0.09 or x=-1.5
Q6088) Solve 8x^2+34x+35=0
Ans: x=-1.75 or x=-2.5
Q6089) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q6090) Solve 26x^2+43x+15=0
Ans: x=-0.5 or x=-1.15
Q6091) Solve 30x^2+31x+7=0
Ans: x=-0.33 or x=-0.7
Q6092) Solve 15x^2+25x+10=0
Ans: x=-0.67 or x=-1
Q6093) Solve 50x^2+33x+4=0
Ans: x=-0.16 or x=-0.5
Q6094) Solve 12x^2+39x+27=0
Ans: x=-1 or x=-2.25
Q6095) Solve 48x^2+41x+6=0
Ans: x=-0.19 or x=-0.67
Q6096) Solve 15x^2+48x+9=0
Ans: x=-0.2 or x=-3
Q6097) Solve 7x^2+38x+31=0
Ans: x=-1 or x=-4.43
Q6098) Solve 2x^2+28x+26=0
Ans: x=-1 or x=-13
Q6099) Solve 45x^2+38x+8=0
Ans: x=-0.4 or x=-0.44
Q6100) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q6101) Solve 17x^2+41x+14=0
Ans: x=-0.41 or x=-2
Q6102) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q6103) Solve 15x^2+37x+22=0
Ans: x=-1 or x=-1.47
Q6104) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q6105) Solve 2x^2+19x+30=0
Ans: x=-2 or x=-7.5
Q6106) Solve 21x^2+23x+2=0
Ans: x=-0.1 or x=-1
Q6107) Solve 42x^2+40x+8=0
Ans: x=-0.29 or x=-0.67
Q6108) Solve 20x^2+41x+20=0
Ans: x=-0.8 or x=-1.25
Q6109) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q6110) Solve 20x^2+38x+18=0
Ans: x=-0.9 or x=-1
Q6111) Solve 31x^2+44x+13=0
Ans: x=-0.42 or x=-1
Q6112) Solve 7x^2+37x+46=0
Ans: x=-2 or x=-3.29
Q6113) Solve 31x^2+47x+16=0
Ans: x=-0.52 or x=-1
Q6114) Solve 11x^2+33x+22=0
Ans: x=-1 or x=-2
Q6115) Solve 12x^2+47x+46=0
Ans: x=-1.92 or x=-2
Q6116) Solve 15x^2+24x+9=0
Ans: x=-0.6 or x=-1
Q6117) Solve 4x^2+21x+5=0
Ans: x=-0.25 or x=-5
Q6118) Solve 5x^2+11x+2=0
Ans: x=-0.2 or x=-2
Q6119) Solve 2x^2+18x+16=0
Ans: x=-1 or x=-8
Q6120) Solve 14x^2+23x+3=0
Ans: x=-0.14 or x=-1.5
Q6121) Solve 2x^2+15x+22=0
Ans: x=-2 or x=-5.5
Q6122) Solve 40x^2+46x+12=0
Ans: x=-0.4 or x=-0.75
Q6123) Solve 6x^2+31x+18=0
Ans: x=-0.67 or x=-4.5
Q6124) Solve 2x^2+19x+44=0
Ans: x=-4 or x=-5.5
Q6125) Solve 12x^2+42x+36=0
Ans: x=-1.5 or x=-2
Q6126) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q6127) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q6128) Solve 30x^2+47x+17=0
Ans: x=-0.57 or x=-1
Q6129) Solve 15x^2+34x+15=0
Ans: x=-0.6 or x=-1.67
Q6130) Solve 9x^2+6x+1=0
Ans: x=-0.33 or x=-0.33
Q6131) Solve 38x^2+39x+10=0
Ans: x=-0.5 or x=-0.53
Q6132) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q6133) Solve 50x^2+35x+3=0
Ans: x=-0.1 or x=-0.6
Q6134) Solve 11x^2+48x+37=0
Ans: x=-1 or x=-3.36
Q6135) Solve 12x^2+22x+8=0
Ans: x=-0.5 or x=-1.33
Q6136) Solve 24x^2+25x+6=0
Ans: x=-0.37 or x=-0.67
Q6137) Solve 14x^2+36x+16=0
Ans: x=-0.57 or x=-2
Q6138) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q6139) Solve 5x^2+16x+11=0
Ans: x=-1 or x=-2.2
Q6140) Solve 4x^2+24x+35=0
Ans: x=-2.5 or x=-3.5
Q6141) Solve 14x^2+25x+11=0
Ans: x=-0.79 or x=-1
Q6142) Solve 10x^2+39x+17=0
Ans: x=-0.5 or x=-3.4
Q6143) Solve 25x^2+35x+6=0
Ans: x=-0.2 or x=-1.2
Q6144) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q6145) Solve x^2+31x+30=0
Ans: x=-1 or x=-30
Q6146) Solve 25x^2+39x+14=0
Ans: x=-0.56 or x=-1
Q6147) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q6148) Solve 4x^2+26x+40=0
Ans: x=-2.5 or x=-4
Q6149) Solve 16x^2+33x+2=0
Ans: x=-0.06 or x=-2
Q6150) Solve 2x^2+21x+19=0
Ans: x=-1 or x=-9.5
Q6151) Solve 36x^2+15x+1=0
Ans: x=-0.08 or x=-0.33
Q6152) Solve 4x^2+18x+18=0
Ans: x=-1.5 or x=-3
Q6153) Solve 12x^2+12x+3=0
Ans: x=-0.5 or x=-0.5
Q6154) Solve 4x^2+40x+19=0
Ans: x=-0.5 or x=-9.5
Q6155) Solve 6x^2+32x+26=0
Ans: x=-1 or x=-4.33
Q6156) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q6157) Solve 22x^2+38x+16=0
Ans: x=-0.73 or x=-1
Q6158) Solve 18x^2+44x+16=0
Ans: x=-0.44 or x=-2
Q6159) Solve 36x^2+49x+5=0
Ans: x=-0.11 or x=-1.25
Q6160) Solve 36x^2+46x+10=0
Ans: x=-0.28 or x=-1
Q6161) Solve 2x^2+7x+5=0
Ans: x=-1 or x=-2.5
Q6162) Solve 4x^2+47x+43=0
Ans: x=-1 or x=-10.75
Q6163) Solve 7x^2+43x+40=0
Ans: x=-1.14 or x=-5
Q6164) Solve 39x^2+29x+2=0
Ans: x=-0.08 or x=-0.67
Q6165) Solve 33x^2+50x+8=0
Ans: x=-0.18 or x=-1.33
Q6166) Solve 16x^2+24x+9=0
Ans: x=-0.75 or x=-0.75
Q6167) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q6168) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q6169) Solve 8x^2+38x+45=0
Ans: x=-2.25 or x=-2.5
Q6170) Solve 5x^2+17x+12=0
Ans: x=-1 or x=-2.4
Q6171) Solve x^2+22x+40=0
Ans: x=-2 or x=-20
Q6172) Solve 2x^2+7x+3=0
Ans: x=-0.5 or x=-3
Q6173) Solve 40x^2+26x+3=0
Ans: x=-0.15 or x=-0.5
Q6174) Solve 3x^2+20x+25=0
Ans: x=-1.67 or x=-5
Q6175) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q6176) Solve 40x^2+13x+1=0
Ans: x=-0.12 or x=-0.2
Q6177) Solve 13x^2+49x+46=0
Ans: x=-1.77 or x=-2
Q6178) Solve 4x^2+37x+40=0
Ans: x=-1.25 or x=-8
Q6179) Solve 4x^2+42x+38=0
Ans: x=-1 or x=-9.5
Q6180) Solve 7x^2+37x+36=0
Ans: x=-1.29 or x=-4
Q6181) Solve 32x^2+44x+14=0
Ans: x=-0.5 or x=-0.87
Q6182) Solve 3x^2+7x+2=0
Ans: x=-0.33 or x=-2
Q6183) Solve 25x^2+50x+25=0
Ans: x=-1 or x=-1
Q6184) Solve 15x^2+50x+40=0
Ans: x=-1.33 or x=-2
Q6185) Solve 5x^2+34x+24=0
Ans: x=-0.8 or x=-6
Q6186) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q6187) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q6188) Solve 7x^2+39x+32=0
Ans: x=-1 or x=-4.57
Q6189) Solve 3x^2+27x+42=0
Ans: x=-2 or x=-7
Q6190) Solve 31x^2+36x+5=0
Ans: x=-0.16 or x=-1
Q6191) Solve x^2+21x+20=0
Ans: x=-1 or x=-20
Q6192) Solve 9x^2+6x+1=0
Ans: x=-0.33 or x=-0.33
Q6193) Solve 10x^2+49x+18=0
Ans: x=-0.4 or x=-4.5
Q6194) Solve 39x^2+28x+5=0
Ans: x=-0.33 or x=-0.38
Q6195) Solve 5x^2+48x+27=0
Ans: x=-0.6 or x=-9
Q6196) Solve 50x^2+40x+8=0
Ans: x=-0.4 or x=-0.4
Q6197) Solve 33x^2+36x+3=0
Ans: x=-0.09 or x=-1
Q6198) Solve 28x^2+39x+5=0
Ans: x=-0.14 or x=-1.25
Q6199) Solve 2x^2+11x+9=0
Ans: x=-1 or x=-4.5
Q6200) Solve 12x^2+44x+32=0
Ans: x=-1 or x=-2.67
Q6201) Solve 24x^2+40x+6=0
Ans: x=-0.17 or x=-1.5
Q6202) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q6203) Solve 21x^2+49x+14=0
Ans: x=-0.33 or x=-2
Q6204) Solve 48x^2+22x+2=0
Ans: x=-0.12 or x=-0.33
Q6205) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q6206) Solve 16x^2+26x+3=0
Ans: x=-0.12 or x=-1.5
Q6207) Solve 34x^2+45x+11=0
Ans: x=-0.32 or x=-1
Q6208) Solve 3x^2+29x+26=0
Ans: x=-1 or x=-8.67
Q6209) Solve 7x^2+23x+16=0
Ans: x=-1 or x=-2.29
Q6210) Solve 3x^2+35x+50=0
Ans: x=-1.67 or x=-10
Q6211) Solve 19x^2+47x+18=0
Ans: x=-0.47 or x=-2
Q6212) Solve 32x^2+40x+12=0
Ans: x=-0.5 or x=-0.75
Q6213) Solve 2x^2+19x+9=0
Ans: x=-0.5 or x=-9
Q6214) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q6215) Solve 34x^2+35x+9=0
Ans: x=-0.5 or x=-0.53
Q6216) Solve 5x^2+28x+23=0
Ans: x=-1 or x=-4.6
Q6217) Solve 7x^2+43x+40=0
Ans: x=-1.14 or x=-5
Q6218) Solve 31x^2+35x+4=0
Ans: x=-0.13 or x=-1
Q6219) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q6220) Solve 18x^2+47x+5=0
Ans: x=-0.11 or x=-2.5
Q6221) Solve 28x^2+39x+11=0
Ans: x=-0.39 or x=-1
Q6222) Solve 2x^2+14x+24=0
Ans: x=-3 or x=-4
Q6223) Solve 34x^2+39x+11=0
Ans: x=-0.5 or x=-0.65
Q6224) Solve 48x^2+36x+6=0
Ans: x=-0.25 or x=-0.5
Q6225) Solve 2x^2+27x+25=0
Ans: x=-1 or x=-12.5
Q6226) Solve 16x^2+45x+26=0
Ans: x=-0.81 or x=-2
Q6227) Solve 15x^2+49x+24=0
Ans: x=-0.6 or x=-2.67
Q6228) Solve 16x^2+36x+18=0
Ans: x=-0.75 or x=-1.5
Q6229) Solve 26x^2+30x+4=0
Ans: x=-0.15 or x=-1
Q6230) Solve 45x^2+41x+4=0
Ans: x=-0.11 or x=-0.8
Q6231) Solve 27x^2+48x+16=0
Ans: x=-0.44 or x=-1.33
Q6232) Solve 28x^2+36x+11=0
Ans: x=-0.5 or x=-0.79
Q6233) Solve 9x^2+45x+50=0
Ans: x=-1.67 or x=-3.33
Q6234) Solve 33x^2+44x+11=0
Ans: x=-0.33 or x=-1
Q6235) Solve 10x^2+45x+50=0
Ans: x=-2 or x=-2.5
Q6236) Solve 39x^2+16x+1=0
Ans: x=-0.08 or x=-0.33
Q6237) Solve 45x^2+32x+3=0
Ans: x=-0.11 or x=-0.6
Q6238) Solve 7x^2+12x+5=0
Ans: x=-0.71 or x=-1
Q6239) Solve 28x^2+15x+2=0
Ans: x=-0.25 or x=-0.29
Q6240) Solve 32x^2+18x+1=0
Ans: x=-0.06 or x=-0.5
Q6241) Solve 9x^2+46x+5=0
Ans: x=-0.11 or x=-5
Q6242) Solve 25x^2+45x+14=0
Ans: x=-0.4 or x=-1.4
Q6243) Solve 4x^2+18x+18=0
Ans: x=-1.5 or x=-3
Q6244) Solve 11x^2+36x+28=0
Ans: x=-1.27 or x=-2
Q6245) Solve 13x^2+47x+24=0
Ans: x=-0.62 or x=-3
Q6246) Solve 31x^2+41x+10=0
Ans: x=-0.32 or x=-1
Q6247) Solve 5x^2+30x+45=0
Ans: x=-3 or x=-3
Q6248) Solve 6x^2+28x+16=0
Ans: x=-0.67 or x=-4
Q6249) Solve 9x^2+21x+6=0
Ans: x=-0.33 or x=-2
Q6250) Solve 40x^2+42x+11=0
Ans: x=-0.5 or x=-0.55
Q6251) Solve 23x^2+47x+24=0
Ans: x=-1 or x=-1.04
Q6252) Solve 8x^2+14x+6=0
Ans: x=-0.75 or x=-1
Q6253) Solve 5x^2+17x+14=0
Ans: x=-1.4 or x=-2
Q6254) Solve 35x^2+27x+4=0
Ans: x=-0.2 or x=-0.57
Q6255) Solve 34x^2+41x+7=0
Ans: x=-0.21 or x=-1
Q6256) Solve 44x^2+43x+8=0
Ans: x=-0.25 or x=-0.73
Q6257) Solve 35x^2+22x+3=0
Ans: x=-0.2 or x=-0.43
Q6258) Solve x^2+25x+46=0
Ans: x=-2 or x=-23
Q6259) Solve 34x^2+41x+7=0
Ans: x=-0.21 or x=-1
Q6260) Solve 6x^2+11x+3=0
Ans: x=-0.33 or x=-1.5
Q6261) Solve x^2+6x+9=0
Ans: x=-3 or x=-3
Q6262) Solve 25x^2+37x+12=0
Ans: x=-0.48 or x=-1
Q6263) Solve 25x^2+32x+7=0
Ans: x=-0.28 or x=-1
Q6264) Solve 48x^2+37x+7=0
Ans: x=-0.33 or x=-0.44
Q6265) Solve 27x^2+31x+4=0
Ans: x=-0.15 or x=-1
Q6266) Solve 44x^2+26x+2=0
Ans: x=-0.09 or x=-0.5
Q6267) Solve 17x^2+20x+3=0
Ans: x=-0.18 or x=-1
Q6268) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q6269) Solve 16x^2+48x+27=0
Ans: x=-0.75 or x=-2.25
Q6270) Solve 42x^2+19x+2=0
Ans: x=-0.17 or x=-0.29
Q6271) Solve 10x^2+41x+31=0
Ans: x=-1 or x=-3.1
Q6272) Solve 48x^2+36x+6=0
Ans: x=-0.25 or x=-0.5
Q6273) Solve 30x^2+21x+3=0
Ans: x=-0.2 or x=-0.5
Q6274) Solve 41x^2+45x+4=0
Ans: x=-0.1 or x=-1
Q6275) Solve 16x^2+49x+3=0
Ans: x=-0.06 or x=-3
Q6276) Solve x^2+37x+36=0
Ans: x=-1 or x=-36
Q6277) Solve 5x^2+10x+5=0
Ans: x=-1 or x=-1
Q6278) Solve 44x^2+41x+6=0
Ans: x=-0.18 or x=-0.75
Q6279) Solve 13x^2+50x+48=0
Ans: x=-1.85 or x=-2
Q6280) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q6281) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q6282) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q6283) Solve 11x^2+13x+2=0
Ans: x=-0.18 or x=-1
Q6284) Solve 8x^2+23x+14=0
Ans: x=-0.87 or x=-2
Q6285) Solve 7x^2+40x+25=0
Ans: x=-0.71 or x=-5
Q6286) Solve 42x^2+50x+8=0
Ans: x=-0.19 or x=-1
Q6287) Solve 37x^2+38x+1=0
Ans: x=-0.03 or x=-1
Q6288) Solve 24x^2+25x+1=0
Ans: x=-0.04 or x=-1
Q6289) Solve 3x^2+12x+9=0
Ans: x=-1 or x=-3
Q6290) Solve 7x^2+37x+10=0
Ans: x=-0.29 or x=-5
Q6291) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q6292) Solve 14x^2+39x+25=0
Ans: x=-1 or x=-1.79
Q6293) Solve 19x^2+41x+22=0
Ans: x=-1 or x=-1.16
Q6294) Solve 36x^2+15x+1=0
Ans: x=-0.08 or x=-0.33
Q6295) Solve 7x^2+29x+30=0
Ans: x=-2 or x=-2.14
Q6296) Solve 18x^2+19x+1=0
Ans: x=-0.06 or x=-1
Q6297) Solve 12x^2+50x+42=0
Ans: x=-1.17 or x=-3
Q6298) Solve 41x^2+44x+3=0
Ans: x=-0.07 or x=-1
Q6299) Solve 50x^2+33x+4=0
Ans: x=-0.16 or x=-0.5
Q6300) Solve 6x^2+15x+9=0
Ans: x=-1 or x=-1.5
Q6301) Solve 3x^2+17x+20=0
Ans: x=-1.67 or x=-4
Q6302) Solve 36x^2+30x+6=0
Ans: x=-0.33 or x=-0.5
Q6303) Solve 20x^2+38x+18=0
Ans: x=-0.9 or x=-1
Q6304) Solve 45x^2+33x+6=0
Ans: x=-0.33 or x=-0.4
Q6305) Solve 48x^2+49x+1=0
Ans: x=-0.02 or x=-1
Q6306) Solve 2x^2+35x+48=0
Ans: x=-1.5 or x=-16
Q6307) Solve 13x^2+48x+44=0
Ans: x=-1.69 or x=-2
Q6308) Solve 38x^2+43x+5=0
Ans: x=-0.13 or x=-1
Q6309) Solve 21x^2+50x+24=0
Ans: x=-0.67 or x=-1.71
Q6310) Solve 36x^2+44x+8=0
Ans: x=-0.22 or x=-1
Q6311) Solve 3x^2+7x+4=0
Ans: x=-1 or x=-1.33
Q6312) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q6313) Solve 21x^2+47x+26=0
Ans: x=-1 or x=-1.24
Q6314) Solve 28x^2+41x+13=0
Ans: x=-0.46 or x=-1
Q6315) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q6316) Solve 27x^2+28x+1=0
Ans: x=-0.04 or x=-1
Q6317) Solve 39x^2+47x+14=0
Ans: x=-0.54 or x=-0.67
Q6318) Solve 32x^2+48x+16=0
Ans: x=-0.5 or x=-1
Q6319) Solve 2x^2+15x+22=0
Ans: x=-2 or x=-5.5
Q6320) Solve 48x^2+41x+6=0
Ans: x=-0.19 or x=-0.67
Q6321) Solve 8x^2+36x+36=0
Ans: x=-1.5 or x=-3
Q6322) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q6323) Solve 17x^2+27x+10=0
Ans: x=-0.59 or x=-1
Q6324) Solve 3x^2+20x+17=0
Ans: x=-1 or x=-5.67
Q6325) Solve 15x^2+43x+26=0
Ans: x=-0.87 or x=-2
Q6326) Solve 20x^2+16x+3=0
Ans: x=-0.3 or x=-0.5
Q6327) Solve 2x^2+28x+48=0
Ans: x=-2 or x=-12
Q6328) Solve 45x^2+41x+4=0
Ans: x=-0.11 or x=-0.8
Q6329) Solve 14x^2+43x+29=0
Ans: x=-1 or x=-2.07
Q6330) Solve x^2+42x+41=0
Ans: x=-1 or x=-41
Q6331) Solve 16x^2+48x+20=0
Ans: x=-0.5 or x=-2.5
Q6332) Solve 17x^2+48x+28=0
Ans: x=-0.82 or x=-2
Q6333) Solve 10x^2+46x+48=0
Ans: x=-1.6 or x=-3
Q6334) Solve 36x^2+32x+7=0
Ans: x=-0.39 or x=-0.5
Q6335) Solve 10x^2+19x+7=0
Ans: x=-0.5 or x=-1.4
Q6336) Solve 10x^2+27x+5=0
Ans: x=-0.2 or x=-2.5
Q6337) Solve 13x^2+35x+22=0
Ans: x=-1 or x=-1.69
Q6338) Solve 18x^2+17x+4=0
Ans: x=-0.44 or x=-0.5
Q6339) Solve 42x^2+38x+8=0
Ans: x=-0.33 or x=-0.57
Q6340) Solve 30x^2+41x+6=0
Ans: x=-0.17 or x=-1.2
Q6341) Solve 16x^2+24x+8=0
Ans: x=-0.5 or x=-1
Q6342) Solve 6x^2+49x+23=0
Ans: x=-0.5 or x=-7.67
Q6343) Solve 14x^2+49x+42=0
Ans: x=-1.5 or x=-2
Q6344) Solve x^2+14x+45=0
Ans: x=-5 or x=-9
Q6345) Solve 24x^2+34x+11=0
Ans: x=-0.5 or x=-0.92
Q6346) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q6347) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q6348) Solve 24x^2+35x+4=0
Ans: x=-0.12 or x=-1.33
Q6349) Solve 8x^2+23x+15=0
Ans: x=-1 or x=-1.87
Q6350) Solve 8x^2+26x+21=0
Ans: x=-1.5 or x=-1.75
Q6351) Solve 6x^2+37x+42=0
Ans: x=-1.5 or x=-4.67
Q6352) Solve 12x^2+26x+14=0
Ans: x=-1 or x=-1.17
Q6353) Solve 25x^2+48x+23=0
Ans: x=-0.92 or x=-1
Q6354) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q6355) Solve 26x^2+39x+13=0
Ans: x=-0.5 or x=-1
Q6356) Solve 6x^2+41x+13=0
Ans: x=-0.33 or x=-6.5
Q6357) Solve 9x^2+18x+8=0
Ans: x=-0.67 or x=-1.33
Q6358) Solve 20x^2+47x+27=0
Ans: x=-1 or x=-1.35
Q6359) Solve 9x^2+34x+25=0
Ans: x=-1 or x=-2.78
Q6360) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q6361) Solve 6x^2+39x+18=0
Ans: x=-0.5 or x=-6
Q6362) Solve 30x^2+32x+8=0
Ans: x=-0.4 or x=-0.67
Q6363) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q6364) Solve 38x^2+31x+6=0
Ans: x=-0.32 or x=-0.5
Q6365) Solve 48x^2+40x+3=0
Ans: x=-0.08 or x=-0.75
Q6366) Solve 19x^2+38x+19=0
Ans: x=-1 or x=-1
Q6367) Solve 14x^2+33x+13=0
Ans: x=-0.5 or x=-1.86
Q6368) Solve 30x^2+32x+8=0
Ans: x=-0.4 or x=-0.67
Q6369) Solve 18x^2+45x+25=0
Ans: x=-0.83 or x=-1.67
Q6370) Solve 21x^2+35x+14=0
Ans: x=-0.67 or x=-1
Q6371) Solve 17x^2+40x+23=0
Ans: x=-1 or x=-1.35
Q6372) Solve 12x^2+22x+6=0
Ans: x=-0.33 or x=-1.5
Q6373) Solve 10x^2+39x+36=0
Ans: x=-1.5 or x=-2.4
Q6374) Solve 11x^2+37x+30=0
Ans: x=-1.36 or x=-2
Q6375) Solve 12x^2+28x+8=0
Ans: x=-0.33 or x=-2
Q6376) Solve 32x^2+28x+3=0
Ans: x=-0.12 or x=-0.75
Q6377) Solve 10x^2+41x+42=0
Ans: x=-2 or x=-2.1
Q6378) Solve 44x^2+45x+1=0
Ans: x=-0.02 or x=-1
Q6379) Solve 33x^2+31x+6=0
Ans: x=-0.27 or x=-0.67
Q6380) Solve 5x^2+32x+48=0
Ans: x=-2.4 or x=-4
Q6381) Solve 8x^2+14x+6=0
Ans: x=-0.75 or x=-1
Q6382) Solve 14x^2+16x+2=0
Ans: x=-0.14 or x=-1
Q6383) Solve 6x^2+37x+17=0
Ans: x=-0.5 or x=-5.67
Q6384) Solve 11x^2+24x+13=0
Ans: x=-1 or x=-1.18
Q6385) Solve 15x^2+48x+33=0
Ans: x=-1 or x=-2.2
Q6386) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q6387) Solve 12x^2+24x+9=0
Ans: x=-0.5 or x=-1.5
Q6388) Solve 36x^2+12x+1=0
Ans: x=-0.17 or x=-0.17
Q6389) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q6390) Solve 3x^2+19x+30=0
Ans: x=-3 or x=-3.33
Q6391) Solve 16x^2+28x+12=0
Ans: x=-0.75 or x=-1
Q6392) Solve 12x^2+31x+19=0
Ans: x=-1 or x=-1.58
Q6393) Solve 3x^2+17x+14=0
Ans: x=-1 or x=-4.67
Q6394) Solve 7x^2+35x+28=0
Ans: x=-1 or x=-4
Q6395) Solve x^2+40x+39=0
Ans: x=-1 or x=-39
Q6396) Solve 40x^2+48x+14=0
Ans: x=-0.5 or x=-0.7
Q6397) Solve 8x^2+17x+9=0
Ans: x=-1 or x=-1.12
Q6398) Solve 8x^2+40x+32=0
Ans: x=-1 or x=-4
Q6399) Solve 40x^2+14x+1=0
Ans: x=-0.1 or x=-0.25
Q6400) Solve 24x^2+50x+19=0
Ans: x=-0.5 or x=-1.58
Q6401) Solve 11x^2+41x+30=0
Ans: x=-1 or x=-2.73
Q6402) Solve 6x^2+30x+36=0
Ans: x=-2 or x=-3
Q6403) Solve 2x^2+7x+3=0
Ans: x=-0.5 or x=-3
Q6404) Solve 5x^2+38x+33=0
Ans: x=-1 or x=-6.6
Q6405) Solve 17x^2+50x+33=0
Ans: x=-1 or x=-1.94
Q6406) Solve 18x^2+47x+19=0
Ans: x=-0.5 or x=-2.11
Q6407) Solve 9x^2+21x+6=0
Ans: x=-0.33 or x=-2
Q6408) Solve 7x^2+37x+36=0
Ans: x=-1.29 or x=-4
Q6409) Solve 14x^2+47x+38=0
Ans: x=-1.36 or x=-2
Q6410) Solve 4x^2+19x+12=0
Ans: x=-0.75 or x=-4
Q6411) Solve 7x^2+8x+1=0
Ans: x=-0.14 or x=-1
Q6412) Solve 7x^2+38x+31=0
Ans: x=-1 or x=-4.43
Q6413) Solve 44x^2+46x+2=0
Ans: x=-0.05 or x=-1
Q6414) Solve 8x^2+43x+15=0
Ans: x=-0.37 or x=-5
Q6415) Solve 27x^2+12x+1=0
Ans: x=-0.11 or x=-0.33
Q6416) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q6417) Solve 22x^2+25x+3=0
Ans: x=-0.14 or x=-1
Q6418) Solve 16x^2+28x+6=0
Ans: x=-0.25 or x=-1.5
Q6419) Solve 12x^2+49x+30=0
Ans: x=-0.75 or x=-3.33
Q6420) Solve 14x^2+41x+26=0
Ans: x=-0.93 or x=-2
Q6421) Solve 3x^2+25x+22=0
Ans: x=-1 or x=-7.33
Q6422) Solve 18x^2+36x+18=0
Ans: x=-1 or x=-1
Q6423) Solve 42x^2+27x+3=0
Ans: x=-0.14 or x=-0.5
Q6424) Solve 5x^2+32x+44=0
Ans: x=-2 or x=-4.4
Q6425) Solve 46x^2+49x+3=0
Ans: x=-0.07 or x=-1
Q6426) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q6427) Solve 35x^2+45x+10=0
Ans: x=-0.29 or x=-1
Q6428) Solve 47x^2+49x+2=0
Ans: x=-0.04 or x=-1
Q6429) Solve 18x^2+44x+16=0
Ans: x=-0.44 or x=-2
Q6430) Solve 2x^2+17x+36=0
Ans: x=-4 or x=-4.5
Q6431) Solve 2x^2+17x+21=0
Ans: x=-1.5 or x=-7
Q6432) Solve 12x^2+40x+33=0
Ans: x=-1.5 or x=-1.83
Q6433) Solve 21x^2+36x+15=0
Ans: x=-0.71 or x=-1
Q6434) Solve x^2+14x+13=0
Ans: x=-1 or x=-13
Q6435) Solve 6x^2+49x+23=0
Ans: x=-0.5 or x=-7.67
Q6436) Solve 36x^2+31x+3=0
Ans: x=-0.11 or x=-0.75
Q6437) Solve 7x^2+32x+36=0
Ans: x=-2 or x=-2.57
Q6438) Solve 30x^2+49x+20=0
Ans: x=-0.8 or x=-0.83
Q6439) Solve 11x^2+42x+31=0
Ans: x=-1 or x=-2.82
Q6440) Solve 21x^2+43x+12=0
Ans: x=-0.33 or x=-1.71
Q6441) Solve 2x^2+20x+32=0
Ans: x=-2 or x=-8
Q6442) Solve 33x^2+40x+12=0
Ans: x=-0.55 or x=-0.67
Q6443) Solve 26x^2+25x+6=0
Ans: x=-0.46 or x=-0.5
Q6444) Solve 5x^2+41x+8=0
Ans: x=-0.2 or x=-8
Q6445) Solve 3x^2+25x+8=0
Ans: x=-0.33 or x=-8
Q6446) Solve 13x^2+32x+12=0
Ans: x=-0.46 or x=-2
Q6447) Solve 44x^2+44x+11=0
Ans: x=-0.5 or x=-0.5
Q6448) Solve 4x^2+41x+45=0
Ans: x=-1.25 or x=-9
Q6449) Solve 40x^2+37x+4=0
Ans: x=-0.12 or x=-0.8
Q6450) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q6451) Solve 7x^2+48x+36=0
Ans: x=-0.86 or x=-6
Q6452) Solve x^2+28x+27=0
Ans: x=-1 or x=-27
Q6453) Solve 22x^2+37x+13=0
Ans: x=-0.5 or x=-1.18
Q6454) Solve 2x^2+32x+30=0
Ans: x=-1 or x=-15
Q6455) Solve 10x^2+7x+1=0
Ans: x=-0.2 or x=-0.5
Q6456) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q6457) Solve 9x^2+46x+40=0
Ans: x=-1.11 or x=-4
Q6458) Solve 2x^2+20x+48=0
Ans: x=-4 or x=-6
Q6459) Solve 11x^2+26x+8=0
Ans: x=-0.36 or x=-2
Q6460) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q6461) Solve 50x^2+45x+10=0
Ans: x=-0.4 or x=-0.5
Q6462) Solve 27x^2+33x+8=0
Ans: x=-0.33 or x=-0.89
Q6463) Solve 9x^2+29x+20=0
Ans: x=-1 or x=-2.22
Q6464) Solve 10x^2+42x+32=0
Ans: x=-1 or x=-3.2
Q6465) Solve 5x^2+34x+24=0
Ans: x=-0.8 or x=-6
Q6466) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q6467) Solve 15x^2+33x+18=0
Ans: x=-1 or x=-1.2
Q6468) Solve 8x^2+48x+40=0
Ans: x=-1 or x=-5
Q6469) Solve 8x^2+11x+3=0
Ans: x=-0.37 or x=-1
Q6470) Solve 10x^2+47x+21=0
Ans: x=-0.5 or x=-4.2
Q6471) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q6472) Solve 42x^2+31x+5=0
Ans: x=-0.24 or x=-0.5
Q6473) Solve 28x^2+41x+13=0
Ans: x=-0.46 or x=-1
Q6474) Solve 6x^2+38x+40=0
Ans: x=-1.33 or x=-5
Q6475) Solve 27x^2+15x+2=0
Ans: x=-0.22 or x=-0.33
Q6476) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q6477) Solve 34x^2+46x+12=0
Ans: x=-0.35 or x=-1
Q6478) Solve 8x^2+28x+12=0
Ans: x=-0.5 or x=-3
Q6479) Solve 3x^2+25x+48=0
Ans: x=-3 or x=-5.33
Q6480) Solve 9x^2+30x+9=0
Ans: x=-0.33 or x=-3
Q6481) Solve 10x^2+21x+2=0
Ans: x=-0.1 or x=-2
Q6482) Solve 48x^2+34x+5=0
Ans: x=-0.21 or x=-0.5
Q6483) Solve 3x^2+26x+16=0
Ans: x=-0.67 or x=-8
Q6484) Solve 24x^2+14x+1=0
Ans: x=-0.08 or x=-0.5
Q6485) Solve 36x^2+46x+14=0
Ans: x=-0.5 or x=-0.78
Q6486) Solve 3x^2+21x+30=0
Ans: x=-2 or x=-5
Q6487) Solve 12x^2+38x+30=0
Ans: x=-1.5 or x=-1.67
Q6488) Solve 19x^2+49x+30=0
Ans: x=-1 or x=-1.58
Q6489) Solve 16x^2+37x+10=0
Ans: x=-0.31 or x=-2
Q6490) Solve 22x^2+17x+3=0
Ans: x=-0.27 or x=-0.5
Q6491) Solve 42x^2+23x+3=0
Ans: x=-0.21 or x=-0.33
Q6492) Solve 4x^2+31x+21=0
Ans: x=-0.75 or x=-7
Q6493) Solve 21x^2+26x+8=0
Ans: x=-0.57 or x=-0.67
Q6494) Solve 10x^2+29x+21=0
Ans: x=-1.4 or x=-1.5
Q6495) Solve 23x^2+47x+2=0
Ans: x=-0.04 or x=-2
Q6496) Solve 44x^2+32x+5=0
Ans: x=-0.23 or x=-0.5
Q6497) Solve 4x^2+23x+15=0
Ans: x=-0.75 or x=-5
Q6498) Solve 15x^2+41x+28=0
Ans: x=-1.33 or x=-1.4
Q6499) Solve 3x^2+37x+44=0
Ans: x=-1.33 or x=-11
Q6500) Solve 8x^2+42x+45=0
Ans: x=-1.5 or x=-3.75
Q6501) Solve 36x^2+49x+10=0
Ans: x=-0.25 or x=-1.11
Q6502) Solve 3x^2+37x+34=0
Ans: x=-1 or x=-11.33
Q6503) Solve 18x^2+42x+12=0
Ans: x=-0.33 or x=-2
Q6504) Solve 35x^2+34x+8=0
Ans: x=-0.4 or x=-0.57
Q6505) Solve 30x^2+34x+4=0
Ans: x=-0.13 or x=-1
Q6506) Solve 6x^2+21x+9=0
Ans: x=-0.5 or x=-3
Q6507) Solve 15x^2+35x+10=0
Ans: x=-0.33 or x=-2
Q6508) Solve 33x^2+17x+2=0
Ans: x=-0.18 or x=-0.33
Q6509) Solve x^2+11x+10=0
Ans: x=-1 or x=-10
Q6510) Solve 2x^2+17x+21=0
Ans: x=-1.5 or x=-7
Q6511) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q6512) Solve 37x^2+42x+5=0
Ans: x=-0.14 or x=-1
Q6513) Solve 21x^2+41x+10=0
Ans: x=-0.29 or x=-1.67
Q6514) Solve 2x^2+21x+19=0
Ans: x=-1 or x=-9.5
Q6515) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q6516) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q6517) Solve 24x^2+38x+10=0
Ans: x=-0.33 or x=-1.25
Q6518) Solve 14x^2+22x+8=0
Ans: x=-0.57 or x=-1
Q6519) Solve 33x^2+43x+14=0
Ans: x=-0.64 or x=-0.67
Q6520) Solve 35x^2+43x+8=0
Ans: x=-0.23 or x=-1
Q6521) Solve 45x^2+34x+5=0
Ans: x=-0.2 or x=-0.56
Q6522) Solve 15x^2+46x+35=0
Ans: x=-1.4 or x=-1.67
Q6523) Solve 22x^2+36x+14=0
Ans: x=-0.64 or x=-1
Q6524) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q6525) Solve 24x^2+43x+5=0
Ans: x=-0.12 or x=-1.67
Q6526) Solve 26x^2+40x+14=0
Ans: x=-0.54 or x=-1
Q6527) Solve 20x^2+45x+10=0
Ans: x=-0.25 or x=-2
Q6528) Solve 18x^2+26x+8=0
Ans: x=-0.44 or x=-1
Q6529) Solve 8x^2+26x+15=0
Ans: x=-0.75 or x=-2.5
Q6530) Solve 11x^2+37x+12=0
Ans: x=-0.36 or x=-3
Q6531) Solve 38x^2+43x+12=0
Ans: x=-0.5 or x=-0.63
Q6532) Solve 8x^2+38x+35=0
Ans: x=-1.25 or x=-3.5
Q6533) Solve 32x^2+38x+6=0
Ans: x=-0.19 or x=-1
Q6534) Solve 36x^2+13x+1=0
Ans: x=-0.11 or x=-0.25
Q6535) Solve 12x^2+31x+9=0
Ans: x=-0.33 or x=-2.25
Q6536) Solve 8x^2+43x+44=0
Ans: x=-1.37 or x=-4
Q6537) Solve 40x^2+29x+3=0
Ans: x=-0.12 or x=-0.6
Q6538) Solve 13x^2+50x+33=0
Ans: x=-0.85 or x=-3
Q6539) Solve 11x^2+19x+8=0
Ans: x=-0.73 or x=-1
Q6540) Solve 16x^2+48x+27=0
Ans: x=-0.75 or x=-2.25
Q6541) Solve 10x^2+40x+40=0
Ans: x=-2 or x=-2
Q6542) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q6543) Solve 2x^2+19x+35=0
Ans: x=-2.5 or x=-7
Q6544) Solve 34x^2+45x+14=0
Ans: x=-0.5 or x=-0.82
Q6545) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q6546) Solve 27x^2+38x+11=0
Ans: x=-0.41 or x=-1
Q6547) Solve 7x^2+27x+26=0
Ans: x=-1.86 or x=-2
Q6548) Solve 21x^2+20x+4=0
Ans: x=-0.29 or x=-0.67
Q6549) Solve 9x^2+45x+50=0
Ans: x=-1.67 or x=-3.33
Q6550) Solve 30x^2+21x+3=0
Ans: x=-0.2 or x=-0.5
Q6551) Solve 8x^2+41x+50=0
Ans: x=-2 or x=-3.12
Q6552) Solve 12x^2+31x+9=0
Ans: x=-0.33 or x=-2.25
Q6553) Solve 20x^2+41x+9=0
Ans: x=-0.25 or x=-1.8
Q6554) Solve 8x^2+38x+17=0
Ans: x=-0.5 or x=-4.25
Q6555) Solve 8x^2+14x+3=0
Ans: x=-0.25 or x=-1.5
Q6556) Solve 7x^2+29x+30=0
Ans: x=-2 or x=-2.14
Q6557) Solve 24x^2+30x+6=0
Ans: x=-0.25 or x=-1
Q6558) Solve 21x^2+46x+13=0
Ans: x=-0.33 or x=-1.86
Q6559) Solve 23x^2+49x+6=0
Ans: x=-0.13 or x=-2
Q6560) Solve 15x^2+40x+25=0
Ans: x=-1 or x=-1.67
Q6561) Solve 7x^2+35x+28=0
Ans: x=-1 or x=-4
Q6562) Solve 42x^2+48x+6=0
Ans: x=-0.14 or x=-1
Q6563) Solve 34x^2+45x+14=0
Ans: x=-0.5 or x=-0.82
Q6564) Solve 5x^2+11x+2=0
Ans: x=-0.2 or x=-2
Q6565) Solve 9x^2+19x+2=0
Ans: x=-0.11 or x=-2
Q6566) Solve x^2+23x+42=0
Ans: x=-2 or x=-21
Q6567) Solve 22x^2+38x+16=0
Ans: x=-0.73 or x=-1
Q6568) Solve 28x^2+43x+10=0
Ans: x=-0.29 or x=-1.25
Q6569) Solve 48x^2+34x+5=0
Ans: x=-0.21 or x=-0.5
Q6570) Solve 9x^2+30x+21=0
Ans: x=-1 or x=-2.33
Q6571) Solve 6x^2+29x+13=0
Ans: x=-0.5 or x=-4.33
Q6572) Solve 7x^2+47x+30=0
Ans: x=-0.71 or x=-6
Q6573) Solve 15x^2+29x+14=0
Ans: x=-0.93 or x=-1
Q6574) Solve 12x^2+47x+33=0
Ans: x=-0.92 or x=-3
Q6575) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q6576) Solve 4x^2+24x+27=0
Ans: x=-1.5 or x=-4.5
Q6577) Solve 12x^2+27x+6=0
Ans: x=-0.25 or x=-2
Q6578) Solve 4x^2+19x+15=0
Ans: x=-1 or x=-3.75
Q6579) Solve 28x^2+38x+12=0
Ans: x=-0.5 or x=-0.86
Q6580) Solve 10x^2+48x+32=0
Ans: x=-0.8 or x=-4
Q6581) Solve 18x^2+47x+30=0
Ans: x=-1.11 or x=-1.5
Q6582) Solve 32x^2+35x+3=0
Ans: x=-0.09 or x=-1
Q6583) Solve 5x^2+44x+39=0
Ans: x=-1 or x=-7.8
Q6584) Solve 17x^2+21x+4=0
Ans: x=-0.24 or x=-1
Q6585) Solve 17x^2+27x+10=0
Ans: x=-0.59 or x=-1
Q6586) Solve 22x^2+48x+26=0
Ans: x=-1 or x=-1.18
Q6587) Solve 6x^2+31x+39=0
Ans: x=-2.17 or x=-3
Q6588) Solve 9x^2+36x+35=0
Ans: x=-1.67 or x=-2.33
Q6589) Solve 10x^2+39x+27=0
Ans: x=-0.9 or x=-3
Q6590) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q6591) Solve 10x^2+43x+46=0
Ans: x=-2 or x=-2.3
Q6592) Solve 39x^2+43x+10=0
Ans: x=-0.33 or x=-0.77
Q6593) Solve 13x^2+32x+19=0
Ans: x=-1 or x=-1.46
Q6594) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q6595) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q6596) Solve 25x^2+28x+3=0
Ans: x=-0.12 or x=-1
Q6597) Solve 5x^2+49x+36=0
Ans: x=-0.8 or x=-9
Q6598) Solve 23x^2+47x+24=0
Ans: x=-1 or x=-1.04
Q6599) Solve 6x^2+37x+35=0
Ans: x=-1.17 or x=-5
Q6600) Solve 16x^2+47x+30=0
Ans: x=-0.94 or x=-2
Q6601) Solve 5x^2+37x+42=0
Ans: x=-1.4 or x=-6
Q6602) Solve 26x^2+36x+10=0
Ans: x=-0.38 or x=-1
Q6603) Solve 21x^2+40x+11=0
Ans: x=-0.33 or x=-1.57
Q6604) Solve 16x^2+32x+15=0
Ans: x=-0.75 or x=-1.25
Q6605) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q6606) Solve 25x^2+50x+21=0
Ans: x=-0.6 or x=-1.4
Q6607) Solve 10x^2+23x+12=0
Ans: x=-0.8 or x=-1.5
Q6608) Solve 15x^2+28x+13=0
Ans: x=-0.87 or x=-1
Q6609) Solve 14x^2+30x+16=0
Ans: x=-1 or x=-1.14
Q6610) Solve 9x^2+26x+16=0
Ans: x=-0.89 or x=-2
Q6611) Solve 13x^2+50x+33=0
Ans: x=-0.85 or x=-3
Q6612) Solve 9x^2+26x+17=0
Ans: x=-1 or x=-1.89
Q6613) Solve 6x^2+29x+9=0
Ans: x=-0.33 or x=-4.5
Q6614) Solve 8x^2+26x+18=0
Ans: x=-1 or x=-2.25
Q6615) Solve 10x^2+43x+46=0
Ans: x=-2 or x=-2.3
Q6616) Solve 15x^2+27x+12=0
Ans: x=-0.8 or x=-1
Q6617) Solve x^2+13x+42=0
Ans: x=-6 or x=-7
Q6618) Solve 9x^2+45x+36=0
Ans: x=-1 or x=-4
Q6619) Solve 16x^2+20x+4=0
Ans: x=-0.25 or x=-1
Q6620) Solve 4x^2+29x+25=0
Ans: x=-1 or x=-6.25
Q6621) Solve 4x^2+38x+34=0
Ans: x=-1 or x=-8.5
Q6622) Solve 12x^2+44x+32=0
Ans: x=-1 or x=-2.67
Q6623) Solve 5x^2+41x+42=0
Ans: x=-1.2 or x=-7
Q6624) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q6625) Solve 42x^2+34x+4=0
Ans: x=-0.14 or x=-0.67
Q6626) Solve 19x^2+25x+6=0
Ans: x=-0.32 or x=-1
Q6627) Solve 30x^2+22x+4=0
Ans: x=-0.33 or x=-0.4
Q6628) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q6629) Solve 36x^2+33x+7=0
Ans: x=-0.33 or x=-0.58
Q6630) Solve 34x^2+41x+7=0
Ans: x=-0.21 or x=-1
Q6631) Solve 27x^2+28x+1=0
Ans: x=-0.04 or x=-1
Q6632) Solve 13x^2+45x+18=0
Ans: x=-0.46 or x=-3
Q6633) Solve 24x^2+47x+23=0
Ans: x=-0.96 or x=-1
Q6634) Solve 6x^2+31x+18=0
Ans: x=-0.67 or x=-4.5
Q6635) Solve 28x^2+36x+8=0
Ans: x=-0.29 or x=-1
Q6636) Solve 8x^2+16x+6=0
Ans: x=-0.5 or x=-1.5
Q6637) Solve 3x^2+25x+50=0
Ans: x=-3.33 or x=-5
Q6638) Solve 27x^2+18x+3=0
Ans: x=-0.33 or x=-0.33
Q6639) Solve 24x^2+16x+2=0
Ans: x=-0.17 or x=-0.5
Q6640) Solve 12x^2+31x+9=0
Ans: x=-0.33 or x=-2.25
Q6641) Solve 5x^2+26x+33=0
Ans: x=-2.2 or x=-3
Q6642) Solve 49x^2+14x+1=0
Ans: x=-0.14 or x=-0.14
Q6643) Solve 15x^2+38x+23=0
Ans: x=-1 or x=-1.53
Q6644) Solve 3x^2+15x+18=0
Ans: x=-2 or x=-3
Q6645) Solve 41x^2+44x+3=0
Ans: x=-0.07 or x=-1
Q6646) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q6647) Solve 14x^2+25x+6=0
Ans: x=-0.29 or x=-1.5
Q6648) Solve 20x^2+49x+11=0
Ans: x=-0.25 or x=-2.2
Q6649) Solve 37x^2+43x+6=0
Ans: x=-0.16 or x=-1
Q6650) Solve 16x^2+38x+15=0
Ans: x=-0.5 or x=-1.87
Q6651) Solve 38x^2+29x+5=0
Ans: x=-0.26 or x=-0.5
Q6652) Solve 42x^2+29x+5=0
Ans: x=-0.33 or x=-0.36
Q6653) Solve 4x^2+46x+42=0
Ans: x=-1 or x=-10.5
Q6654) Solve 19x^2+49x+30=0
Ans: x=-1 or x=-1.58
Q6655) Solve 19x^2+33x+14=0
Ans: x=-0.74 or x=-1
Q6656) Solve 48x^2+31x+5=0
Ans: x=-0.31 or x=-0.33
Q6657) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q6658) Solve 44x^2+15x+1=0
Ans: x=-0.09 or x=-0.25
Q6659) Solve 26x^2+31x+5=0
Ans: x=-0.19 or x=-1
Q6660) Solve 37x^2+44x+7=0
Ans: x=-0.19 or x=-1
Q6661) Solve 12x^2+43x+35=0
Ans: x=-1.25 or x=-2.33
Q6662) Solve 14x^2+45x+36=0
Ans: x=-1.5 or x=-1.71
Q6663) Solve x^2+10x+9=0
Ans: x=-1 or x=-9
Q6664) Solve 38x^2+21x+1=0
Ans: x=-0.05 or x=-0.5
Q6665) Solve 9x^2+28x+3=0
Ans: x=-0.11 or x=-3
Q6666) Solve 12x^2+8x+1=0
Ans: x=-0.17 or x=-0.5
Q6667) Solve 33x^2+23x+4=0
Ans: x=-0.33 or x=-0.36
Q6668) Solve 48x^2+40x+8=0
Ans: x=-0.33 or x=-0.5
Q6669) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q6670) Solve 42x^2+20x+2=0
Ans: x=-0.14 or x=-0.33
Q6671) Solve 3x^2+48x+45=0
Ans: x=-1 or x=-15
Q6672) Solve 6x^2+37x+45=0
Ans: x=-1.67 or x=-4.5
Q6673) Solve 5x^2+9x+4=0
Ans: x=-0.8 or x=-1
Q6674) Solve 25x^2+33x+8=0
Ans: x=-0.32 or x=-1
Q6675) Solve 23x^2+41x+18=0
Ans: x=-0.78 or x=-1
Q6676) Solve 13x^2+14x+1=0
Ans: x=-0.08 or x=-1
Q6677) Solve 7x^2+19x+10=0
Ans: x=-0.71 or x=-2
Q6678) Solve 9x^2+19x+10=0
Ans: x=-1 or x=-1.11
Q6679) Solve 5x^2+34x+45=0
Ans: x=-1.8 or x=-5
Q6680) Solve 9x^2+37x+38=0
Ans: x=-2 or x=-2.11
Q6681) Solve 8x^2+41x+50=0
Ans: x=-2 or x=-3.12
Q6682) Solve 10x^2+18x+8=0
Ans: x=-0.8 or x=-1
Q6683) Solve x^2+6x+5=0
Ans: x=-1 or x=-5
Q6684) Solve 49x^2+21x+2=0
Ans: x=-0.14 or x=-0.29
Q6685) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q6686) Solve 12x^2+24x+12=0
Ans: x=-1 or x=-1
Q6687) Solve 36x^2+45x+11=0
Ans: x=-0.33 or x=-0.92
Q6688) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q6689) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q6690) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q6691) Solve 37x^2+38x+1=0
Ans: x=-0.03 or x=-1
Q6692) Solve 23x^2+46x+23=0
Ans: x=-1 or x=-1
Q6693) Solve 8x^2+48x+22=0
Ans: x=-0.5 or x=-5.5
Q6694) Solve 19x^2+36x+17=0
Ans: x=-0.89 or x=-1
Q6695) Solve 14x^2+34x+20=0
Ans: x=-1 or x=-1.43
Q6696) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q6697) Solve 9x^2+13x+4=0
Ans: x=-0.44 or x=-1
Q6698) Solve 22x^2+50x+28=0
Ans: x=-1 or x=-1.27
Q6699) Solve 2x^2+27x+25=0
Ans: x=-1 or x=-12.5
Q6700) Solve 5x^2+14x+8=0
Ans: x=-0.8 or x=-2
Q6701) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q6702) Solve 2x^2+13x+21=0
Ans: x=-3 or x=-3.5
Q6703) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q6704) Solve 20x^2+48x+19=0
Ans: x=-0.5 or x=-1.9
Q6705) Solve 8x^2+19x+11=0
Ans: x=-1 or x=-1.37
Q6706) Solve 14x^2+38x+24=0
Ans: x=-1 or x=-1.71
Q6707) Solve 9x^2+27x+18=0
Ans: x=-1 or x=-2
Q6708) Solve 31x^2+45x+14=0
Ans: x=-0.45 or x=-1
Q6709) Solve 3x^2+29x+40=0
Ans: x=-1.67 or x=-8
Q6710) Solve 25x^2+27x+2=0
Ans: x=-0.08 or x=-1
Q6711) Solve 48x^2+50x+2=0
Ans: x=-0.04 or x=-1
Q6712) Solve 4x^2+42x+38=0
Ans: x=-1 or x=-9.5
Q6713) Solve 18x^2+50x+28=0
Ans: x=-0.78 or x=-2
Q6714) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q6715) Solve 36x^2+25x+4=0
Ans: x=-0.25 or x=-0.44
Q6716) Solve 7x^2+20x+13=0
Ans: x=-1 or x=-1.86
Q6717) Solve 45x^2+44x+7=0
Ans: x=-0.2 or x=-0.78
Q6718) Solve 6x^2+31x+28=0
Ans: x=-1.17 or x=-4
Q6719) Solve 18x^2+49x+31=0
Ans: x=-1 or x=-1.72
Q6720) Solve 20x^2+32x+3=0
Ans: x=-0.1 or x=-1.5
Q6721) Solve 45x^2+29x+4=0
Ans: x=-0.2 or x=-0.44
Q6722) Solve 7x^2+39x+32=0
Ans: x=-1 or x=-4.57
Q6723) Solve 11x^2+42x+31=0
Ans: x=-1 or x=-2.82
Q6724) Solve x^2+12x+11=0
Ans: x=-1 or x=-11
Q6725) Solve 44x^2+48x+13=0
Ans: x=-0.5 or x=-0.59
Q6726) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q6727) Solve 18x^2+9x+1=0
Ans: x=-0.17 or x=-0.33
Q6728) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q6729) Solve 14x^2+20x+6=0
Ans: x=-0.43 or x=-1
Q6730) Solve x^2+14x+40=0
Ans: x=-4 or x=-10
Q6731) Solve 5x^2+29x+38=0
Ans: x=-2 or x=-3.8
Q6732) Solve 19x^2+40x+21=0
Ans: x=-1 or x=-1.11
Q6733) Solve 16x^2+42x+20=0
Ans: x=-0.62 or x=-2
Q6734) Solve 22x^2+49x+27=0
Ans: x=-1 or x=-1.23
Q6735) Solve 2x^2+28x+26=0
Ans: x=-1 or x=-13
Q6736) Solve 6x^2+41x+44=0
Ans: x=-1.33 or x=-5.5
Q6737) Solve 8x^2+44x+48=0
Ans: x=-1.5 or x=-4
Q6738) Solve 4x^2+26x+36=0
Ans: x=-2 or x=-4.5
Q6739) Solve 5x^2+41x+42=0
Ans: x=-1.2 or x=-7
Q6740) Solve x^2+18x+45=0
Ans: x=-3 or x=-15
Q6741) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q6742) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q6743) Solve 11x^2+45x+34=0
Ans: x=-1 or x=-3.09
Q6744) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q6745) Solve 30x^2+17x+2=0
Ans: x=-0.17 or x=-0.4
Q6746) Solve 9x^2+39x+40=0
Ans: x=-1.67 or x=-2.67
Q6747) Solve 23x^2+28x+5=0
Ans: x=-0.22 or x=-1
Q6748) Solve 16x^2+34x+18=0
Ans: x=-1 or x=-1.12
Q6749) Solve 38x^2+49x+11=0
Ans: x=-0.29 or x=-1
Q6750) Solve 18x^2+33x+9=0
Ans: x=-0.33 or x=-1.5
Q6751) Solve 31x^2+42x+11=0
Ans: x=-0.35 or x=-1
Q6752) Solve 6x^2+16x+10=0
Ans: x=-1 or x=-1.67
Q6753) Solve 46x^2+48x+2=0
Ans: x=-0.04 or x=-1
Q6754) Solve 35x^2+32x+5=0
Ans: x=-0.2 or x=-0.71
Q6755) Solve 36x^2+37x+1=0
Ans: x=-0.03 or x=-1
Q6756) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q6757) Solve 4x^2+23x+19=0
Ans: x=-1 or x=-4.75
Q6758) Solve 26x^2+33x+10=0
Ans: x=-0.5 or x=-0.77
Q6759) Solve 7x^2+17x+6=0
Ans: x=-0.43 or x=-2
Q6760) Solve 12x^2+31x+20=0
Ans: x=-1.25 or x=-1.33
Q6761) Solve 36x^2+40x+4=0
Ans: x=-0.11 or x=-1
Q6762) Solve 23x^2+41x+18=0
Ans: x=-0.78 or x=-1
Q6763) Solve 42x^2+41x+5=0
Ans: x=-0.14 or x=-0.83
Q6764) Solve x^2+27x+26=0
Ans: x=-1 or x=-26
Q6765) Solve 3x^2+27x+42=0
Ans: x=-2 or x=-7
Q6766) Solve 32x^2+38x+6=0
Ans: x=-0.19 or x=-1
Q6767) Solve 22x^2+48x+8=0
Ans: x=-0.18 or x=-2
Q6768) Solve 16x^2+30x+14=0
Ans: x=-0.87 or x=-1
Q6769) Solve 15x^2+18x+3=0
Ans: x=-0.2 or x=-1
Q6770) Solve 48x^2+50x+2=0
Ans: x=-0.04 or x=-1
Q6771) Solve 12x^2+19x+7=0
Ans: x=-0.58 or x=-1
Q6772) Solve 15x^2+38x+11=0
Ans: x=-0.33 or x=-2.2
Q6773) Solve 9x^2+44x+35=0
Ans: x=-1 or x=-3.89
Q6774) Solve 3x^2+26x+23=0
Ans: x=-1 or x=-7.67
Q6775) Solve 3x^2+22x+39=0
Ans: x=-3 or x=-4.33
Q6776) Solve 17x^2+45x+22=0
Ans: x=-0.65 or x=-2
Q6777) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q6778) Solve 14x^2+19x+5=0
Ans: x=-0.36 or x=-1
Q6779) Solve 42x^2+47x+10=0
Ans: x=-0.29 or x=-0.83
Q6780) Solve 35x^2+41x+12=0
Ans: x=-0.57 or x=-0.6
Q6781) Solve 21x^2+47x+20=0
Ans: x=-0.57 or x=-1.67
Q6782) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q6783) Solve 30x^2+49x+19=0
Ans: x=-0.63 or x=-1
Q6784) Solve 18x^2+33x+12=0
Ans: x=-0.5 or x=-1.33
Q6785) Solve 31x^2+33x+2=0
Ans: x=-0.06 or x=-1
Q6786) Solve 10x^2+34x+24=0
Ans: x=-1 or x=-2.4
Q6787) Solve 10x^2+28x+16=0
Ans: x=-0.8 or x=-2
Q6788) Solve 16x^2+31x+15=0
Ans: x=-0.94 or x=-1
Q6789) Solve 12x^2+33x+21=0
Ans: x=-1 or x=-1.75
Q6790) Solve 21x^2+31x+8=0
Ans: x=-0.33 or x=-1.14
Q6791) Solve 34x^2+39x+11=0
Ans: x=-0.5 or x=-0.65
Q6792) Solve 18x^2+43x+24=0
Ans: x=-0.89 or x=-1.5
Q6793) Solve 15x^2+37x+14=0
Ans: x=-0.47 or x=-2
Q6794) Solve 40x^2+49x+9=0
Ans: x=-0.22 or x=-1
Q6795) Solve 39x^2+44x+5=0
Ans: x=-0.13 or x=-1
Q6796) Solve 7x^2+31x+34=0
Ans: x=-2 or x=-2.43
Q6797) Solve x^2+19x+48=0
Ans: x=-3 or x=-16
Q6798) Solve 24x^2+32x+8=0
Ans: x=-0.33 or x=-1
Q6799) Solve x^2+16x+15=0
Ans: x=-1 or x=-15
Q6800) Solve 4x^2+24x+36=0
Ans: x=-3 or x=-3
Q6801) Solve x^2+15x+50=0
Ans: x=-5 or x=-10
Q6802) Solve 34x^2+37x+10=0
Ans: x=-0.5 or x=-0.59
Q6803) Solve 15x^2+11x+2=0
Ans: x=-0.33 or x=-0.4
Q6804) Solve 13x^2+40x+3=0
Ans: x=-0.08 or x=-3
Q6805) Solve 18x^2+47x+19=0
Ans: x=-0.5 or x=-2.11
Q6806) Solve 48x^2+44x+10=0
Ans: x=-0.42 or x=-0.5
Q6807) Solve 6x^2+16x+10=0
Ans: x=-1 or x=-1.67
Q6808) Solve 20x^2+21x+4=0
Ans: x=-0.25 or x=-0.8
Q6809) Solve 12x^2+27x+15=0
Ans: x=-1 or x=-1.25
Q6810) Solve 40x^2+31x+6=0
Ans: x=-0.37 or x=-0.4
Q6811) Solve 7x^2+39x+50=0
Ans: x=-2 or x=-3.57
Q6812) Solve x^2+19x+48=0
Ans: x=-3 or x=-16
Q6813) Solve 10x^2+39x+29=0
Ans: x=-1 or x=-2.9
Q6814) Solve 17x^2+39x+22=0
Ans: x=-1 or x=-1.29
Q6815) Solve 8x^2+41x+5=0
Ans: x=-0.12 or x=-5
Q6816) Solve 19x^2+29x+10=0
Ans: x=-0.53 or x=-1
Q6817) Solve 10x^2+48x+32=0
Ans: x=-0.8 or x=-4
Q6818) Solve 25x^2+40x+15=0
Ans: x=-0.6 or x=-1
Q6819) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q6820) Solve 28x^2+44x+16=0
Ans: x=-0.57 or x=-1
Q6821) Solve 7x^2+31x+34=0
Ans: x=-2 or x=-2.43
Q6822) Solve 42x^2+32x+6=0
Ans: x=-0.33 or x=-0.43
Q6823) Solve 8x^2+18x+4=0
Ans: x=-0.25 or x=-2
Q6824) Solve 7x^2+24x+9=0
Ans: x=-0.43 or x=-3
Q6825) Solve 23x^2+49x+26=0
Ans: x=-1 or x=-1.13
Q6826) Solve 38x^2+29x+5=0
Ans: x=-0.26 or x=-0.5
Q6827) Solve 6x^2+19x+3=0
Ans: x=-0.17 or x=-3
Q6828) Solve 18x^2+21x+3=0
Ans: x=-0.17 or x=-1
Q6829) Solve 3x^2+18x+27=0
Ans: x=-3 or x=-3
Q6830) Solve 9x^2+35x+26=0
Ans: x=-1 or x=-2.89
Q6831) Solve 10x^2+44x+42=0
Ans: x=-1.4 or x=-3
Q6832) Solve 40x^2+22x+3=0
Ans: x=-0.25 or x=-0.3
Q6833) Solve 21x^2+41x+10=0
Ans: x=-0.29 or x=-1.67
Q6834) Solve 2x^2+42x+40=0
Ans: x=-1 or x=-20
Q6835) Solve 16x^2+24x+9=0
Ans: x=-0.75 or x=-0.75
Q6836) Solve 16x^2+35x+6=0
Ans: x=-0.19 or x=-2
Q6837) Solve 15x^2+11x+2=0
Ans: x=-0.33 or x=-0.4
Q6838) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q6839) Solve 3x^2+23x+14=0
Ans: x=-0.67 or x=-7
Q6840) Solve 16x^2+39x+23=0
Ans: x=-1 or x=-1.44
Q6841) Solve 13x^2+16x+3=0
Ans: x=-0.23 or x=-1
Q6842) Solve 12x^2+38x+16=0
Ans: x=-0.5 or x=-2.67
Q6843) Solve 8x^2+42x+34=0
Ans: x=-1 or x=-4.25
Q6844) Solve 35x^2+48x+13=0
Ans: x=-0.37 or x=-1
Q6845) Solve 14x^2+39x+16=0
Ans: x=-0.5 or x=-2.29
Q6846) Solve 7x^2+31x+24=0
Ans: x=-1 or x=-3.43
Q6847) Solve 36x^2+29x+5=0
Ans: x=-0.25 or x=-0.56
Q6848) Solve x^2+16x+39=0
Ans: x=-3 or x=-13
Q6849) Solve 41x^2+45x+4=0
Ans: x=-0.1 or x=-1
Q6850) Solve 25x^2+40x+12=0
Ans: x=-0.4 or x=-1.2
Q6851) Solve 6x^2+38x+12=0
Ans: x=-0.33 or x=-6
Q6852) Solve 16x^2+41x+25=0
Ans: x=-1 or x=-1.56
Q6853) Solve 16x^2+24x+5=0
Ans: x=-0.25 or x=-1.25
Q6854) Solve 23x^2+24x+1=0
Ans: x=-0.04 or x=-1
Q6855) Solve 15x^2+31x+10=0
Ans: x=-0.4 or x=-1.67
Q6856) Solve 12x^2+43x+31=0
Ans: x=-1 or x=-2.58
Q6857) Solve 19x^2+47x+18=0
Ans: x=-0.47 or x=-2
Q6858) Solve 27x^2+48x+20=0
Ans: x=-0.67 or x=-1.11
Q6859) Solve 3x^2+10x+7=0
Ans: x=-1 or x=-2.33
Q6860) Solve 40x^2+50x+10=0
Ans: x=-0.25 or x=-1
Q6861) Solve 40x^2+44x+4=0
Ans: x=-0.1 or x=-1
Q6862) Solve 13x^2+21x+8=0
Ans: x=-0.62 or x=-1
Q6863) Solve 10x^2+49x+36=0
Ans: x=-0.9 or x=-4
Q6864) Solve 4x^2+43x+39=0
Ans: x=-1 or x=-9.75
Q6865) Solve 18x^2+21x+3=0
Ans: x=-0.17 or x=-1
Q6866) Solve 8x^2+31x+23=0
Ans: x=-1 or x=-2.87
Q6867) Solve 36x^2+30x+4=0
Ans: x=-0.17 or x=-0.67
Q6868) Solve 8x^2+36x+40=0
Ans: x=-2 or x=-2.5
Q6869) Solve 12x^2+30x+12=0
Ans: x=-0.5 or x=-2
Q6870) Solve 30x^2+43x+11=0
Ans: x=-0.33 or x=-1.1
Q6871) Solve 4x^2+17x+18=0
Ans: x=-2 or x=-2.25
Q6872) Solve 9x^2+25x+16=0
Ans: x=-1 or x=-1.78
Q6873) Solve 24x^2+50x+4=0
Ans: x=-0.08 or x=-2
Q6874) Solve 46x^2+41x+9=0
Ans: x=-0.39 or x=-0.5
Q6875) Solve 8x^2+29x+15=0
Ans: x=-0.62 or x=-3
Q6876) Solve 36x^2+42x+12=0
Ans: x=-0.5 or x=-0.67
Q6877) Solve 12x^2+11x+2=0
Ans: x=-0.25 or x=-0.67
Q6878) Solve 14x^2+25x+6=0
Ans: x=-0.29 or x=-1.5
Q6879) Solve 14x^2+17x+3=0
Ans: x=-0.21 or x=-1
Q6880) Solve 11x^2+23x+2=0
Ans: x=-0.09 or x=-2
Q6881) Solve 42x^2+49x+14=0
Ans: x=-0.5 or x=-0.67
Q6882) Solve 34x^2+41x+12=0
Ans: x=-0.5 or x=-0.71
Q6883) Solve 25x^2+27x+2=0
Ans: x=-0.08 or x=-1
Q6884) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q6885) Solve 4x^2+15x+9=0
Ans: x=-0.75 or x=-3
Q6886) Solve 27x^2+46x+19=0
Ans: x=-0.7 or x=-1
Q6887) Solve 18x^2+40x+8=0
Ans: x=-0.22 or x=-2
Q6888) Solve 7x^2+25x+18=0
Ans: x=-1 or x=-2.57
Q6889) Solve 12x^2+18x+6=0
Ans: x=-0.5 or x=-1
Q6890) Solve 43x^2+50x+7=0
Ans: x=-0.16 or x=-1
Q6891) Solve 15x^2+48x+36=0
Ans: x=-1.2 or x=-2
Q6892) Solve x^2+19x+18=0
Ans: x=-1 or x=-18
Q6893) Solve 39x^2+50x+16=0
Ans: x=-0.62 or x=-0.67
Q6894) Solve 12x^2+31x+9=0
Ans: x=-0.33 or x=-2.25
Q6895) Solve 5x^2+38x+21=0
Ans: x=-0.6 or x=-7
Q6896) Solve 9x^2+29x+20=0
Ans: x=-1 or x=-2.22
Q6897) Solve 20x^2+31x+12=0
Ans: x=-0.75 or x=-0.8
Q6898) Solve 33x^2+31x+6=0
Ans: x=-0.27 or x=-0.67
Q6899) Solve 8x^2+50x+33=0
Ans: x=-0.75 or x=-5.5
Q6900) Solve 13x^2+31x+10=0
Ans: x=-0.38 or x=-2
Q6901) Solve 6x^2+12x+6=0
Ans: x=-1 or x=-1
Q6902) Solve 9x^2+41x+20=0
Ans: x=-0.56 or x=-4
Q6903) Solve x^2+26x+25=0
Ans: x=-1 or x=-25
Q6904) Solve 44x^2+39x+7=0
Ans: x=-0.25 or x=-0.64
Q6905) Solve 4x^2+21x+27=0
Ans: x=-2.25 or x=-3
Q6906) Solve 9x^2+36x+20=0
Ans: x=-0.67 or x=-3.33
Q6907) Solve 5x^2+20x+15=0
Ans: x=-1 or x=-3
Q6908) Solve 36x^2+26x+4=0
Ans: x=-0.22 or x=-0.5
Q6909) Solve 31x^2+37x+6=0
Ans: x=-0.19 or x=-1
Q6910) Solve 9x^2+20x+4=0
Ans: x=-0.22 or x=-2
Q6911) Solve 42x^2+50x+12=0
Ans: x=-0.33 or x=-0.86
Q6912) Solve 12x^2+41x+22=0
Ans: x=-0.67 or x=-2.75
Q6913) Solve 40x^2+42x+2=0
Ans: x=-0.05 or x=-1
Q6914) Solve 5x^2+28x+32=0
Ans: x=-1.6 or x=-4
Q6915) Solve 19x^2+45x+26=0
Ans: x=-1 or x=-1.37
Q6916) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q6917) Solve x^2+4x+4=0
Ans: x=-2 or x=-2
Q6918) Solve 8x^2+46x+45=0
Ans: x=-1.25 or x=-4.5
Q6919) Solve 10x^2+24x+8=0
Ans: x=-0.4 or x=-2
Q6920) Solve 10x^2+48x+38=0
Ans: x=-1 or x=-3.8
Q6921) Solve 29x^2+31x+2=0
Ans: x=-0.07 or x=-1
Q6922) Solve 11x^2+15x+4=0
Ans: x=-0.36 or x=-1
Q6923) Solve 13x^2+42x+32=0
Ans: x=-1.23 or x=-2
Q6924) Solve 6x^2+35x+46=0
Ans: x=-2 or x=-3.83
Q6925) Solve 25x^2+40x+7=0
Ans: x=-0.2 or x=-1.4
Q6926) Solve 7x^2+33x+38=0
Ans: x=-2 or x=-2.71
Q6927) Solve 7x^2+32x+25=0
Ans: x=-1 or x=-3.57
Q6928) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q6929) Solve 20x^2+17x+3=0
Ans: x=-0.25 or x=-0.6
Q6930) Solve 3x^2+47x+44=0
Ans: x=-1 or x=-14.67
Q6931) Solve 11x^2+46x+8=0
Ans: x=-0.18 or x=-4
Q6932) Solve 44x^2+38x+8=0
Ans: x=-0.36 or x=-0.5
Q6933) Solve 4x^2+20x+25=0
Ans: x=-2.5 or x=-2.5
Q6934) Solve 27x^2+42x+16=0
Ans: x=-0.67 or x=-0.89
Q6935) Solve 4x^2+21x+26=0
Ans: x=-2 or x=-3.25
Q6936) Solve 15x^2+43x+28=0
Ans: x=-1 or x=-1.87
Q6937) Solve 8x^2+6x+1=0
Ans: x=-0.25 or x=-0.5
Q6938) Solve 21x^2+46x+13=0
Ans: x=-0.33 or x=-1.86
Q6939) Solve x^2+12x+20=0
Ans: x=-2 or x=-10
Q6940) Solve 14x^2+45x+36=0
Ans: x=-1.5 or x=-1.71
Q6941) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q6942) Solve 39x^2+44x+5=0
Ans: x=-0.13 or x=-1
Q6943) Solve 11x^2+47x+42=0
Ans: x=-1.27 or x=-3
Q6944) Solve 6x^2+28x+16=0
Ans: x=-0.67 or x=-4
Q6945) Solve 3x^2+19x+30=0
Ans: x=-3 or x=-3.33
Q6946) Solve 9x^2+36x+20=0
Ans: x=-0.67 or x=-3.33
Q6947) Solve 46x^2+47x+1=0
Ans: x=-0.02 or x=-1
Q6948) Solve 12x^2+42x+18=0
Ans: x=-0.5 or x=-3
Q6949) Solve 8x^2+17x+9=0
Ans: x=-1 or x=-1.12
Q6950) Solve 45x^2+44x+7=0
Ans: x=-0.2 or x=-0.78
Q6951) Solve 46x^2+29x+3=0
Ans: x=-0.13 or x=-0.5
Q6952) Solve 46x^2+39x+8=0
Ans: x=-0.35 or x=-0.5
Q6953) Solve 32x^2+46x+15=0
Ans: x=-0.5 or x=-0.94
Q6954) Solve 38x^2+46x+8=0
Ans: x=-0.21 or x=-1
Q6955) Solve 23x^2+38x+15=0
Ans: x=-0.65 or x=-1
Q6956) Solve 23x^2+46x+23=0
Ans: x=-1 or x=-1
Q6957) Solve 48x^2+19x+1=0
Ans: x=-0.06 or x=-0.33
Q6958) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q6959) Solve 6x^2+19x+10=0
Ans: x=-0.67 or x=-2.5
Q6960) Solve 46x^2+49x+3=0
Ans: x=-0.07 or x=-1
Q6961) Solve 12x^2+31x+14=0
Ans: x=-0.58 or x=-2
Q6962) Solve 9x^2+21x+6=0
Ans: x=-0.33 or x=-2
Q6963) Solve 42x^2+31x+2=0
Ans: x=-0.07 or x=-0.67
Q6964) Solve 15x^2+40x+25=0
Ans: x=-1 or x=-1.67
Q6965) Solve 34x^2+33x+8=0
Ans: x=-0.47 or x=-0.5
Q6966) Solve 25x^2+32x+7=0
Ans: x=-0.28 or x=-1
Q6967) Solve 45x^2+42x+9=0
Ans: x=-0.33 or x=-0.6
Q6968) Solve 10x^2+39x+17=0
Ans: x=-0.5 or x=-3.4
Q6969) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q6970) Solve 11x^2+43x+30=0
Ans: x=-0.91 or x=-3
Q6971) Solve 22x^2+35x+3=0
Ans: x=-0.09 or x=-1.5
Q6972) Solve 14x^2+25x+9=0
Ans: x=-0.5 or x=-1.29
Q6973) Solve 26x^2+27x+7=0
Ans: x=-0.5 or x=-0.54
Q6974) Solve 10x^2+43x+39=0
Ans: x=-1.3 or x=-3
Q6975) Solve 13x^2+16x+3=0
Ans: x=-0.23 or x=-1
Q6976) Solve 7x^2+14x+7=0
Ans: x=-1 or x=-1
Q6977) Solve 15x^2+39x+24=0
Ans: x=-1 or x=-1.6
Q6978) Solve 17x^2+42x+16=0
Ans: x=-0.47 or x=-2
Q6979) Solve 21x^2+38x+16=0
Ans: x=-0.67 or x=-1.14
Q6980) Solve 48x^2+24x+3=0
Ans: x=-0.25 or x=-0.25
Q6981) Solve 24x^2+14x+1=0
Ans: x=-0.08 or x=-0.5
Q6982) Solve 8x^2+41x+50=0
Ans: x=-2 or x=-3.12
Q6983) Solve 15x^2+23x+4=0
Ans: x=-0.2 or x=-1.33
Q6984) Solve 28x^2+22x+4=0
Ans: x=-0.29 or x=-0.5
Q6985) Solve 7x^2+17x+10=0
Ans: x=-1 or x=-1.43
Q6986) Solve 30x^2+23x+2=0
Ans: x=-0.1 or x=-0.67
Q6987) Solve 10x^2+21x+9=0
Ans: x=-0.6 or x=-1.5
Q6988) Solve 20x^2+43x+21=0
Ans: x=-0.75 or x=-1.4
Q6989) Solve 13x^2+48x+44=0
Ans: x=-1.69 or x=-2
Q6990) Solve 28x^2+43x+10=0
Ans: x=-0.29 or x=-1.25
Q6991) Solve 18x^2+38x+20=0
Ans: x=-1 or x=-1.11
Q6992) Solve 14x^2+41x+17=0
Ans: x=-0.5 or x=-2.43
Q6993) Solve 3x^2+11x+10=0
Ans: x=-1.67 or x=-2
Q6994) Solve 4x^2+25x+25=0
Ans: x=-1.25 or x=-5
Q6995) Solve 44x^2+45x+9=0
Ans: x=-0.27 or x=-0.75
Q6996) Solve 17x^2+45x+22=0
Ans: x=-0.65 or x=-2
Q6997) Solve 9x^2+16x+7=0
Ans: x=-0.78 or x=-1
Q6998) Solve 5x^2+34x+45=0
Ans: x=-1.8 or x=-5
Q6999) Solve 18x^2+46x+20=0
Ans: x=-0.56 or x=-2
Q7000) Solve 7x^2+16x+9=0
Ans: x=-1 or x=-1.29
Q7001) Solve 2x^2+23x+30=0
Ans: x=-1.5 or x=-10
Q7002) Solve 33x^2+49x+18=0
Ans: x=-0.67 or x=-0.82
Q7003) Solve 8x^2+38x+9=0
Ans: x=-0.25 or x=-4.5
Q7004) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q7005) Solve 22x^2+41x+19=0
Ans: x=-0.86 or x=-1
Q7006) Solve 4x^2+22x+30=0
Ans: x=-2.5 or x=-3
Q7007) Solve 5x^2+32x+12=0
Ans: x=-0.4 or x=-6
Q7008) Solve 40x^2+33x+5=0
Ans: x=-0.2 or x=-0.62
Q7009) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q7010) Solve 2x^2+32x+30=0
Ans: x=-1 or x=-15
Q7011) Solve 50x^2+45x+10=0
Ans: x=-0.4 or x=-0.5
Q7012) Solve 46x^2+48x+2=0
Ans: x=-0.04 or x=-1
Q7013) Solve 30x^2+32x+8=0
Ans: x=-0.4 or x=-0.67
Q7014) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q7015) Solve 8x^2+18x+9=0
Ans: x=-0.75 or x=-1.5
Q7016) Solve 26x^2+17x+2=0
Ans: x=-0.15 or x=-0.5
Q7017) Solve 9x^2+37x+4=0
Ans: x=-0.11 or x=-4
Q7018) Solve 22x^2+45x+23=0
Ans: x=-1 or x=-1.05
Q7019) Solve 21x^2+22x+1=0
Ans: x=-0.05 or x=-1
Q7020) Solve 10x^2+17x+6=0
Ans: x=-0.5 or x=-1.2
Q7021) Solve 4x^2+22x+30=0
Ans: x=-2.5 or x=-3
Q7022) Solve 7x^2+44x+45=0
Ans: x=-1.29 or x=-5
Q7023) Solve 10x^2+33x+20=0
Ans: x=-0.8 or x=-2.5
Q7024) Solve 39x^2+47x+14=0
Ans: x=-0.54 or x=-0.67
Q7025) Solve 10x^2+40x+40=0
Ans: x=-2 or x=-2
Q7026) Solve 12x^2+31x+7=0
Ans: x=-0.25 or x=-2.33
Q7027) Solve 14x^2+43x+29=0
Ans: x=-1 or x=-2.07
Q7028) Solve 3x^2+17x+24=0
Ans: x=-2.67 or x=-3
Q7029) Solve 25x^2+40x+12=0
Ans: x=-0.4 or x=-1.2
Q7030) Solve 13x^2+45x+18=0
Ans: x=-0.46 or x=-3
Q7031) Solve 5x^2+39x+34=0
Ans: x=-1 or x=-6.8
Q7032) Solve 10x^2+37x+7=0
Ans: x=-0.2 or x=-3.5
Q7033) Solve 9x^2+45x+44=0
Ans: x=-1.33 or x=-3.67
Q7034) Solve 35x^2+27x+4=0
Ans: x=-0.2 or x=-0.57
Q7035) Solve 8x^2+42x+10=0
Ans: x=-0.25 or x=-5
Q7036) Solve 14x^2+46x+32=0
Ans: x=-1 or x=-2.29
Q7037) Solve 19x^2+27x+8=0
Ans: x=-0.42 or x=-1
Q7038) Solve 5x^2+15x+10=0
Ans: x=-1 or x=-2
Q7039) Solve 4x^2+25x+39=0
Ans: x=-3 or x=-3.25
Q7040) Solve 45x^2+42x+9=0
Ans: x=-0.33 or x=-0.6
Q7041) Solve 12x^2+50x+42=0
Ans: x=-1.17 or x=-3
Q7042) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q7043) Solve 24x^2+34x+7=0
Ans: x=-0.25 or x=-1.17
Q7044) Solve 3x^2+18x+27=0
Ans: x=-3 or x=-3
Q7045) Solve 18x^2+39x+15=0
Ans: x=-0.5 or x=-1.67
Q7046) Solve 18x^2+27x+4=0
Ans: x=-0.17 or x=-1.33
Q7047) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q7048) Solve 6x^2+46x+28=0
Ans: x=-0.67 or x=-7
Q7049) Solve 12x^2+28x+15=0
Ans: x=-0.83 or x=-1.5
Q7050) Solve 5x^2+16x+12=0
Ans: x=-1.2 or x=-2
Q7051) Solve 39x^2+22x+3=0
Ans: x=-0.23 or x=-0.33
Q7052) Solve 21x^2+47x+22=0
Ans: x=-0.67 or x=-1.57
Q7053) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q7054) Solve 19x^2+36x+17=0
Ans: x=-0.89 or x=-1
Q7055) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q7056) Solve 26x^2+49x+15=0
Ans: x=-0.38 or x=-1.5
Q7057) Solve 6x^2+25x+19=0
Ans: x=-1 or x=-3.17
Q7058) Solve 9x^2+18x+8=0
Ans: x=-0.67 or x=-1.33
Q7059) Solve 48x^2+38x+5=0
Ans: x=-0.17 or x=-0.62
Q7060) Solve 10x^2+46x+48=0
Ans: x=-1.6 or x=-3
Q7061) Solve 24x^2+48x+24=0
Ans: x=-1 or x=-1
Q7062) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q7063) Solve 12x^2+44x+35=0
Ans: x=-1.17 or x=-2.5
Q7064) Solve 28x^2+48x+20=0
Ans: x=-0.71 or x=-1
Q7065) Solve 44x^2+37x+3=0
Ans: x=-0.09 or x=-0.75
Q7066) Solve 8x^2+24x+18=0
Ans: x=-1.5 or x=-1.5
Q7067) Solve 40x^2+24x+2=0
Ans: x=-0.1 or x=-0.5
Q7068) Solve 5x^2+28x+39=0
Ans: x=-2.6 or x=-3
Q7069) Solve 9x^2+16x+7=0
Ans: x=-0.78 or x=-1
Q7070) Solve x^2+40x+39=0
Ans: x=-1 or x=-39
Q7071) Solve 13x^2+31x+10=0
Ans: x=-0.38 or x=-2
Q7072) Solve 14x^2+36x+22=0
Ans: x=-1 or x=-1.57
Q7073) Solve 6x^2+16x+8=0
Ans: x=-0.67 or x=-2
Q7074) Solve 19x^2+41x+6=0
Ans: x=-0.16 or x=-2
Q7075) Solve 3x^2+23x+34=0
Ans: x=-2 or x=-5.67
Q7076) Solve 13x^2+37x+24=0
Ans: x=-1 or x=-1.85
Q7077) Solve 24x^2+50x+4=0
Ans: x=-0.08 or x=-2
Q7078) Solve 33x^2+34x+8=0
Ans: x=-0.36 or x=-0.67
Q7079) Solve 12x^2+49x+44=0
Ans: x=-1.33 or x=-2.75
Q7080) Solve 3x^2+20x+32=0
Ans: x=-2.67 or x=-4
Q7081) Solve 5x^2+12x+7=0
Ans: x=-1 or x=-1.4
Q7082) Solve 4x^2+9x+2=0
Ans: x=-0.25 or x=-2
Q7083) Solve 32x^2+44x+15=0
Ans: x=-0.62 or x=-0.75
Q7084) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q7085) Solve x^2+8x+16=0
Ans: x=-4 or x=-4
Q7086) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q7087) Solve 6x^2+39x+45=0
Ans: x=-1.5 or x=-5
Q7088) Solve 36x^2+36x+8=0
Ans: x=-0.33 or x=-0.67
Q7089) Solve 6x^2+25x+25=0
Ans: x=-1.67 or x=-2.5
Q7090) Solve 12x^2+25x+12=0
Ans: x=-0.75 or x=-1.33
Q7091) Solve 20x^2+47x+27=0
Ans: x=-1 or x=-1.35
Q7092) Solve 14x^2+33x+19=0
Ans: x=-1 or x=-1.36
Q7093) Solve 24x^2+50x+21=0
Ans: x=-0.58 or x=-1.5
Q7094) Solve 22x^2+39x+9=0
Ans: x=-0.27 or x=-1.5
Q7095) Solve 33x^2+41x+8=0
Ans: x=-0.24 or x=-1
Q7096) Solve 2x^2+25x+23=0
Ans: x=-1 or x=-11.5
Q7097) Solve 8x^2+38x+30=0
Ans: x=-1 or x=-3.75
Q7098) Solve 2x^2+17x+35=0
Ans: x=-3.5 or x=-5
Q7099) Solve 9x^2+30x+24=0
Ans: x=-1.33 or x=-2
Q7100) Solve 37x^2+40x+3=0
Ans: x=-0.08 or x=-1
Q7101) Solve 25x^2+20x+4=0
Ans: x=-0.4 or x=-0.4
Q7102) Solve 14x^2+27x+9=0
Ans: x=-0.43 or x=-1.5
Q7103) Solve 12x^2+48x+36=0
Ans: x=-1 or x=-3
Q7104) Solve 25x^2+50x+25=0
Ans: x=-1 or x=-1
Q7105) Solve 4x^2+24x+35=0
Ans: x=-2.5 or x=-3.5
Q7106) Solve 39x^2+50x+11=0
Ans: x=-0.28 or x=-1
Q7107) Solve 2x^2+24x+22=0
Ans: x=-1 or x=-11
Q7108) Solve 32x^2+36x+7=0
Ans: x=-0.25 or x=-0.87
Q7109) Solve 22x^2+50x+12=0
Ans: x=-0.27 or x=-2
Q7110) Solve 4x^2+6x+2=0
Ans: x=-0.5 or x=-1
Q7111) Solve 2x^2+16x+24=0
Ans: x=-2 or x=-6
Q7112) Solve 49x^2+49x+6=0
Ans: x=-0.14 or x=-0.86
Q7113) Solve x^2+14x+40=0
Ans: x=-4 or x=-10
Q7114) Solve 13x^2+50x+33=0
Ans: x=-0.85 or x=-3
Q7115) Solve 18x^2+47x+30=0
Ans: x=-1.11 or x=-1.5
Q7116) Solve 15x^2+31x+16=0
Ans: x=-1 or x=-1.07
Q7117) Solve 32x^2+50x+17=0
Ans: x=-0.5 or x=-1.06
Q7118) Solve 28x^2+36x+8=0
Ans: x=-0.29 or x=-1
Q7119) Solve 3x^2+12x+9=0
Ans: x=-1 or x=-3
Q7120) Solve 24x^2+29x+5=0
Ans: x=-0.21 or x=-1
Q7121) Solve 14x^2+27x+13=0
Ans: x=-0.93 or x=-1
Q7122) Solve 14x^2+34x+20=0
Ans: x=-1 or x=-1.43
Q7123) Solve x^2+13x+12=0
Ans: x=-1 or x=-12
Q7124) Solve 7x^2+47x+30=0
Ans: x=-0.71 or x=-6
Q7125) Solve 43x^2+47x+4=0
Ans: x=-0.09 or x=-1
Q7126) Solve 14x^2+37x+24=0
Ans: x=-1.14 or x=-1.5
Q7127) Solve 23x^2+30x+7=0
Ans: x=-0.3 or x=-1
Q7128) Solve 14x^2+49x+35=0
Ans: x=-1 or x=-2.5
Q7129) Solve 15x^2+36x+21=0
Ans: x=-1 or x=-1.4
Q7130) Solve 44x^2+38x+8=0
Ans: x=-0.36 or x=-0.5
Q7131) Solve 12x^2+38x+16=0
Ans: x=-0.5 or x=-2.67
Q7132) Solve 10x^2+41x+21=0
Ans: x=-0.6 or x=-3.5
Q7133) Solve 22x^2+45x+18=0
Ans: x=-0.55 or x=-1.5
Q7134) Solve 42x^2+26x+4=0
Ans: x=-0.29 or x=-0.33
Q7135) Solve 24x^2+46x+17=0
Ans: x=-0.5 or x=-1.42
Q7136) Solve 12x^2+46x+34=0
Ans: x=-1 or x=-2.83
Q7137) Solve 11x^2+36x+25=0
Ans: x=-1 or x=-2.27
Q7138) Solve 20x^2+39x+19=0
Ans: x=-0.95 or x=-1
Q7139) Solve 26x^2+37x+11=0
Ans: x=-0.42 or x=-1
Q7140) Solve 8x^2+42x+49=0
Ans: x=-1.75 or x=-3.5
Q7141) Solve 10x^2+17x+3=0
Ans: x=-0.2 or x=-1.5
Q7142) Solve 49x^2+14x+1=0
Ans: x=-0.14 or x=-0.14
Q7143) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q7144) Solve 24x^2+44x+12=0
Ans: x=-0.33 or x=-1.5
Q7145) Solve 19x^2+44x+25=0
Ans: x=-1 or x=-1.32
Q7146) Solve 36x^2+48x+15=0
Ans: x=-0.5 or x=-0.83
Q7147) Solve 12x^2+28x+11=0
Ans: x=-0.5 or x=-1.83
Q7148) Solve 7x^2+11x+4=0
Ans: x=-0.57 or x=-1
Q7149) Solve 14x^2+29x+11=0
Ans: x=-0.5 or x=-1.57
Q7150) Solve 8x^2+16x+6=0
Ans: x=-0.5 or x=-1.5
Q7151) Solve 19x^2+42x+8=0
Ans: x=-0.21 or x=-2
Q7152) Solve 6x^2+22x+16=0
Ans: x=-1 or x=-2.67
Q7153) Solve 50x^2+49x+12=0
Ans: x=-0.48 or x=-0.5
Q7154) Solve 30x^2+43x+4=0
Ans: x=-0.1 or x=-1.33
Q7155) Solve 22x^2+37x+6=0
Ans: x=-0.18 or x=-1.5
Q7156) Solve 25x^2+30x+9=0
Ans: x=-0.6 or x=-0.6
Q7157) Solve 6x^2+32x+10=0
Ans: x=-0.33 or x=-5
Q7158) Solve 8x^2+25x+17=0
Ans: x=-1 or x=-2.12
Q7159) Solve 6x^2+25x+11=0
Ans: x=-0.5 or x=-3.67
Q7160) Solve 20x^2+27x+7=0
Ans: x=-0.35 or x=-1
Q7161) Solve 17x^2+41x+14=0
Ans: x=-0.41 or x=-2
Q7162) Solve 21x^2+50x+29=0
Ans: x=-1 or x=-1.38
Q7163) Solve 35x^2+36x+9=0
Ans: x=-0.43 or x=-0.6
Q7164) Solve 5x^2+26x+5=0
Ans: x=-0.2 or x=-5
Q7165) Solve 8x^2+25x+17=0
Ans: x=-1 or x=-2.12
Q7166) Solve 12x^2+27x+6=0
Ans: x=-0.25 or x=-2
Q7167) Solve 17x^2+43x+18=0
Ans: x=-0.53 or x=-2
Q7168) Solve 5x^2+32x+12=0
Ans: x=-0.4 or x=-6
Q7169) Solve 9x^2+41x+46=0
Ans: x=-2 or x=-2.56
Q7170) Solve 50x^2+33x+4=0
Ans: x=-0.16 or x=-0.5
Q7171) Solve 2x^2+27x+25=0
Ans: x=-1 or x=-12.5
Q7172) Solve 12x^2+44x+35=0
Ans: x=-1.17 or x=-2.5
Q7173) Solve 6x^2+31x+33=0
Ans: x=-1.5 or x=-3.67
Q7174) Solve 15x^2+34x+8=0
Ans: x=-0.27 or x=-2
Q7175) Solve 40x^2+32x+6=0
Ans: x=-0.3 or x=-0.5
Q7176) Solve 6x^2+23x+15=0
Ans: x=-0.83 or x=-3
Q7177) Solve 4x^2+36x+45=0
Ans: x=-1.5 or x=-7.5
Q7178) Solve 6x^2+28x+30=0
Ans: x=-1.67 or x=-3
Q7179) Solve 11x^2+23x+12=0
Ans: x=-1 or x=-1.09
Q7180) Solve 11x^2+44x+33=0
Ans: x=-1 or x=-3
Q7181) Solve 42x^2+31x+2=0
Ans: x=-0.07 or x=-0.67
Q7182) Solve 15x^2+19x+6=0
Ans: x=-0.6 or x=-0.67
Q7183) Solve 9x^2+24x+16=0
Ans: x=-1.33 or x=-1.33
Q7184) Solve 22x^2+34x+12=0
Ans: x=-0.55 or x=-1
Q7185) Solve 28x^2+43x+15=0
Ans: x=-0.54 or x=-1
Q7186) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q7187) Solve 48x^2+42x+9=0
Ans: x=-0.37 or x=-0.5
Q7188) Solve 12x^2+38x+6=0
Ans: x=-0.17 or x=-3
Q7189) Solve 16x^2+50x+39=0
Ans: x=-1.5 or x=-1.62
Q7190) Solve 8x^2+27x+19=0
Ans: x=-1 or x=-2.37
Q7191) Solve 14x^2+32x+8=0
Ans: x=-0.29 or x=-2
Q7192) Solve 10x^2+39x+36=0
Ans: x=-1.5 or x=-2.4
Q7193) Solve 12x^2+28x+16=0
Ans: x=-1 or x=-1.33
Q7194) Solve 5x^2+22x+21=0
Ans: x=-1.4 or x=-3
Q7195) Solve 4x^2+13x+9=0
Ans: x=-1 or x=-2.25
Q7196) Solve 16x^2+24x+8=0
Ans: x=-0.5 or x=-1
Q7197) Solve 18x^2+30x+8=0
Ans: x=-0.33 or x=-1.33
Q7198) Solve 11x^2+43x+42=0
Ans: x=-1.91 or x=-2
Q7199) Solve 4x^2+29x+45=0
Ans: x=-2.25 or x=-5
Q7200) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q7201) Solve 13x^2+44x+31=0
Ans: x=-1 or x=-2.38
Q7202) Solve 4x^2+25x+36=0
Ans: x=-2.25 or x=-4
Q7203) Solve 3x^2+25x+50=0
Ans: x=-3.33 or x=-5
Q7204) Solve 36x^2+50x+14=0
Ans: x=-0.39 or x=-1
Q7205) Solve 2x^2+20x+32=0
Ans: x=-2 or x=-8
Q7206) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q7207) Solve 9x^2+35x+34=0
Ans: x=-1.89 or x=-2
Q7208) Solve 20x^2+41x+9=0
Ans: x=-0.25 or x=-1.8
Q7209) Solve 6x^2+30x+36=0
Ans: x=-2 or x=-3
Q7210) Solve 6x^2+33x+27=0
Ans: x=-1 or x=-4.5
Q7211) Solve 42x^2+31x+5=0
Ans: x=-0.24 or x=-0.5
Q7212) Solve 16x^2+40x+9=0
Ans: x=-0.25 or x=-2.25
Q7213) Solve 16x^2+46x+19=0
Ans: x=-0.5 or x=-2.37
Q7214) Solve 35x^2+39x+10=0
Ans: x=-0.4 or x=-0.71
Q7215) Solve 16x^2+26x+10=0
Ans: x=-0.62 or x=-1
Q7216) Solve 32x^2+38x+11=0
Ans: x=-0.5 or x=-0.69
Q7217) Solve 26x^2+32x+6=0
Ans: x=-0.23 or x=-1
Q7218) Solve 8x^2+16x+8=0
Ans: x=-1 or x=-1
Q7219) Solve 12x^2+31x+7=0
Ans: x=-0.25 or x=-2.33
Q7220) Solve 48x^2+46x+11=0
Ans: x=-0.46 or x=-0.5
Q7221) Solve 14x^2+45x+34=0
Ans: x=-1.21 or x=-2
Q7222) Solve 22x^2+24x+2=0
Ans: x=-0.09 or x=-1
Q7223) Solve 14x^2+16x+2=0
Ans: x=-0.14 or x=-1
Q7224) Solve 2x^2+19x+30=0
Ans: x=-2 or x=-7.5
Q7225) Solve 9x^2+35x+26=0
Ans: x=-1 or x=-2.89
Q7226) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q7227) Solve 15x^2+38x+7=0
Ans: x=-0.2 or x=-2.33
Q7228) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q7229) Solve 11x^2+19x+8=0
Ans: x=-0.73 or x=-1
Q7230) Solve 12x^2+20x+7=0
Ans: x=-0.5 or x=-1.17
Q7231) Solve 18x^2+43x+14=0
Ans: x=-0.39 or x=-2
Q7232) Solve 21x^2+37x+12=0
Ans: x=-0.43 or x=-1.33
Q7233) Solve 16x^2+48x+36=0
Ans: x=-1.5 or x=-1.5
Q7234) Solve 16x^2+34x+4=0
Ans: x=-0.12 or x=-2
Q7235) Solve 11x^2+39x+34=0
Ans: x=-1.55 or x=-2
Q7236) Solve 48x^2+48x+12=0
Ans: x=-0.5 or x=-0.5
Q7237) Solve x^2+38x+37=0
Ans: x=-1 or x=-37
Q7238) Solve 13x^2+46x+21=0
Ans: x=-0.54 or x=-3
Q7239) Solve 30x^2+50x+20=0
Ans: x=-0.67 or x=-1
Q7240) Solve 18x^2+41x+16=0
Ans: x=-0.5 or x=-1.78
Q7241) Solve 50x^2+41x+8=0
Ans: x=-0.32 or x=-0.5
Q7242) Solve 10x^2+50x+40=0
Ans: x=-1 or x=-4
Q7243) Solve 50x^2+45x+4=0
Ans: x=-0.1 or x=-0.8
Q7244) Solve 18x^2+36x+16=0
Ans: x=-0.67 or x=-1.33
Q7245) Solve 3x^2+38x+35=0
Ans: x=-1 or x=-11.67
Q7246) Solve 32x^2+32x+6=0
Ans: x=-0.25 or x=-0.75
Q7247) Solve 9x^2+12x+4=0
Ans: x=-0.67 or x=-0.67
Q7248) Solve 14x^2+32x+18=0
Ans: x=-1 or x=-1.29
Q7249) Solve 4x^2+20x+16=0
Ans: x=-1 or x=-4
Q7250) Solve 21x^2+26x+8=0
Ans: x=-0.57 or x=-0.67
Q7251) Solve 36x^2+32x+7=0
Ans: x=-0.39 or x=-0.5
Q7252) Solve 15x^2+35x+10=0
Ans: x=-0.33 or x=-2
Q7253) Solve 46x^2+29x+3=0
Ans: x=-0.13 or x=-0.5
Q7254) Solve 18x^2+33x+14=0
Ans: x=-0.67 or x=-1.17
Q7255) Solve 42x^2+47x+11=0
Ans: x=-0.33 or x=-0.79
Q7256) Solve 15x^2+8x+1=0
Ans: x=-0.2 or x=-0.33
Q7257) Solve 21x^2+17x+2=0
Ans: x=-0.14 or x=-0.67
Q7258) Solve 26x^2+43x+17=0
Ans: x=-0.65 or x=-1
Q7259) Solve 16x^2+42x+17=0
Ans: x=-0.5 or x=-2.12
Q7260) Solve 3x^2+47x+30=0
Ans: x=-0.67 or x=-15
Q7261) Solve 5x^2+16x+3=0
Ans: x=-0.2 or x=-3
Q7262) Solve 24x^2+35x+9=0
Ans: x=-0.33 or x=-1.12
Q7263) Solve 10x^2+49x+36=0
Ans: x=-0.9 or x=-4
Q7264) Solve 7x^2+36x+45=0
Ans: x=-2.14 or x=-3
Q7265) Solve 5x^2+30x+45=0
Ans: x=-3 or x=-3
Q7266) Solve 16x^2+44x+30=0
Ans: x=-1.25 or x=-1.5
Q7267) Solve 15x^2+47x+32=0
Ans: x=-1 or x=-2.13
Q7268) Solve 5x^2+44x+32=0
Ans: x=-0.8 or x=-8
Q7269) Solve 10x^2+11x+3=0
Ans: x=-0.5 or x=-0.6
Q7270) Solve x^2+14x+48=0
Ans: x=-6 or x=-8
Q7271) Solve 8x^2+30x+7=0
Ans: x=-0.25 or x=-3.5
Q7272) Solve 50x^2+31x+3=0
Ans: x=-0.12 or x=-0.5
Q7273) Solve 6x^2+31x+38=0
Ans: x=-2 or x=-3.17
Q7274) Solve 12x^2+36x+24=0
Ans: x=-1 or x=-2
Q7275) Solve 10x^2+21x+8=0
Ans: x=-0.5 or x=-1.6
Q7276) Solve 30x^2+47x+7=0
Ans: x=-0.17 or x=-1.4
Q7277) Solve 10x^2+38x+28=0
Ans: x=-1 or x=-2.8
Q7278) Solve 19x^2+42x+23=0
Ans: x=-1 or x=-1.21
Q7279) Solve 11x^2+20x+9=0
Ans: x=-0.82 or x=-1
Q7280) Solve 18x^2+45x+27=0
Ans: x=-1 or x=-1.5
Q7281) Solve 2x^2+35x+48=0
Ans: x=-1.5 or x=-16
Q7282) Solve 2x^2+15x+7=0
Ans: x=-0.5 or x=-7
Q7283) Solve 14x^2+40x+24=0
Ans: x=-0.86 or x=-2
Q7284) Solve x^2+7x+10=0
Ans: x=-2 or x=-5
Q7285) Solve 39x^2+47x+8=0
Ans: x=-0.21 or x=-1
Q7286) Solve 11x^2+37x+30=0
Ans: x=-1.36 or x=-2
Q7287) Solve 34x^2+25x+4=0
Ans: x=-0.24 or x=-0.5
Q7288) Solve 28x^2+41x+15=0
Ans: x=-0.71 or x=-0.75
Q7289) Solve 36x^2+45x+14=0
Ans: x=-0.58 or x=-0.67
Q7290) Solve 18x^2+48x+14=0
Ans: x=-0.33 or x=-2.33
Q7291) Solve 36x^2+46x+10=0
Ans: x=-0.28 or x=-1
Q7292) Solve 7x^2+29x+24=0
Ans: x=-1.14 or x=-3
Q7293) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q7294) Solve 4x^2+45x+41=0
Ans: x=-1 or x=-10.25
Q7295) Solve 9x^2+30x+21=0
Ans: x=-1 or x=-2.33
Q7296) Solve 34x^2+31x+7=0
Ans: x=-0.41 or x=-0.5
Q7297) Solve 48x^2+19x+1=0
Ans: x=-0.06 or x=-0.33
Q7298) Solve 13x^2+41x+28=0
Ans: x=-1 or x=-2.15
Q7299) Solve 14x^2+23x+8=0
Ans: x=-0.5 or x=-1.14
Q7300) Solve 16x^2+30x+14=0
Ans: x=-0.87 or x=-1
Q7301) Solve 25x^2+10x+1=0
Ans: x=-0.2 or x=-0.2
Q7302) Solve 9x^2+45x+44=0
Ans: x=-1.33 or x=-3.67
Q7303) Solve 15x^2+21x+6=0
Ans: x=-0.4 or x=-1
Q7304) Solve 26x^2+44x+18=0
Ans: x=-0.69 or x=-1
Q7305) Solve 2x^2+11x+12=0
Ans: x=-1.5 or x=-4
Q7306) Solve 16x^2+24x+8=0
Ans: x=-0.5 or x=-1
Q7307) Solve 11x^2+41x+24=0
Ans: x=-0.73 or x=-3
Q7308) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q7309) Solve 24x^2+14x+1=0
Ans: x=-0.08 or x=-0.5
Q7310) Solve 8x^2+28x+24=0
Ans: x=-1.5 or x=-2
Q7311) Solve 4x^2+12x+8=0
Ans: x=-1 or x=-2
Q7312) Solve 16x^2+46x+33=0
Ans: x=-1.37 or x=-1.5
Q7313) Solve 36x^2+24x+4=0
Ans: x=-0.33 or x=-0.33
Q7314) Solve 13x^2+48x+44=0
Ans: x=-1.69 or x=-2
Q7315) Solve 18x^2+50x+28=0
Ans: x=-0.78 or x=-2
Q7316) Solve 5x^2+28x+32=0
Ans: x=-1.6 or x=-4
Q7317) Solve 2x^2+24x+40=0
Ans: x=-2 or x=-10
Q7318) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q7319) Solve 5x^2+8x+3=0
Ans: x=-0.6 or x=-1
Q7320) Solve 12x^2+36x+27=0
Ans: x=-1.5 or x=-1.5
Q7321) Solve 16x^2+28x+6=0
Ans: x=-0.25 or x=-1.5
Q7322) Solve 9x^2+24x+12=0
Ans: x=-0.67 or x=-2
Q7323) Solve 19x^2+48x+29=0
Ans: x=-1 or x=-1.53
Q7324) Solve 24x^2+37x+14=0
Ans: x=-0.67 or x=-0.87
Q7325) Solve 2x^2+21x+34=0
Ans: x=-2 or x=-8.5
Q7326) Solve 8x^2+33x+34=0
Ans: x=-2 or x=-2.12
Q7327) Solve 11x^2+12x+1=0
Ans: x=-0.09 or x=-1
Q7328) Solve 7x^2+29x+22=0
Ans: x=-1 or x=-3.14
Q7329) Solve 4x^2+35x+31=0
Ans: x=-1 or x=-7.75
Q7330) Solve 19x^2+39x+20=0
Ans: x=-1 or x=-1.05
Q7331) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q7332) Solve 18x^2+44x+16=0
Ans: x=-0.44 or x=-2
Q7333) Solve 7x^2+30x+23=0
Ans: x=-1 or x=-3.29
Q7334) Solve 4x^2+45x+11=0
Ans: x=-0.25 or x=-11
Q7335) Solve 21x^2+37x+12=0
Ans: x=-0.43 or x=-1.33
Q7336) Solve 6x^2+20x+6=0
Ans: x=-0.33 or x=-3
Q7337) Solve 6x^2+25x+14=0
Ans: x=-0.67 or x=-3.5
Q7338) Solve 3x^2+14x+16=0
Ans: x=-2 or x=-2.67
Q7339) Solve 40x^2+22x+1=0
Ans: x=-0.05 or x=-0.5
Q7340) Solve 30x^2+16x+2=0
Ans: x=-0.2 or x=-0.33
Q7341) Solve 5x^2+34x+24=0
Ans: x=-0.8 or x=-6
Q7342) Solve 7x^2+41x+34=0
Ans: x=-1 or x=-4.86
Q7343) Solve x^2+37x+36=0
Ans: x=-1 or x=-36
Q7344) Solve 30x^2+37x+11=0
Ans: x=-0.5 or x=-0.73
Q7345) Solve 7x^2+34x+24=0
Ans: x=-0.86 or x=-4
Q7346) Solve 9x^2+36x+27=0
Ans: x=-1 or x=-3
Q7347) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q7348) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q7349) Solve 17x^2+39x+22=0
Ans: x=-1 or x=-1.29
Q7350) Solve 10x^2+28x+18=0
Ans: x=-1 or x=-1.8
Q7351) Solve 15x^2+48x+36=0
Ans: x=-1.2 or x=-2
Q7352) Solve 24x^2+34x+10=0
Ans: x=-0.42 or x=-1
Q7353) Solve 3x^2+22x+19=0
Ans: x=-1 or x=-6.33
Q7354) Solve 18x^2+29x+3=0
Ans: x=-0.11 or x=-1.5
Q7355) Solve 21x^2+29x+8=0
Ans: x=-0.38 or x=-1
Q7356) Solve 3x^2+4x+1=0
Ans: x=-0.33 or x=-1
Q7357) Solve 26x^2+48x+22=0
Ans: x=-0.85 or x=-1
Q7358) Solve 24x^2+26x+7=0
Ans: x=-0.5 or x=-0.58
Q7359) Solve 2x^2+26x+24=0
Ans: x=-1 or x=-12
Q7360) Solve 24x^2+10x+1=0
Ans: x=-0.17 or x=-0.25
Q7361) Solve 28x^2+43x+10=0
Ans: x=-0.29 or x=-1.25
Q7362) Solve 16x^2+42x+20=0
Ans: x=-0.62 or x=-2
Q7363) Solve 12x^2+33x+21=0
Ans: x=-1 or x=-1.75
Q7364) Solve 10x^2+31x+21=0
Ans: x=-1 or x=-2.1
Q7365) Solve 8x^2+34x+21=0
Ans: x=-0.75 or x=-3.5
Q7366) Solve 18x^2+32x+14=0
Ans: x=-0.78 or x=-1
Q7367) Solve 12x^2+46x+30=0
Ans: x=-0.83 or x=-3
Q7368) Solve 6x^2+39x+33=0
Ans: x=-1 or x=-5.5
Q7369) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q7370) Solve 16x^2+12x+2=0
Ans: x=-0.25 or x=-0.5
Q7371) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q7372) Solve 24x^2+20x+4=0
Ans: x=-0.33 or x=-0.5
Q7373) Solve 24x^2+29x+5=0
Ans: x=-0.21 or x=-1
Q7374) Solve 13x^2+16x+3=0
Ans: x=-0.23 or x=-1
Q7375) Solve 3x^2+31x+10=0
Ans: x=-0.33 or x=-10
Q7376) Solve 2x^2+20x+48=0
Ans: x=-4 or x=-6
Q7377) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q7378) Solve 21x^2+40x+11=0
Ans: x=-0.33 or x=-1.57
Q7379) Solve 44x^2+40x+9=0
Ans: x=-0.41 or x=-0.5
Q7380) Solve 32x^2+49x+17=0
Ans: x=-0.53 or x=-1
Q7381) Solve 11x^2+48x+45=0
Ans: x=-1.36 or x=-3
Q7382) Solve 8x^2+31x+21=0
Ans: x=-0.87 or x=-3
Q7383) Solve 36x^2+36x+5=0
Ans: x=-0.17 or x=-0.83
Q7384) Solve 3x^2+15x+12=0
Ans: x=-1 or x=-4
Q7385) Solve 2x^2+20x+48=0
Ans: x=-4 or x=-6
Q7386) Solve 2x^2+8x+8=0
Ans: x=-2 or x=-2
Q7387) Solve 16x^2+50x+39=0
Ans: x=-1.5 or x=-1.62
Q7388) Solve 32x^2+44x+14=0
Ans: x=-0.5 or x=-0.87
Q7389) Solve 12x^2+19x+5=0
Ans: x=-0.33 or x=-1.25
Q7390) Solve 10x^2+49x+18=0
Ans: x=-0.4 or x=-4.5
Q7391) Solve 4x^2+11x+6=0
Ans: x=-0.75 or x=-2
Q7392) Solve 4x^2+49x+12=0
Ans: x=-0.25 or x=-12
Q7393) Solve 5x^2+23x+18=0
Ans: x=-1 or x=-3.6
Q7394) Solve 10x^2+46x+48=0
Ans: x=-1.6 or x=-3
Q7395) Solve 21x^2+46x+24=0
Ans: x=-0.86 or x=-1.33
Q7396) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q7397) Solve 6x^2+27x+12=0
Ans: x=-0.5 or x=-4
Q7398) Solve 48x^2+20x+2=0
Ans: x=-0.17 or x=-0.25
Q7399) Solve 10x^2+21x+11=0
Ans: x=-1 or x=-1.1
Q7400) Solve 12x^2+38x+26=0
Ans: x=-1 or x=-2.17
Q7401) Solve 8x^2+26x+20=0
Ans: x=-1.25 or x=-2
Q7402) Solve 24x^2+26x+2=0
Ans: x=-0.08 or x=-1
Q7403) Solve 4x^2+44x+21=0
Ans: x=-0.5 or x=-10.5
Q7404) Solve 6x^2+49x+43=0
Ans: x=-1 or x=-7.17
Q7405) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q7406) Solve 22x^2+45x+18=0
Ans: x=-0.55 or x=-1.5
Q7407) Solve 12x^2+41x+24=0
Ans: x=-0.75 or x=-2.67
Q7408) Solve 2x^2+37x+18=0
Ans: x=-0.5 or x=-18
Q7409) Solve 3x^2+40x+48=0
Ans: x=-1.33 or x=-12
Q7410) Solve 8x^2+30x+18=0
Ans: x=-0.75 or x=-3
Q7411) Solve 27x^2+15x+2=0
Ans: x=-0.22 or x=-0.33
Q7412) Solve 14x^2+31x+17=0
Ans: x=-1 or x=-1.21
Q7413) Solve 5x^2+18x+16=0
Ans: x=-1.6 or x=-2
Q7414) Solve 28x^2+24x+5=0
Ans: x=-0.36 or x=-0.5
Q7415) Solve 17x^2+24x+7=0
Ans: x=-0.41 or x=-1
Q7416) Solve 15x^2+43x+30=0
Ans: x=-1.2 or x=-1.67
Q7417) Solve 26x^2+31x+9=0
Ans: x=-0.5 or x=-0.69
Q7418) Solve 6x^2+43x+20=0
Ans: x=-0.5 or x=-6.67
Q7419) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q7420) Solve 23x^2+41x+18=0
Ans: x=-0.78 or x=-1
Q7421) Solve 11x^2+41x+30=0
Ans: x=-1 or x=-2.73
Q7422) Solve 18x^2+35x+17=0
Ans: x=-0.94 or x=-1
Q7423) Solve 13x^2+48x+35=0
Ans: x=-1 or x=-2.69
Q7424) Solve 12x^2+44x+40=0
Ans: x=-1.67 or x=-2
Q7425) Solve 30x^2+49x+20=0
Ans: x=-0.8 or x=-0.83
Q7426) Solve 28x^2+39x+8=0
Ans: x=-0.25 or x=-1.14
Q7427) Solve 5x^2+49x+44=0
Ans: x=-1 or x=-8.8
Q7428) Solve 12x^2+33x+21=0
Ans: x=-1 or x=-1.75
Q7429) Solve 6x^2+31x+40=0
Ans: x=-2.5 or x=-2.67
Q7430) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q7431) Solve 28x^2+37x+12=0
Ans: x=-0.57 or x=-0.75
Q7432) Solve 8x^2+46x+21=0
Ans: x=-0.5 or x=-5.25
Q7433) Solve 12x^2+34x+22=0
Ans: x=-1 or x=-1.83
Q7434) Solve 19x^2+40x+21=0
Ans: x=-1 or x=-1.11
Q7435) Solve 6x^2+38x+32=0
Ans: x=-1 or x=-5.33
Q7436) Solve 9x^2+27x+20=0
Ans: x=-1.33 or x=-1.67
Q7437) Solve 12x^2+16x+4=0
Ans: x=-0.33 or x=-1
Q7438) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q7439) Solve 11x^2+50x+24=0
Ans: x=-0.55 or x=-4
Q7440) Solve 32x^2+36x+10=0
Ans: x=-0.5 or x=-0.62
Q7441) Solve 18x^2+39x+18=0
Ans: x=-0.67 or x=-1.5
Q7442) Solve 28x^2+37x+12=0
Ans: x=-0.57 or x=-0.75
Q7443) Solve 48x^2+38x+5=0
Ans: x=-0.17 or x=-0.62
Q7444) Solve 26x^2+43x+15=0
Ans: x=-0.5 or x=-1.15
Q7445) Solve 14x^2+37x+15=0
Ans: x=-0.5 or x=-2.14
Q7446) Solve 14x^2+39x+16=0
Ans: x=-0.5 or x=-2.29
Q7447) Solve 2x^2+21x+40=0
Ans: x=-2.5 or x=-8
Q7448) Solve 9x^2+30x+21=0
Ans: x=-1 or x=-2.33
Q7449) Solve 32x^2+20x+2=0
Ans: x=-0.12 or x=-0.5
Q7450) Solve 23x^2+49x+6=0
Ans: x=-0.13 or x=-2
Q7451) Solve 2x^2+14x+20=0
Ans: x=-2 or x=-5
Q7452) Solve 45x^2+19x+2=0
Ans: x=-0.2 or x=-0.22
Q7453) Solve x^2+19x+18=0
Ans: x=-1 or x=-18
Q7454) Solve 4x^2+41x+10=0
Ans: x=-0.25 or x=-10
Q7455) Solve 34x^2+31x+7=0
Ans: x=-0.41 or x=-0.5
Q7456) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q7457) Solve 11x^2+46x+35=0
Ans: x=-1 or x=-3.18
Q7458) Solve x^2+12x+20=0
Ans: x=-2 or x=-10
Q7459) Solve 50x^2+47x+11=0
Ans: x=-0.44 or x=-0.5
Q7460) Solve 18x^2+43x+17=0
Ans: x=-0.5 or x=-1.89
Q7461) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q7462) Solve 32x^2+28x+3=0
Ans: x=-0.12 or x=-0.75
Q7463) Solve 6x^2+44x+14=0
Ans: x=-0.33 or x=-7
Q7464) Solve 23x^2+42x+19=0
Ans: x=-0.83 or x=-1
Q7465) Solve 6x^2+34x+28=0
Ans: x=-1 or x=-4.67
Q7466) Solve 8x^2+37x+39=0
Ans: x=-1.62 or x=-3
Q7467) Solve 18x^2+18x+4=0
Ans: x=-0.33 or x=-0.67
Q7468) Solve 30x^2+37x+9=0
Ans: x=-0.33 or x=-0.9
Q7469) Solve 7x^2+19x+10=0
Ans: x=-0.71 or x=-2
Q7470) Solve 14x^2+34x+12=0
Ans: x=-0.43 or x=-2
Q7471) Solve 34x^2+19x+1=0
Ans: x=-0.06 or x=-0.5
Q7472) Solve 29x^2+36x+7=0
Ans: x=-0.24 or x=-1
Q7473) Solve 9x^2+41x+20=0
Ans: x=-0.56 or x=-4
Q7474) Solve 3x^2+14x+16=0
Ans: x=-2 or x=-2.67
Q7475) Solve 15x^2+43x+26=0
Ans: x=-0.87 or x=-2
Q7476) Solve 12x^2+45x+33=0
Ans: x=-1 or x=-2.75
Q7477) Solve 10x^2+37x+33=0
Ans: x=-1.5 or x=-2.2
Q7478) Solve 15x^2+49x+34=0
Ans: x=-1 or x=-2.27
Q7479) Solve 12x^2+47x+40=0
Ans: x=-1.25 or x=-2.67
Q7480) Solve 18x^2+31x+6=0
Ans: x=-0.22 or x=-1.5
Q7481) Solve 4x^2+31x+21=0
Ans: x=-0.75 or x=-7
Q7482) Solve 24x^2+38x+10=0
Ans: x=-0.33 or x=-1.25
Q7483) Solve 18x^2+20x+2=0
Ans: x=-0.11 or x=-1
Q7484) Solve 20x^2+30x+10=0
Ans: x=-0.5 or x=-1
Q7485) Solve 10x^2+24x+8=0
Ans: x=-0.4 or x=-2
Q7486) Solve 5x^2+43x+38=0
Ans: x=-1 or x=-7.6
Q7487) Solve 6x^2+32x+42=0
Ans: x=-2.33 or x=-3
Q7488) Solve 13x^2+49x+36=0
Ans: x=-1 or x=-2.77
Q7489) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q7490) Solve x^2+6x+5=0
Ans: x=-1 or x=-5
Q7491) Solve 20x^2+30x+10=0
Ans: x=-0.5 or x=-1
Q7492) Solve 16x^2+26x+3=0
Ans: x=-0.12 or x=-1.5
Q7493) Solve x^2+37x+36=0
Ans: x=-1 or x=-36
Q7494) Solve 21x^2+36x+15=0
Ans: x=-0.71 or x=-1
Q7495) Solve 4x^2+30x+36=0
Ans: x=-1.5 or x=-6
Q7496) Solve 11x^2+35x+6=0
Ans: x=-0.18 or x=-3
Q7497) Solve 9x^2+39x+30=0
Ans: x=-1 or x=-3.33
Q7498) Solve 21x^2+40x+19=0
Ans: x=-0.9 or x=-1
Q7499) Solve 14x^2+45x+31=0
Ans: x=-1 or x=-2.21
Q7500) Solve 6x^2+29x+30=0
Ans: x=-1.5 or x=-3.33
Q7501) Solve 36x^2+13x+1=0
Ans: x=-0.11 or x=-0.25
Q7502) Solve 42x^2+44x+2=0
Ans: x=-0.05 or x=-1
Q7503) Solve 12x^2+35x+8=0
Ans: x=-0.25 or x=-2.67
Q7504) Solve 44x^2+34x+6=0
Ans: x=-0.27 or x=-0.5
Q7505) Solve 9x^2+16x+7=0
Ans: x=-0.78 or x=-1
Q7506) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q7507) Solve 4x^2+37x+33=0
Ans: x=-1 or x=-8.25
Q7508) Solve 32x^2+22x+3=0
Ans: x=-0.19 or x=-0.5
Q7509) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q7510) Solve 5x^2+35x+30=0
Ans: x=-1 or x=-6
Q7511) Solve 15x^2+47x+34=0
Ans: x=-1.13 or x=-2
Q7512) Solve 10x^2+49x+36=0
Ans: x=-0.9 or x=-4
Q7513) Solve 31x^2+49x+18=0
Ans: x=-0.58 or x=-1
Q7514) Solve 7x^2+37x+30=0
Ans: x=-1 or x=-4.29
Q7515) Solve 4x^2+28x+45=0
Ans: x=-2.5 or x=-4.5
Q7516) Solve 4x^2+38x+34=0
Ans: x=-1 or x=-8.5
Q7517) Solve 13x^2+20x+7=0
Ans: x=-0.54 or x=-1
Q7518) Solve 8x^2+25x+18=0
Ans: x=-1.12 or x=-2
Q7519) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q7520) Solve 4x^2+46x+42=0
Ans: x=-1 or x=-10.5
Q7521) Solve 5x^2+35x+30=0
Ans: x=-1 or x=-6
Q7522) Solve 8x^2+38x+24=0
Ans: x=-0.75 or x=-4
Q7523) Solve 10x^2+40x+30=0
Ans: x=-1 or x=-3
Q7524) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q7525) Solve 7x^2+47x+30=0
Ans: x=-0.71 or x=-6
Q7526) Solve 8x^2+23x+15=0
Ans: x=-1 or x=-1.87
Q7527) Solve 15x^2+18x+3=0
Ans: x=-0.2 or x=-1
Q7528) Solve 5x^2+42x+49=0
Ans: x=-1.4 or x=-7
Q7529) Solve 8x^2+37x+39=0
Ans: x=-1.62 or x=-3
Q7530) Solve 6x^2+31x+33=0
Ans: x=-1.5 or x=-3.67
Q7531) Solve 36x^2+44x+13=0
Ans: x=-0.5 or x=-0.72
Q7532) Solve 3x^2+22x+35=0
Ans: x=-2.33 or x=-5
Q7533) Solve 42x^2+50x+12=0
Ans: x=-0.33 or x=-0.86
Q7534) Solve 7x^2+30x+23=0
Ans: x=-1 or x=-3.29
Q7535) Solve 21x^2+20x+4=0
Ans: x=-0.29 or x=-0.67
Q7536) Solve 9x^2+25x+14=0
Ans: x=-0.78 or x=-2
Q7537) Solve 32x^2+42x+10=0
Ans: x=-0.31 or x=-1
Q7538) Solve 19x^2+32x+13=0
Ans: x=-0.68 or x=-1
Q7539) Solve 13x^2+36x+20=0
Ans: x=-0.77 or x=-2
Q7540) Solve 14x^2+31x+17=0
Ans: x=-1 or x=-1.21
Q7541) Solve 12x^2+38x+16=0
Ans: x=-0.5 or x=-2.67
Q7542) Solve 50x^2+47x+11=0
Ans: x=-0.44 or x=-0.5
Q7543) Solve 4x^2+31x+27=0
Ans: x=-1 or x=-6.75
Q7544) Solve 5x^2+50x+45=0
Ans: x=-1 or x=-9
Q7545) Solve 8x^2+19x+11=0
Ans: x=-1 or x=-1.37
Q7546) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q7547) Solve 4x^2+27x+44=0
Ans: x=-2.75 or x=-4
Q7548) Solve 16x^2+48x+11=0
Ans: x=-0.25 or x=-2.75
Q7549) Solve 9x^2+20x+11=0
Ans: x=-1 or x=-1.22
Q7550) Solve 10x^2+19x+6=0
Ans: x=-0.4 or x=-1.5
Q7551) Solve 10x^2+13x+3=0
Ans: x=-0.3 or x=-1
Q7552) Solve 38x^2+25x+3=0
Ans: x=-0.16 or x=-0.5
Q7553) Solve 5x^2+17x+14=0
Ans: x=-1.4 or x=-2
Q7554) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q7555) Solve 4x^2+42x+20=0
Ans: x=-0.5 or x=-10
Q7556) Solve 50x^2+50x+8=0
Ans: x=-0.2 or x=-0.8
Q7557) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q7558) Solve 48x^2+38x+4=0
Ans: x=-0.12 or x=-0.67
Q7559) Solve 3x^2+22x+32=0
Ans: x=-2 or x=-5.33
Q7560) Solve 14x^2+41x+26=0
Ans: x=-0.93 or x=-2
Q7561) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q7562) Solve 15x^2+47x+14=0
Ans: x=-0.33 or x=-2.8
Q7563) Solve 15x^2+40x+20=0
Ans: x=-0.67 or x=-2
Q7564) Solve 38x^2+31x+6=0
Ans: x=-0.32 or x=-0.5
Q7565) Solve 15x^2+49x+24=0
Ans: x=-0.6 or x=-2.67
Q7566) Solve 8x^2+47x+39=0
Ans: x=-1 or x=-4.87
Q7567) Solve 2x^2+9x+9=0
Ans: x=-1.5 or x=-3
Q7568) Solve 2x^2+47x+23=0
Ans: x=-0.5 or x=-23
Q7569) Solve 7x^2+33x+20=0
Ans: x=-0.71 or x=-4
Q7570) Solve 8x^2+19x+6=0
Ans: x=-0.37 or x=-2
Q7571) Solve 45x^2+42x+9=0
Ans: x=-0.33 or x=-0.6
Q7572) Solve 8x^2+41x+36=0
Ans: x=-1.12 or x=-4
Q7573) Solve 3x^2+46x+43=0
Ans: x=-1 or x=-14.33
Q7574) Solve 18x^2+25x+8=0
Ans: x=-0.5 or x=-0.89
Q7575) Solve 13x^2+42x+29=0
Ans: x=-1 or x=-2.23
Q7576) Solve 15x^2+23x+8=0
Ans: x=-0.53 or x=-1
Q7577) Solve 5x^2+17x+12=0
Ans: x=-1 or x=-2.4
Q7578) Solve 9x^2+20x+11=0
Ans: x=-1 or x=-1.22
Q7579) Solve 33x^2+50x+13=0
Ans: x=-0.33 or x=-1.18
Q7580) Solve 5x^2+16x+3=0
Ans: x=-0.2 or x=-3
Q7581) Solve 15x^2+44x+29=0
Ans: x=-1 or x=-1.93
Q7582) Solve 10x^2+37x+16=0
Ans: x=-0.5 or x=-3.2
Q7583) Solve 48x^2+30x+3=0
Ans: x=-0.12 or x=-0.5
Q7584) Solve 17x^2+40x+12=0
Ans: x=-0.35 or x=-2
Q7585) Solve 40x^2+22x+1=0
Ans: x=-0.05 or x=-0.5
Q7586) Solve 9x^2+44x+32=0
Ans: x=-0.89 or x=-4
Q7587) Solve 40x^2+36x+8=0
Ans: x=-0.4 or x=-0.5
Q7588) Solve 31x^2+50x+19=0
Ans: x=-0.61 or x=-1
Q7589) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q7590) Solve 22x^2+43x+16=0
Ans: x=-0.5 or x=-1.45
Q7591) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q7592) Solve 15x^2+50x+15=0
Ans: x=-0.33 or x=-3
Q7593) Solve 10x^2+45x+50=0
Ans: x=-2 or x=-2.5
Q7594) Solve x^2+23x+42=0
Ans: x=-2 or x=-21
Q7595) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q7596) Solve 3x^2+17x+20=0
Ans: x=-1.67 or x=-4
Q7597) Solve 9x^2+37x+4=0
Ans: x=-0.11 or x=-4
Q7598) Solve 9x^2+24x+7=0
Ans: x=-0.33 or x=-2.33
Q7599) Solve 9x^2+33x+18=0
Ans: x=-0.67 or x=-3
Q7600) Solve 5x^2+29x+42=0
Ans: x=-2.8 or x=-3
Q7601) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q7602) Solve x^2+33x+32=0
Ans: x=-1 or x=-32
Q7603) Solve 24x^2+50x+25=0
Ans: x=-0.83 or x=-1.25
Q7604) Solve 15x^2+41x+14=0
Ans: x=-0.4 or x=-2.33
Q7605) Solve 16x^2+18x+5=0
Ans: x=-0.5 or x=-0.62
Q7606) Solve 13x^2+26x+13=0
Ans: x=-1 or x=-1
Q7607) Solve x^2+28x+27=0
Ans: x=-1 or x=-27
Q7608) Solve 28x^2+37x+12=0
Ans: x=-0.57 or x=-0.75
Q7609) Solve 12x^2+19x+4=0
Ans: x=-0.25 or x=-1.33
Q7610) Solve 3x^2+26x+48=0
Ans: x=-2.67 or x=-6
Q7611) Solve x^2+7x+10=0
Ans: x=-2 or x=-5
Q7612) Solve 2x^2+15x+25=0
Ans: x=-2.5 or x=-5
Q7613) Solve 30x^2+41x+13=0
Ans: x=-0.5 or x=-0.87
Q7614) Solve x^2+32x+31=0
Ans: x=-1 or x=-31
Q7615) Solve 38x^2+46x+8=0
Ans: x=-0.21 or x=-1
Q7616) Solve 16x^2+40x+9=0
Ans: x=-0.25 or x=-2.25
Q7617) Solve 49x^2+28x+4=0
Ans: x=-0.29 or x=-0.29
Q7618) Solve 3x^2+46x+15=0
Ans: x=-0.33 or x=-15
Q7619) Solve 14x^2+47x+15=0
Ans: x=-0.36 or x=-3
Q7620) Solve 4x^2+36x+45=0
Ans: x=-1.5 or x=-7.5
Q7621) Solve 7x^2+27x+20=0
Ans: x=-1 or x=-2.86
Q7622) Solve 13x^2+20x+7=0
Ans: x=-0.54 or x=-1
Q7623) Solve 34x^2+40x+6=0
Ans: x=-0.18 or x=-1
Q7624) Solve 5x^2+26x+21=0
Ans: x=-1 or x=-4.2
Q7625) Solve 7x^2+32x+16=0
Ans: x=-0.57 or x=-4
Q7626) Solve 28x^2+39x+5=0
Ans: x=-0.14 or x=-1.25
Q7627) Solve 5x^2+22x+8=0
Ans: x=-0.4 or x=-4
Q7628) Solve 40x^2+34x+7=0
Ans: x=-0.35 or x=-0.5
Q7629) Solve 7x^2+43x+40=0
Ans: x=-1.14 or x=-5
Q7630) Solve 16x^2+39x+14=0
Ans: x=-0.44 or x=-2
Q7631) Solve 4x^2+31x+21=0
Ans: x=-0.75 or x=-7
Q7632) Solve 12x^2+24x+9=0
Ans: x=-0.5 or x=-1.5
Q7633) Solve 9x^2+22x+13=0
Ans: x=-1 or x=-1.44
Q7634) Solve 23x^2+25x+2=0
Ans: x=-0.09 or x=-1
Q7635) Solve 4x^2+17x+15=0
Ans: x=-1.25 or x=-3
Q7636) Solve 21x^2+46x+24=0
Ans: x=-0.86 or x=-1.33
Q7637) Solve 4x^2+27x+38=0
Ans: x=-2 or x=-4.75
Q7638) Solve 5x^2+29x+36=0
Ans: x=-1.8 or x=-4
Q7639) Solve 12x^2+39x+9=0
Ans: x=-0.25 or x=-3
Q7640) Solve 4x^2+18x+18=0
Ans: x=-1.5 or x=-3
Q7641) Solve 4x^2+15x+9=0
Ans: x=-0.75 or x=-3
Q7642) Solve 43x^2+48x+5=0
Ans: x=-0.12 or x=-1
Q7643) Solve 19x^2+49x+30=0
Ans: x=-1 or x=-1.58
Q7644) Solve 45x^2+44x+7=0
Ans: x=-0.2 or x=-0.78
Q7645) Solve 9x^2+36x+11=0
Ans: x=-0.33 or x=-3.67
Q7646) Solve 14x^2+31x+17=0
Ans: x=-1 or x=-1.21
Q7647) Solve 38x^2+44x+6=0
Ans: x=-0.16 or x=-1
Q7648) Solve 15x^2+28x+5=0
Ans: x=-0.2 or x=-1.67
Q7649) Solve 36x^2+27x+5=0
Ans: x=-0.33 or x=-0.42
Q7650) Solve 12x^2+43x+31=0
Ans: x=-1 or x=-2.58
Q7651) Solve 6x^2+28x+16=0
Ans: x=-0.67 or x=-4
Q7652) Solve 33x^2+49x+16=0
Ans: x=-0.48 or x=-1
Q7653) Solve 33x^2+32x+7=0
Ans: x=-0.33 or x=-0.64
Q7654) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q7655) Solve 44x^2+35x+6=0
Ans: x=-0.25 or x=-0.55
Q7656) Solve 10x^2+43x+46=0
Ans: x=-2 or x=-2.3
Q7657) Solve 25x^2+30x+5=0
Ans: x=-0.2 or x=-1
Q7658) Solve 6x^2+29x+30=0
Ans: x=-1.5 or x=-3.33
Q7659) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q7660) Solve 2x^2+13x+11=0
Ans: x=-1 or x=-5.5
Q7661) Solve 38x^2+45x+13=0
Ans: x=-0.5 or x=-0.68
Q7662) Solve x^2+3x+2=0
Ans: x=-1 or x=-2
Q7663) Solve 33x^2+50x+13=0
Ans: x=-0.33 or x=-1.18
Q7664) Solve 14x^2+41x+26=0
Ans: x=-0.93 or x=-2
Q7665) Solve 2x^2+18x+36=0
Ans: x=-3 or x=-6
Q7666) Solve 4x^2+27x+23=0
Ans: x=-1 or x=-5.75
Q7667) Solve 10x^2+43x+12=0
Ans: x=-0.3 or x=-4
Q7668) Solve 9x^2+37x+38=0
Ans: x=-2 or x=-2.11
Q7669) Solve 36x^2+36x+5=0
Ans: x=-0.17 or x=-0.83
Q7670) Solve 46x^2+33x+5=0
Ans: x=-0.22 or x=-0.5
Q7671) Solve 5x^2+17x+12=0
Ans: x=-1 or x=-2.4
Q7672) Solve 18x^2+42x+20=0
Ans: x=-0.67 or x=-1.67
Q7673) Solve 10x^2+48x+38=0
Ans: x=-1 or x=-3.8
Q7674) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q7675) Solve 3x^2+11x+6=0
Ans: x=-0.67 or x=-3
Q7676) Solve 5x^2+23x+12=0
Ans: x=-0.6 or x=-4
Q7677) Solve 14x^2+41x+30=0
Ans: x=-1.43 or x=-1.5
Q7678) Solve 24x^2+50x+11=0
Ans: x=-0.25 or x=-1.83
Q7679) Solve 44x^2+41x+6=0
Ans: x=-0.18 or x=-0.75
Q7680) Solve 27x^2+42x+16=0
Ans: x=-0.67 or x=-0.89
Q7681) Solve 9x^2+29x+20=0
Ans: x=-1 or x=-2.22
Q7682) Solve 32x^2+48x+16=0
Ans: x=-0.5 or x=-1
Q7683) Solve 21x^2+26x+5=0
Ans: x=-0.24 or x=-1
Q7684) Solve 32x^2+30x+7=0
Ans: x=-0.44 or x=-0.5
Q7685) Solve 12x^2+41x+15=0
Ans: x=-0.42 or x=-3
Q7686) Solve 9x^2+45x+14=0
Ans: x=-0.33 or x=-4.67
Q7687) Solve 6x^2+20x+14=0
Ans: x=-1 or x=-2.33
Q7688) Solve 6x^2+17x+7=0
Ans: x=-0.5 or x=-2.33
Q7689) Solve x^2+40x+39=0
Ans: x=-1 or x=-39
Q7690) Solve 45x^2+38x+8=0
Ans: x=-0.4 or x=-0.44
Q7691) Solve x^2+13x+22=0
Ans: x=-2 or x=-11
Q7692) Solve 50x^2+40x+6=0
Ans: x=-0.2 or x=-0.6
Q7693) Solve 18x^2+36x+16=0
Ans: x=-0.67 or x=-1.33
Q7694) Solve 10x^2+25x+10=0
Ans: x=-0.5 or x=-2
Q7695) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q7696) Solve 23x^2+47x+2=0
Ans: x=-0.04 or x=-2
Q7697) Solve 32x^2+28x+5=0
Ans: x=-0.25 or x=-0.62
Q7698) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q7699) Solve 10x^2+41x+18=0
Ans: x=-0.5 or x=-3.6
Q7700) Solve 2x^2+23x+11=0
Ans: x=-0.5 or x=-11
Q7701) Solve 12x^2+37x+28=0
Ans: x=-1.33 or x=-1.75
Q7702) Solve 11x^2+38x+27=0
Ans: x=-1 or x=-2.45
Q7703) Solve 30x^2+34x+8=0
Ans: x=-0.33 or x=-0.8
Q7704) Solve 23x^2+29x+6=0
Ans: x=-0.26 or x=-1
Q7705) Solve 30x^2+35x+5=0
Ans: x=-0.17 or x=-1
Q7706) Solve 5x^2+32x+35=0
Ans: x=-1.4 or x=-5
Q7707) Solve 5x^2+29x+36=0
Ans: x=-1.8 or x=-4
Q7708) Solve 4x^2+46x+22=0
Ans: x=-0.5 or x=-11
Q7709) Solve 3x^2+13x+10=0
Ans: x=-1 or x=-3.33
Q7710) Solve 48x^2+46x+10=0
Ans: x=-0.33 or x=-0.62
Q7711) Solve 9x^2+35x+26=0
Ans: x=-1 or x=-2.89
Q7712) Solve 14x^2+49x+21=0
Ans: x=-0.5 or x=-3
Q7713) Solve 4x^2+12x+8=0
Ans: x=-1 or x=-2
Q7714) Solve 22x^2+43x+16=0
Ans: x=-0.5 or x=-1.45
Q7715) Solve x^2+7x+6=0
Ans: x=-1 or x=-6
Q7716) Solve 48x^2+50x+7=0
Ans: x=-0.17 or x=-0.87
Q7717) Solve 34x^2+47x+13=0
Ans: x=-0.38 or x=-1
Q7718) Solve 30x^2+35x+10=0
Ans: x=-0.5 or x=-0.67
Q7719) Solve 15x^2+49x+38=0
Ans: x=-1.27 or x=-2
Q7720) Solve 14x^2+39x+25=0
Ans: x=-1 or x=-1.79
Q7721) Solve 12x^2+46x+20=0
Ans: x=-0.5 or x=-3.33
Q7722) Solve 15x^2+22x+8=0
Ans: x=-0.67 or x=-0.8
Q7723) Solve 19x^2+20x+1=0
Ans: x=-0.05 or x=-1
Q7724) Solve 6x^2+24x+18=0
Ans: x=-1 or x=-3
Q7725) Solve 9x^2+41x+46=0
Ans: x=-2 or x=-2.56
Q7726) Solve 4x^2+21x+27=0
Ans: x=-2.25 or x=-3
Q7727) Solve 11x^2+44x+44=0
Ans: x=-2 or x=-2
Q7728) Solve 7x^2+10x+3=0
Ans: x=-0.43 or x=-1
Q7729) Solve 27x^2+48x+5=0
Ans: x=-0.11 or x=-1.67
Q7730) Solve 17x^2+40x+12=0
Ans: x=-0.35 or x=-2
Q7731) Solve 3x^2+26x+23=0
Ans: x=-1 or x=-7.67
Q7732) Solve 20x^2+43x+23=0
Ans: x=-1 or x=-1.15
Q7733) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q7734) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q7735) Solve 19x^2+45x+26=0
Ans: x=-1 or x=-1.37
Q7736) Solve 15x^2+8x+1=0
Ans: x=-0.2 or x=-0.33
Q7737) Solve 3x^2+6x+3=0
Ans: x=-1 or x=-1
Q7738) Solve 18x^2+23x+5=0
Ans: x=-0.28 or x=-1
Q7739) Solve x^2+11x+10=0
Ans: x=-1 or x=-10
Q7740) Solve 18x^2+13x+2=0
Ans: x=-0.22 or x=-0.5
Q7741) Solve 18x^2+22x+4=0
Ans: x=-0.22 or x=-1
Q7742) Solve x^2+41x+40=0
Ans: x=-1 or x=-40
Q7743) Solve 44x^2+45x+9=0
Ans: x=-0.27 or x=-0.75
Q7744) Solve 16x^2+36x+18=0
Ans: x=-0.75 or x=-1.5
Q7745) Solve 12x^2+42x+30=0
Ans: x=-1 or x=-2.5
Q7746) Solve 22x^2+47x+21=0
Ans: x=-0.64 or x=-1.5
Q7747) Solve 40x^2+24x+2=0
Ans: x=-0.1 or x=-0.5
Q7748) Solve 12x^2+50x+48=0
Ans: x=-1.5 or x=-2.67
Q7749) Solve 3x^2+25x+50=0
Ans: x=-3.33 or x=-5
Q7750) Solve 22x^2+49x+24=0
Ans: x=-0.73 or x=-1.5
Q7751) Solve 28x^2+22x+4=0
Ans: x=-0.29 or x=-0.5
Q7752) Solve 11x^2+21x+10=0
Ans: x=-0.91 or x=-1
Q7753) Solve 7x^2+28x+21=0
Ans: x=-1 or x=-3
Q7754) Solve 45x^2+33x+2=0
Ans: x=-0.07 or x=-0.67
Q7755) Solve 33x^2+46x+13=0
Ans: x=-0.39 or x=-1
Q7756) Solve 18x^2+19x+1=0
Ans: x=-0.06 or x=-1
Q7757) Solve 28x^2+37x+12=0
Ans: x=-0.57 or x=-0.75
Q7758) Solve 9x^2+10x+1=0
Ans: x=-0.11 or x=-1
Q7759) Solve 10x^2+32x+6=0
Ans: x=-0.2 or x=-3
Q7760) Solve 24x^2+46x+22=0
Ans: x=-0.92 or x=-1
Q7761) Solve 25x^2+34x+9=0
Ans: x=-0.36 or x=-1
Q7762) Solve 8x^2+21x+10=0
Ans: x=-0.62 or x=-2
Q7763) Solve 27x^2+27x+6=0
Ans: x=-0.33 or x=-0.67
Q7764) Solve 2x^2+37x+18=0
Ans: x=-0.5 or x=-18
Q7765) Solve 45x^2+47x+12=0
Ans: x=-0.44 or x=-0.6
Q7766) Solve 23x^2+42x+19=0
Ans: x=-0.83 or x=-1
Q7767) Solve 20x^2+44x+24=0
Ans: x=-1 or x=-1.2
Q7768) Solve 12x^2+49x+15=0
Ans: x=-0.33 or x=-3.75
Q7769) Solve 12x^2+50x+8=0
Ans: x=-0.17 or x=-4
Q7770) Solve 30x^2+17x+2=0
Ans: x=-0.17 or x=-0.4
Q7771) Solve 16x^2+42x+17=0
Ans: x=-0.5 or x=-2.12
Q7772) Solve 8x^2+38x+45=0
Ans: x=-2.25 or x=-2.5
Q7773) Solve 24x^2+34x+7=0
Ans: x=-0.25 or x=-1.17
Q7774) Solve 9x^2+49x+20=0
Ans: x=-0.44 or x=-5
Q7775) Solve 12x^2+35x+18=0
Ans: x=-0.67 or x=-2.25
Q7776) Solve 23x^2+39x+16=0
Ans: x=-0.7 or x=-1
Q7777) Solve 32x^2+42x+10=0
Ans: x=-0.31 or x=-1
Q7778) Solve 10x^2+39x+14=0
Ans: x=-0.4 or x=-3.5
Q7779) Solve 10x^2+42x+36=0
Ans: x=-1.2 or x=-3
Q7780) Solve 9x^2+33x+28=0
Ans: x=-1.33 or x=-2.33
Q7781) Solve 9x^2+38x+33=0
Ans: x=-1.22 or x=-3
Q7782) Solve 35x^2+19x+2=0
Ans: x=-0.14 or x=-0.4
Q7783) Solve 30x^2+38x+8=0
Ans: x=-0.27 or x=-1
Q7784) Solve 12x^2+28x+8=0
Ans: x=-0.33 or x=-2
Q7785) Solve 7x^2+45x+50=0
Ans: x=-1.43 or x=-5
Q7786) Solve 38x^2+44x+6=0
Ans: x=-0.16 or x=-1
Q7787) Solve 14x^2+36x+22=0
Ans: x=-1 or x=-1.57
Q7788) Solve 8x^2+26x+20=0
Ans: x=-1.25 or x=-2
Q7789) Solve 9x^2+46x+40=0
Ans: x=-1.11 or x=-4
Q7790) Solve x^2+20x+36=0
Ans: x=-2 or x=-18
Q7791) Solve 12x^2+14x+4=0
Ans: x=-0.5 or x=-0.67
Q7792) Solve 16x^2+42x+17=0
Ans: x=-0.5 or x=-2.12
Q7793) Solve 7x^2+44x+37=0
Ans: x=-1 or x=-5.29
Q7794) Solve 6x^2+42x+36=0
Ans: x=-1 or x=-6
Q7795) Solve 6x^2+45x+39=0
Ans: x=-1 or x=-6.5
Q7796) Solve 34x^2+35x+1=0
Ans: x=-0.03 or x=-1
Q7797) Solve 15x^2+38x+16=0
Ans: x=-0.53 or x=-2
Q7798) Solve 7x^2+19x+10=0
Ans: x=-0.71 or x=-2
Q7799) Solve 15x^2+38x+11=0
Ans: x=-0.33 or x=-2.2
Q7800) Solve 11x^2+46x+48=0
Ans: x=-2 or x=-2.18
Q7801) Solve 14x^2+29x+2=0
Ans: x=-0.07 or x=-2
Q7802) Solve 7x^2+39x+32=0
Ans: x=-1 or x=-4.57
Q7803) Solve 26x^2+19x+3=0
Ans: x=-0.23 or x=-0.5
Q7804) Solve 50x^2+45x+7=0
Ans: x=-0.2 or x=-0.7
Q7805) Solve 6x^2+41x+19=0
Ans: x=-0.5 or x=-6.33
Q7806) Solve 34x^2+45x+14=0
Ans: x=-0.5 or x=-0.82
Q7807) Solve 12x^2+44x+32=0
Ans: x=-1 or x=-2.67
Q7808) Solve 16x^2+40x+25=0
Ans: x=-1.25 or x=-1.25
Q7809) Solve 15x^2+41x+22=0
Ans: x=-0.73 or x=-2
Q7810) Solve 4x^2+19x+21=0
Ans: x=-1.75 or x=-3
Q7811) Solve 9x^2+28x+19=0
Ans: x=-1 or x=-2.11
Q7812) Solve 22x^2+44x+22=0
Ans: x=-1 or x=-1
Q7813) Solve 16x^2+49x+3=0
Ans: x=-0.06 or x=-3
Q7814) Solve 5x^2+12x+4=0
Ans: x=-0.4 or x=-2
Q7815) Solve 10x^2+7x+1=0
Ans: x=-0.2 or x=-0.5
Q7816) Solve 2x^2+9x+7=0
Ans: x=-1 or x=-3.5
Q7817) Solve 27x^2+40x+13=0
Ans: x=-0.48 or x=-1
Q7818) Solve 35x^2+19x+2=0
Ans: x=-0.14 or x=-0.4
Q7819) Solve 4x^2+17x+18=0
Ans: x=-2 or x=-2.25
Q7820) Solve 42x^2+23x+3=0
Ans: x=-0.21 or x=-0.33
Q7821) Solve 18x^2+47x+5=0
Ans: x=-0.11 or x=-2.5
Q7822) Solve 6x^2+38x+32=0
Ans: x=-1 or x=-5.33
Q7823) Solve 20x^2+27x+9=0
Ans: x=-0.6 or x=-0.75
Q7824) Solve 9x^2+32x+28=0
Ans: x=-1.56 or x=-2
Q7825) Solve 9x^2+48x+28=0
Ans: x=-0.67 or x=-4.67
Q7826) Solve 16x^2+47x+31=0
Ans: x=-1 or x=-1.94
Q7827) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q7828) Solve 2x^2+18x+36=0
Ans: x=-3 or x=-6
Q7829) Solve 9x^2+41x+46=0
Ans: x=-2 or x=-2.56
Q7830) Solve 40x^2+39x+9=0
Ans: x=-0.37 or x=-0.6
Q7831) Solve 17x^2+40x+12=0
Ans: x=-0.35 or x=-2
Q7832) Solve 32x^2+38x+11=0
Ans: x=-0.5 or x=-0.69
Q7833) Solve 9x^2+20x+4=0
Ans: x=-0.22 or x=-2
Q7834) Solve 35x^2+49x+14=0
Ans: x=-0.4 or x=-1
Q7835) Solve 3x^2+35x+50=0
Ans: x=-1.67 or x=-10
Q7836) Solve 11x^2+21x+10=0
Ans: x=-0.91 or x=-1
Q7837) Solve 24x^2+10x+1=0
Ans: x=-0.17 or x=-0.25
Q7838) Solve 20x^2+41x+9=0
Ans: x=-0.25 or x=-1.8
Q7839) Solve 2x^2+18x+40=0
Ans: x=-4 or x=-5
Q7840) Solve 40x^2+31x+6=0
Ans: x=-0.37 or x=-0.4
Q7841) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q7842) Solve 18x^2+32x+14=0
Ans: x=-0.78 or x=-1
Q7843) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q7844) Solve 10x^2+15x+5=0
Ans: x=-0.5 or x=-1
Q7845) Solve 2x^2+18x+28=0
Ans: x=-2 or x=-7
Q7846) Solve 7x^2+28x+28=0
Ans: x=-2 or x=-2
Q7847) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q7848) Solve 5x^2+26x+32=0
Ans: x=-2 or x=-3.2
Q7849) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q7850) Solve 5x^2+33x+40=0
Ans: x=-1.6 or x=-5
Q7851) Solve 11x^2+49x+38=0
Ans: x=-1 or x=-3.45
Q7852) Solve 20x^2+19x+3=0
Ans: x=-0.2 or x=-0.75
Q7853) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q7854) Solve 7x^2+50x+7=0
Ans: x=-0.14 or x=-7
Q7855) Solve 3x^2+22x+39=0
Ans: x=-3 or x=-4.33
Q7856) Solve 6x^2+20x+6=0
Ans: x=-0.33 or x=-3
Q7857) Solve 5x^2+23x+24=0
Ans: x=-1.6 or x=-3
Q7858) Solve 6x^2+31x+39=0
Ans: x=-2.17 or x=-3
Q7859) Solve 22x^2+35x+12=0
Ans: x=-0.5 or x=-1.09
Q7860) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q7861) Solve 36x^2+37x+7=0
Ans: x=-0.25 or x=-0.78
Q7862) Solve 30x^2+17x+2=0
Ans: x=-0.17 or x=-0.4
Q7863) Solve 14x^2+42x+28=0
Ans: x=-1 or x=-2
Q7864) Solve 20x^2+37x+17=0
Ans: x=-0.85 or x=-1
Q7865) Solve 7x^2+44x+12=0
Ans: x=-0.29 or x=-6
Q7866) Solve 4x^2+28x+48=0
Ans: x=-3 or x=-4
Q7867) Solve 27x^2+39x+14=0
Ans: x=-0.67 or x=-0.78
Q7868) Solve 13x^2+43x+34=0
Ans: x=-1.31 or x=-2
Q7869) Solve 30x^2+47x+16=0
Ans: x=-0.5 or x=-1.07
Q7870) Solve 12x^2+44x+35=0
Ans: x=-1.17 or x=-2.5
Q7871) Solve 4x^2+39x+35=0
Ans: x=-1 or x=-8.75
Q7872) Solve 5x^2+34x+48=0
Ans: x=-2 or x=-4.8
Q7873) Solve 5x^2+29x+42=0
Ans: x=-2.8 or x=-3
Q7874) Solve 9x^2+30x+25=0
Ans: x=-1.67 or x=-1.67
Q7875) Solve 8x^2+38x+45=0
Ans: x=-2.25 or x=-2.5
Q7876) Solve 9x^2+26x+17=0
Ans: x=-1 or x=-1.89
Q7877) Solve 11x^2+16x+5=0
Ans: x=-0.45 or x=-1
Q7878) Solve 7x^2+29x+22=0
Ans: x=-1 or x=-3.14
Q7879) Solve 38x^2+41x+11=0
Ans: x=-0.5 or x=-0.58
Q7880) Solve 38x^2+46x+8=0
Ans: x=-0.21 or x=-1
Q7881) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q7882) Solve 18x^2+29x+10=0
Ans: x=-0.5 or x=-1.11
Q7883) Solve 3x^2+21x+30=0
Ans: x=-2 or x=-5
Q7884) Solve 32x^2+46x+14=0
Ans: x=-0.44 or x=-1
Q7885) Solve 38x^2+43x+5=0
Ans: x=-0.13 or x=-1
Q7886) Solve x^2+41x+40=0
Ans: x=-1 or x=-40
Q7887) Solve 12x^2+44x+40=0
Ans: x=-1.67 or x=-2
Q7888) Solve 14x^2+33x+19=0
Ans: x=-1 or x=-1.36
Q7889) Solve 2x^2+26x+44=0
Ans: x=-2 or x=-11
Q7890) Solve 24x^2+40x+6=0
Ans: x=-0.17 or x=-1.5
Q7891) Solve 26x^2+33x+10=0
Ans: x=-0.5 or x=-0.77
Q7892) Solve 16x^2+41x+25=0
Ans: x=-1 or x=-1.56
Q7893) Solve 11x^2+40x+21=0
Ans: x=-0.64 or x=-3
Q7894) Solve 26x^2+31x+5=0
Ans: x=-0.19 or x=-1
Q7895) Solve 50x^2+25x+2=0
Ans: x=-0.1 or x=-0.4
Q7896) Solve 18x^2+30x+12=0
Ans: x=-0.67 or x=-1
Q7897) Solve 15x^2+32x+9=0
Ans: x=-0.33 or x=-1.8
Q7898) Solve 2x^2+14x+24=0
Ans: x=-3 or x=-4
Q7899) Solve 33x^2+50x+13=0
Ans: x=-0.33 or x=-1.18
Q7900) Solve 20x^2+46x+18=0
Ans: x=-0.5 or x=-1.8
Q7901) Solve 8x^2+34x+30=0
Ans: x=-1.25 or x=-3
Q7902) Solve 7x^2+33x+20=0
Ans: x=-0.71 or x=-4
Q7903) Solve 14x^2+9x+1=0
Ans: x=-0.14 or x=-0.5
Q7904) Solve 20x^2+20x+5=0
Ans: x=-0.5 or x=-0.5
Q7905) Solve 44x^2+34x+6=0
Ans: x=-0.27 or x=-0.5
Q7906) Solve 8x^2+31x+21=0
Ans: x=-0.87 or x=-3
Q7907) Solve 4x^2+24x+35=0
Ans: x=-2.5 or x=-3.5
Q7908) Solve 30x^2+47x+17=0
Ans: x=-0.57 or x=-1
Q7909) Solve 6x^2+13x+2=0
Ans: x=-0.17 or x=-2
Q7910) Solve 41x^2+47x+6=0
Ans: x=-0.15 or x=-1
Q7911) Solve 12x^2+41x+34=0
Ans: x=-1.42 or x=-2
Q7912) Solve 28x^2+33x+9=0
Ans: x=-0.43 or x=-0.75
Q7913) Solve 10x^2+16x+6=0
Ans: x=-0.6 or x=-1
Q7914) Solve 6x^2+14x+8=0
Ans: x=-1 or x=-1.33
Q7915) Solve 7x^2+9x+2=0
Ans: x=-0.29 or x=-1
Q7916) Solve 35x^2+47x+6=0
Ans: x=-0.14 or x=-1.2
Q7917) Solve 32x^2+20x+3=0
Ans: x=-0.25 or x=-0.37
Q7918) Solve 7x^2+48x+41=0
Ans: x=-1 or x=-5.86
Q7919) Solve 6x^2+39x+18=0
Ans: x=-0.5 or x=-6
Q7920) Solve 4x^2+10x+4=0
Ans: x=-0.5 or x=-2
Q7921) Solve 8x^2+35x+27=0
Ans: x=-1 or x=-3.37
Q7922) Solve 34x^2+27x+5=0
Ans: x=-0.29 or x=-0.5
Q7923) Solve 38x^2+48x+10=0
Ans: x=-0.26 or x=-1
Q7924) Solve 15x^2+19x+4=0
Ans: x=-0.27 or x=-1
Q7925) Solve 12x^2+17x+6=0
Ans: x=-0.67 or x=-0.75
Q7926) Solve 8x^2+14x+5=0
Ans: x=-0.5 or x=-1.25
Q7927) Solve 30x^2+23x+4=0
Ans: x=-0.27 or x=-0.5
Q7928) Solve 26x^2+45x+19=0
Ans: x=-0.73 or x=-1
Q7929) Solve x^2+15x+14=0
Ans: x=-1 or x=-14
Q7930) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q7931) Solve 48x^2+32x+4=0
Ans: x=-0.17 or x=-0.5
Q7932) Solve 32x^2+48x+10=0
Ans: x=-0.25 or x=-1.25
Q7933) Solve 11x^2+28x+17=0
Ans: x=-1 or x=-1.55
Q7934) Solve 40x^2+30x+5=0
Ans: x=-0.25 or x=-0.5
Q7935) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q7936) Solve 45x^2+33x+6=0
Ans: x=-0.33 or x=-0.4
Q7937) Solve 20x^2+18x+4=0
Ans: x=-0.4 or x=-0.5
Q7938) Solve 3x^2+18x+27=0
Ans: x=-3 or x=-3
Q7939) Solve 10x^2+40x+40=0
Ans: x=-2 or x=-2
Q7940) Solve 21x^2+38x+16=0
Ans: x=-0.67 or x=-1.14
Q7941) Solve 10x^2+47x+21=0
Ans: x=-0.5 or x=-4.2
Q7942) Solve 40x^2+38x+9=0
Ans: x=-0.45 or x=-0.5
Q7943) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q7944) Solve 9x^2+35x+34=0
Ans: x=-1.89 or x=-2
Q7945) Solve 11x^2+14x+3=0
Ans: x=-0.27 or x=-1
Q7946) Solve 3x^2+8x+5=0
Ans: x=-1 or x=-1.67
Q7947) Solve 11x^2+34x+24=0
Ans: x=-1.09 or x=-2
Q7948) Solve 38x^2+23x+2=0
Ans: x=-0.11 or x=-0.5
Q7949) Solve 33x^2+45x+12=0
Ans: x=-0.36 or x=-1
Q7950) Solve 18x^2+48x+32=0
Ans: x=-1.33 or x=-1.33
Q7951) Solve 23x^2+37x+14=0
Ans: x=-0.61 or x=-1
Q7952) Solve 39x^2+43x+10=0
Ans: x=-0.33 or x=-0.77
Q7953) Solve 2x^2+14x+24=0
Ans: x=-3 or x=-4
Q7954) Solve 10x^2+27x+11=0
Ans: x=-0.5 or x=-2.2
Q7955) Solve 11x^2+42x+27=0
Ans: x=-0.82 or x=-3
Q7956) Solve 6x^2+5x+1=0
Ans: x=-0.33 or x=-0.5
Q7957) Solve 10x^2+42x+8=0
Ans: x=-0.2 or x=-4
Q7958) Solve 36x^2+45x+9=0
Ans: x=-0.25 or x=-1
Q7959) Solve 6x^2+29x+20=0
Ans: x=-0.83 or x=-4
Q7960) Solve 9x^2+49x+20=0
Ans: x=-0.44 or x=-5
Q7961) Solve 3x^2+11x+10=0
Ans: x=-1.67 or x=-2
Q7962) Solve 9x^2+27x+14=0
Ans: x=-0.67 or x=-2.33
Q7963) Solve 10x^2+39x+36=0
Ans: x=-1.5 or x=-2.4
Q7964) Solve 7x^2+37x+36=0
Ans: x=-1.29 or x=-4
Q7965) Solve 10x^2+36x+32=0
Ans: x=-1.6 or x=-2
Q7966) Solve 3x^2+30x+48=0
Ans: x=-2 or x=-8
Q7967) Solve 9x^2+10x+1=0
Ans: x=-0.11 or x=-1
Q7968) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q7969) Solve 50x^2+35x+5=0
Ans: x=-0.2 or x=-0.5
Q7970) Solve 13x^2+48x+35=0
Ans: x=-1 or x=-2.69
Q7971) Solve 4x^2+38x+48=0
Ans: x=-1.5 or x=-8
Q7972) Solve 16x^2+42x+20=0
Ans: x=-0.62 or x=-2
Q7973) Solve x^2+20x+19=0
Ans: x=-1 or x=-19
Q7974) Solve 3x^2+29x+40=0
Ans: x=-1.67 or x=-8
Q7975) Solve 12x^2+45x+27=0
Ans: x=-0.75 or x=-3
Q7976) Solve 50x^2+41x+8=0
Ans: x=-0.32 or x=-0.5
Q7977) Solve 25x^2+41x+16=0
Ans: x=-0.64 or x=-1
Q7978) Solve x^2+6x+8=0
Ans: x=-2 or x=-4
Q7979) Solve 15x^2+22x+7=0
Ans: x=-0.47 or x=-1
Q7980) Solve 12x^2+49x+44=0
Ans: x=-1.33 or x=-2.75
Q7981) Solve 12x^2+7x+1=0
Ans: x=-0.25 or x=-0.33
Q7982) Solve 15x^2+49x+34=0
Ans: x=-1 or x=-2.27
Q7983) Solve 12x^2+35x+23=0
Ans: x=-1 or x=-1.92
Q7984) Solve 18x^2+37x+15=0
Ans: x=-0.56 or x=-1.5
Q7985) Solve 11x^2+45x+4=0
Ans: x=-0.09 or x=-4
Q7986) Solve 7x^2+22x+16=0
Ans: x=-1.14 or x=-2
Q7987) Solve 19x^2+27x+8=0
Ans: x=-0.42 or x=-1
Q7988) Solve 31x^2+42x+11=0
Ans: x=-0.35 or x=-1
Q7989) Solve 19x^2+46x+16=0
Ans: x=-0.42 or x=-2
Q7990) Solve 35x^2+39x+10=0
Ans: x=-0.4 or x=-0.71
Q7991) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q7992) Solve 2x^2+21x+27=0
Ans: x=-1.5 or x=-9
Q7993) Solve 8x^2+36x+28=0
Ans: x=-1 or x=-3.5
Q7994) Solve 7x^2+30x+8=0
Ans: x=-0.29 or x=-4
Q7995) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q7996) Solve 40x^2+44x+4=0
Ans: x=-0.1 or x=-1
Q7997) Solve 37x^2+45x+8=0
Ans: x=-0.22 or x=-1
Q7998) Solve 35x^2+46x+11=0
Ans: x=-0.31 or x=-1
Q7999) Solve 2x^2+27x+36=0
Ans: x=-1.5 or x=-12
Q8000) Solve 12x^2+50x+50=0
Ans: x=-1.67 or x=-2.5
Q8001) Solve 17x^2+40x+12=0
Ans: x=-0.35 or x=-2
Q8002) Solve 7x^2+26x+24=0
Ans: x=-1.71 or x=-2
Q8003) Solve 20x^2+33x+7=0
Ans: x=-0.25 or x=-1.4
Q8004) Solve 14x^2+43x+20=0
Ans: x=-0.57 or x=-2.5
Q8005) Solve 3x^2+15x+12=0
Ans: x=-1 or x=-4
Q8006) Solve 42x^2+44x+2=0
Ans: x=-0.05 or x=-1
Q8007) Solve 48x^2+41x+6=0
Ans: x=-0.19 or x=-0.67
Q8008) Solve 6x^2+47x+22=0
Ans: x=-0.5 or x=-7.33
Q8009) Solve 23x^2+45x+22=0
Ans: x=-0.96 or x=-1
Q8010) Solve 12x^2+12x+3=0
Ans: x=-0.5 or x=-0.5
Q8011) Solve x^2+18x+17=0
Ans: x=-1 or x=-17
Q8012) Solve 18x^2+45x+13=0
Ans: x=-0.33 or x=-2.17
Q8013) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Q8014) Solve 18x^2+41x+21=0
Ans: x=-0.78 or x=-1.5
Q8015) Solve 14x^2+23x+9=0
Ans: x=-0.64 or x=-1
Q8016) Solve 5x^2+18x+16=0
Ans: x=-1.6 or x=-2
Q8017) Solve 17x^2+44x+20=0
Ans: x=-0.59 or x=-2
Q8018) Solve 15x^2+44x+28=0
Ans: x=-0.93 or x=-2
Q8019) Solve 30x^2+32x+2=0
Ans: x=-0.07 or x=-1
Q8020) Solve 22x^2+49x+10=0
Ans: x=-0.23 or x=-2
Q8021) Solve x^2+8x+16=0
Ans: x=-4 or x=-4
Q8022) Solve 40x^2+14x+1=0
Ans: x=-0.1 or x=-0.25
Q8023) Solve 29x^2+44x+15=0
Ans: x=-0.52 or x=-1
Q8024) Solve 11x^2+43x+30=0
Ans: x=-0.91 or x=-3
Q8025) Solve 5x^2+32x+44=0
Ans: x=-2 or x=-4.4
Q8026) Solve 14x^2+36x+22=0
Ans: x=-1 or x=-1.57
Q8027) Solve 10x^2+38x+36=0
Ans: x=-1.8 or x=-2
Q8028) Solve 48x^2+48x+9=0
Ans: x=-0.25 or x=-0.75
Q8029) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q8030) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q8031) Solve 12x^2+27x+6=0
Ans: x=-0.25 or x=-2
Q8032) Solve 12x^2+13x+3=0
Ans: x=-0.33 or x=-0.75
Q8033) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q8034) Solve 4x^2+36x+17=0
Ans: x=-0.5 or x=-8.5
Q8035) Solve 18x^2+49x+33=0
Ans: x=-1.22 or x=-1.5
Q8036) Solve 12x^2+49x+4=0
Ans: x=-0.08 or x=-4
Q8037) Solve 8x^2+30x+22=0
Ans: x=-1 or x=-2.75
Q8038) Solve 14x^2+46x+32=0
Ans: x=-1 or x=-2.29
Q8039) Solve 3x^2+38x+24=0
Ans: x=-0.67 or x=-12
Q8040) Solve 15x^2+48x+33=0
Ans: x=-1 or x=-2.2
Q8041) Solve 15x^2+37x+20=0
Ans: x=-0.8 or x=-1.67
Q8042) Solve 19x^2+45x+14=0
Ans: x=-0.37 or x=-2
Q8043) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q8044) Solve 34x^2+50x+16=0
Ans: x=-0.47 or x=-1
Q8045) Solve 4x^2+29x+7=0
Ans: x=-0.25 or x=-7
Q8046) Solve 31x^2+41x+10=0
Ans: x=-0.32 or x=-1
Q8047) Solve 11x^2+13x+2=0
Ans: x=-0.18 or x=-1
Q8048) Solve 30x^2+41x+14=0
Ans: x=-0.67 or x=-0.7
Q8049) Solve 6x^2+39x+33=0
Ans: x=-1 or x=-5.5
Q8050) Solve 24x^2+28x+8=0
Ans: x=-0.5 or x=-0.67
Q8051) Solve 48x^2+28x+4=0
Ans: x=-0.25 or x=-0.33
Q8052) Solve 11x^2+39x+34=0
Ans: x=-1.55 or x=-2
Q8053) Solve 32x^2+18x+1=0
Ans: x=-0.06 or x=-0.5
Q8054) Solve 32x^2+48x+16=0
Ans: x=-0.5 or x=-1
Q8055) Solve 48x^2+46x+5=0
Ans: x=-0.12 or x=-0.83
Q8056) Solve 45x^2+39x+6=0
Ans: x=-0.2 or x=-0.67
Q8057) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q8058) Solve 28x^2+31x+3=0
Ans: x=-0.11 or x=-1
Q8059) Solve 40x^2+31x+6=0
Ans: x=-0.37 or x=-0.4
Q8060) Solve 12x^2+38x+28=0
Ans: x=-1.17 or x=-2
Q8061) Solve 14x^2+50x+44=0
Ans: x=-1.57 or x=-2
Q8062) Solve 40x^2+32x+6=0
Ans: x=-0.3 or x=-0.5
Q8063) Solve 7x^2+24x+9=0
Ans: x=-0.43 or x=-3
Q8064) Solve 29x^2+36x+7=0
Ans: x=-0.24 or x=-1
Q8065) Solve 14x^2+21x+7=0
Ans: x=-0.5 or x=-1
Q8066) Solve 12x^2+35x+22=0
Ans: x=-0.92 or x=-2
Q8067) Solve 29x^2+42x+13=0
Ans: x=-0.45 or x=-1
Q8068) Solve 2x^2+19x+9=0
Ans: x=-0.5 or x=-9
Q8069) Solve 12x^2+27x+6=0
Ans: x=-0.25 or x=-2
Q8070) Solve 9x^2+33x+18=0
Ans: x=-0.67 or x=-3
Q8071) Solve 4x^2+19x+21=0
Ans: x=-1.75 or x=-3
Q8072) Solve 36x^2+47x+11=0
Ans: x=-0.31 or x=-1
Q8073) Solve 4x^2+13x+9=0
Ans: x=-1 or x=-2.25
Q8074) Solve 8x^2+16x+6=0
Ans: x=-0.5 or x=-1.5
Q8075) Solve 19x^2+28x+9=0
Ans: x=-0.47 or x=-1
Q8076) Solve 14x^2+24x+10=0
Ans: x=-0.71 or x=-1
Q8077) Solve 24x^2+35x+9=0
Ans: x=-0.33 or x=-1.12
Q8078) Solve 38x^2+48x+10=0
Ans: x=-0.26 or x=-1
Q8079) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q8080) Solve 4x^2+22x+24=0
Ans: x=-1.5 or x=-4
Q8081) Solve 38x^2+48x+10=0
Ans: x=-0.26 or x=-1
Q8082) Solve 40x^2+22x+3=0
Ans: x=-0.25 or x=-0.3
Q8083) Solve 7x^2+32x+16=0
Ans: x=-0.57 or x=-4
Q8084) Solve 5x^2+31x+42=0
Ans: x=-2 or x=-4.2
Q8085) Solve 28x^2+16x+1=0
Ans: x=-0.07 or x=-0.5
Q8086) Solve 11x^2+47x+12=0
Ans: x=-0.27 or x=-4
Q8087) Solve 6x^2+34x+20=0
Ans: x=-0.67 or x=-5
Q8088) Solve 32x^2+46x+15=0
Ans: x=-0.5 or x=-0.94
Q8089) Solve 11x^2+35x+26=0
Ans: x=-1.18 or x=-2
Q8090) Solve 34x^2+38x+4=0
Ans: x=-0.12 or x=-1
Q8091) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q8092) Solve 30x^2+43x+14=0
Ans: x=-0.5 or x=-0.93
Q8093) Solve 24x^2+40x+16=0
Ans: x=-0.67 or x=-1
Q8094) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q8095) Solve 4x^2+32x+28=0
Ans: x=-1 or x=-7
Q8096) Solve 16x^2+20x+4=0
Ans: x=-0.25 or x=-1
Q8097) Solve 8x^2+17x+9=0
Ans: x=-1 or x=-1.12
Q8098) Solve 12x^2+32x+20=0
Ans: x=-1 or x=-1.67
Q8099) Solve 5x^2+29x+36=0
Ans: x=-1.8 or x=-4
Q8100) Solve 15x^2+39x+24=0
Ans: x=-1 or x=-1.6
Q8101) Solve 24x^2+16x+2=0
Ans: x=-0.17 or x=-0.5
Q8102) Solve 10x^2+46x+48=0
Ans: x=-1.6 or x=-3
Q8103) Solve 9x^2+45x+26=0
Ans: x=-0.67 or x=-4.33
Q8104) Solve 34x^2+47x+15=0
Ans: x=-0.5 or x=-0.88
Q8105) Solve 6x^2+26x+28=0
Ans: x=-2 or x=-2.33
Q8106) Solve 4x^2+27x+23=0
Ans: x=-1 or x=-5.75
Q8107) Solve 4x^2+22x+18=0
Ans: x=-1 or x=-4.5
Q8108) Solve 16x^2+48x+11=0
Ans: x=-0.25 or x=-2.75
Q8109) Solve 19x^2+47x+28=0
Ans: x=-1 or x=-1.47
Q8110) Solve 33x^2+46x+16=0
Ans: x=-0.67 or x=-0.73
Q8111) Solve 11x^2+14x+3=0
Ans: x=-0.27 or x=-1
Q8112) Solve 6x^2+37x+50=0
Ans: x=-2 or x=-4.17
Q8113) Solve 50x^2+29x+2=0
Ans: x=-0.08 or x=-0.5
Q8114) Solve 8x^2+26x+11=0
Ans: x=-0.5 or x=-2.75
Q8115) Solve 8x^2+18x+10=0
Ans: x=-1 or x=-1.25
Q8116) Solve 21x^2+50x+24=0
Ans: x=-0.67 or x=-1.71
Q8117) Solve 14x^2+18x+4=0
Ans: x=-0.29 or x=-1
Q8118) Solve 19x^2+50x+31=0
Ans: x=-1 or x=-1.63
Q8119) Solve 5x^2+32x+27=0
Ans: x=-1 or x=-5.4
Q8120) Solve 3x^2+28x+44=0
Ans: x=-2 or x=-7.33
Q8121) Solve 2x^2+45x+43=0
Ans: x=-1 or x=-21.5
Q8122) Solve 6x^2+29x+13=0
Ans: x=-0.5 or x=-4.33
Q8123) Solve 12x^2+35x+22=0
Ans: x=-0.92 or x=-2
Q8124) Solve 4x^2+21x+5=0
Ans: x=-0.25 or x=-5
Q8125) Solve 31x^2+35x+4=0
Ans: x=-0.13 or x=-1
Q8126) Solve 16x^2+46x+33=0
Ans: x=-1.37 or x=-1.5
Q8127) Solve 2x^2+21x+45=0
Ans: x=-3 or x=-7.5
Q8128) Solve 8x^2+46x+21=0
Ans: x=-0.5 or x=-5.25
Q8129) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q8130) Solve 23x^2+43x+20=0
Ans: x=-0.87 or x=-1
Q8131) Solve 20x^2+47x+24=0
Ans: x=-0.75 or x=-1.6
Q8132) Solve 21x^2+47x+10=0
Ans: x=-0.24 or x=-2
Q8133) Solve 8x^2+28x+20=0
Ans: x=-1 or x=-2.5
Q8134) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q8135) Solve 29x^2+44x+15=0
Ans: x=-0.52 or x=-1
Q8136) Solve 29x^2+31x+2=0
Ans: x=-0.07 or x=-1
Q8137) Solve 6x^2+22x+20=0
Ans: x=-1.67 or x=-2
Q8138) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q8139) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q8140) Solve 6x^2+43x+42=0
Ans: x=-1.17 or x=-6
Q8141) Solve 42x^2+47x+11=0
Ans: x=-0.33 or x=-0.79
Q8142) Solve 20x^2+49x+9=0
Ans: x=-0.2 or x=-2.25
Q8143) Solve 14x^2+25x+6=0
Ans: x=-0.29 or x=-1.5
Q8144) Solve 2x^2+49x+47=0
Ans: x=-1 or x=-23.5
Q8145) Solve 5x^2+21x+22=0
Ans: x=-2 or x=-2.2
Q8146) Solve 9x^2+33x+28=0
Ans: x=-1.33 or x=-2.33
Q8147) Solve 10x^2+35x+25=0
Ans: x=-1 or x=-2.5
Q8148) Solve 24x^2+43x+5=0
Ans: x=-0.12 or x=-1.67
Q8149) Solve 20x^2+34x+12=0
Ans: x=-0.5 or x=-1.2
Q8150) Solve 39x^2+32x+4=0
Ans: x=-0.15 or x=-0.67
Q8151) Solve 34x^2+43x+9=0
Ans: x=-0.26 or x=-1
Q8152) Solve 6x^2+43x+37=0
Ans: x=-1 or x=-6.17
Q8153) Solve 4x^2+45x+11=0
Ans: x=-0.25 or x=-11
Q8154) Solve 4x^2+16x+12=0
Ans: x=-1 or x=-3
Q8155) Solve 24x^2+44x+12=0
Ans: x=-0.33 or x=-1.5
Q8156) Solve 10x^2+14x+4=0
Ans: x=-0.4 or x=-1
Q8157) Solve 36x^2+25x+4=0
Ans: x=-0.25 or x=-0.44
Q8158) Solve 15x^2+49x+26=0
Ans: x=-0.67 or x=-2.6
Q8159) Solve 2x^2+7x+6=0
Ans: x=-1.5 or x=-2
Q8160) Solve 7x^2+11x+4=0
Ans: x=-0.57 or x=-1
Q8161) Solve 12x^2+43x+31=0
Ans: x=-1 or x=-2.58
Q8162) Solve 33x^2+38x+5=0
Ans: x=-0.15 or x=-1
Q8163) Solve 15x^2+41x+28=0
Ans: x=-1.33 or x=-1.4
Q8164) Solve 6x^2+32x+42=0
Ans: x=-2.33 or x=-3
Q8165) Solve 8x^2+28x+12=0
Ans: x=-0.5 or x=-3
Q8166) Solve 12x^2+19x+7=0
Ans: x=-0.58 or x=-1
Q8167) Solve 5x^2+34x+45=0
Ans: x=-1.8 or x=-5
Q8168) Solve 4x^2+28x+24=0
Ans: x=-1 or x=-6
Q8169) Solve 37x^2+44x+7=0
Ans: x=-0.19 or x=-1
Q8170) Solve 7x^2+33x+20=0
Ans: x=-0.71 or x=-4
Q8171) Solve 30x^2+35x+10=0
Ans: x=-0.5 or x=-0.67
Q8172) Solve 11x^2+28x+17=0
Ans: x=-1 or x=-1.55
Q8173) Solve 10x^2+47x+9=0
Ans: x=-0.2 or x=-4.5
Q8174) Solve 30x^2+33x+3=0
Ans: x=-0.1 or x=-1
Q8175) Solve 10x^2+23x+6=0
Ans: x=-0.3 or x=-2
Q8176) Solve 22x^2+13x+1=0
Ans: x=-0.09 or x=-0.5
Q8177) Solve 11x^2+49x+48=0
Ans: x=-1.45 or x=-3
Q8178) Solve 8x^2+37x+42=0
Ans: x=-2 or x=-2.62
Q8179) Solve 15x^2+23x+6=0
Ans: x=-0.33 or x=-1.2
Q8180) Solve 7x^2+45x+18=0
Ans: x=-0.43 or x=-6
Q8181) Solve 27x^2+30x+3=0
Ans: x=-0.11 or x=-1
Q8182) Solve 12x^2+42x+30=0
Ans: x=-1 or x=-2.5
Q8183) Solve 39x^2+41x+2=0
Ans: x=-0.05 or x=-1
Q8184) Solve 5x^2+37x+14=0
Ans: x=-0.4 or x=-7
Q8185) Solve 11x^2+39x+34=0
Ans: x=-1.55 or x=-2
Q8186) Solve 20x^2+20x+5=0
Ans: x=-0.5 or x=-0.5
Q8187) Solve 4x^2+26x+12=0
Ans: x=-0.5 or x=-6
Q8188) Solve 9x^2+38x+40=0
Ans: x=-2 or x=-2.22
Q8189) Solve 21x^2+35x+14=0
Ans: x=-0.67 or x=-1
Q8190) Solve 2x^2+13x+21=0
Ans: x=-3 or x=-3.5
Q8191) Solve 10x^2+41x+33=0
Ans: x=-1.1 or x=-3
Q8192) Solve 6x^2+29x+13=0
Ans: x=-0.5 or x=-4.33
Q8193) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q8194) Solve 17x^2+46x+24=0
Ans: x=-0.71 or x=-2
Q8195) Solve 14x^2+37x+15=0
Ans: x=-0.5 or x=-2.14
Q8196) Solve 45x^2+34x+5=0
Ans: x=-0.2 or x=-0.56
Q8197) Solve 9x^2+43x+28=0
Ans: x=-0.78 or x=-4
Q8198) Solve 11x^2+43x+32=0
Ans: x=-1 or x=-2.91
Q8199) Solve 24x^2+41x+11=0
Ans: x=-0.33 or x=-1.37
Q8200) Solve 30x^2+49x+6=0
Ans: x=-0.13 or x=-1.5
Q8201) Solve 5x^2+22x+17=0
Ans: x=-1 or x=-3.4
Q8202) Solve 16x^2+40x+24=0
Ans: x=-1 or x=-1.5
Q8203) Solve 26x^2+15x+1=0
Ans: x=-0.08 or x=-0.5
Q8204) Solve 15x^2+33x+6=0
Ans: x=-0.2 or x=-2
Q8205) Solve 17x^2+36x+4=0
Ans: x=-0.12 or x=-2
Q8206) Solve 33x^2+46x+16=0
Ans: x=-0.67 or x=-0.73
Q8207) Solve 14x^2+32x+18=0
Ans: x=-1 or x=-1.29
Q8208) Solve 32x^2+32x+6=0
Ans: x=-0.25 or x=-0.75
Q8209) Solve 34x^2+19x+1=0
Ans: x=-0.06 or x=-0.5
Q8210) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q8211) Solve 28x^2+34x+6=0
Ans: x=-0.21 or x=-1
Q8212) Solve x^2+16x+39=0
Ans: x=-3 or x=-13
Q8213) Solve 48x^2+30x+3=0
Ans: x=-0.12 or x=-0.5
Q8214) Solve 8x^2+34x+30=0
Ans: x=-1.25 or x=-3
Q8215) Solve 21x^2+43x+2=0
Ans: x=-0.05 or x=-2
Q8216) Solve 12x^2+45x+33=0
Ans: x=-1 or x=-2.75
Q8217) Solve 10x^2+47x+37=0
Ans: x=-1 or x=-3.7
Q8218) Solve 16x^2+26x+9=0
Ans: x=-0.5 or x=-1.12
Q8219) Solve 40x^2+42x+8=0
Ans: x=-0.25 or x=-0.8
Q8220) Solve x^2+8x+12=0
Ans: x=-2 or x=-6
Q8221) Solve 9x^2+22x+8=0
Ans: x=-0.44 or x=-2
Q8222) Solve 25x^2+35x+6=0
Ans: x=-0.2 or x=-1.2
Q8223) Solve 4x^2+17x+15=0
Ans: x=-1.25 or x=-3
Q8224) Solve 8x^2+40x+42=0
Ans: x=-1.5 or x=-3.5
Q8225) Solve 6x^2+13x+2=0
Ans: x=-0.17 or x=-2
Q8226) Solve 20x^2+46x+12=0
Ans: x=-0.3 or x=-2
Q8227) Solve 42x^2+47x+5=0
Ans: x=-0.12 or x=-1
Q8228) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q8229) Solve 10x^2+31x+24=0
Ans: x=-1.5 or x=-1.6
Q8230) Solve 3x^2+8x+4=0
Ans: x=-0.67 or x=-2
Q8231) Solve 34x^2+42x+8=0
Ans: x=-0.24 or x=-1
Q8232) Solve 3x^2+25x+28=0
Ans: x=-1.33 or x=-7
Q8233) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q8234) Solve 13x^2+41x+6=0
Ans: x=-0.15 or x=-3
Q8235) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q8236) Solve x^2+23x+42=0
Ans: x=-2 or x=-21
Q8237) Solve 16x^2+47x+31=0
Ans: x=-1 or x=-1.94
Q8238) Solve 5x^2+28x+39=0
Ans: x=-2.6 or x=-3
Q8239) Solve 22x^2+41x+12=0
Ans: x=-0.36 or x=-1.5
Q8240) Solve 24x^2+32x+10=0
Ans: x=-0.5 or x=-0.83
Q8241) Solve 6x^2+50x+44=0
Ans: x=-1 or x=-7.33
Q8242) Solve 12x^2+26x+10=0
Ans: x=-0.5 or x=-1.67
Q8243) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q8244) Solve 10x^2+7x+1=0
Ans: x=-0.2 or x=-0.5
Q8245) Solve 48x^2+49x+11=0
Ans: x=-0.33 or x=-0.69
Q8246) Solve 8x^2+46x+45=0
Ans: x=-1.25 or x=-4.5
Q8247) Solve 18x^2+48x+24=0
Ans: x=-0.67 or x=-2
Q8248) Solve 6x^2+23x+10=0
Ans: x=-0.5 or x=-3.33
Q8249) Solve 4x^2+16x+12=0
Ans: x=-1 or x=-3
Q8250) Solve 8x^2+37x+29=0
Ans: x=-1 or x=-3.62
Q8251) Solve 38x^2+43x+12=0
Ans: x=-0.5 or x=-0.63
Q8252) Solve 5x^2+49x+44=0
Ans: x=-1 or x=-8.8
Q8253) Solve 8x^2+29x+26=0
Ans: x=-1.62 or x=-2
Q8254) Solve 9x^2+42x+13=0
Ans: x=-0.33 or x=-4.33
Q8255) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q8256) Solve 35x^2+36x+1=0
Ans: x=-0.03 or x=-1
Q8257) Solve 24x^2+47x+13=0
Ans: x=-0.33 or x=-1.62
Q8258) Solve 36x^2+34x+8=0
Ans: x=-0.44 or x=-0.5
Q8259) Solve 17x^2+43x+18=0
Ans: x=-0.53 or x=-2
Q8260) Solve 26x^2+41x+3=0
Ans: x=-0.08 or x=-1.5
Q8261) Solve 8x^2+40x+42=0
Ans: x=-1.5 or x=-3.5
Q8262) Solve 13x^2+22x+9=0
Ans: x=-0.69 or x=-1
Q8263) Solve 32x^2+22x+3=0
Ans: x=-0.19 or x=-0.5
Q8264) Solve 4x^2+20x+21=0
Ans: x=-1.5 or x=-3.5
Q8265) Solve 4x^2+12x+8=0
Ans: x=-1 or x=-2
Q8266) Solve 5x^2+41x+8=0
Ans: x=-0.2 or x=-8
Q8267) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q8268) Solve 12x^2+41x+35=0
Ans: x=-1.67 or x=-1.75
Q8269) Solve 12x^2+46x+30=0
Ans: x=-0.83 or x=-3
Q8270) Solve 50x^2+50x+8=0
Ans: x=-0.2 or x=-0.8
Q8271) Solve 19x^2+27x+8=0
Ans: x=-0.42 or x=-1
Q8272) Solve 21x^2+23x+2=0
Ans: x=-0.1 or x=-1
Q8273) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q8274) Solve 42x^2+37x+8=0
Ans: x=-0.38 or x=-0.5
Q8275) Solve 12x^2+26x+4=0
Ans: x=-0.17 or x=-2
Q8276) Solve 2x^2+22x+48=0
Ans: x=-3 or x=-8
Q8277) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q8278) Solve 8x^2+36x+16=0
Ans: x=-0.5 or x=-4
Q8279) Solve 16x^2+49x+3=0
Ans: x=-0.06 or x=-3
Q8280) Solve 14x^2+21x+7=0
Ans: x=-0.5 or x=-1
Q8281) Solve 18x^2+36x+16=0
Ans: x=-0.67 or x=-1.33
Q8282) Solve 20x^2+27x+7=0
Ans: x=-0.35 or x=-1
Q8283) Solve 35x^2+43x+12=0
Ans: x=-0.43 or x=-0.8
Q8284) Solve 18x^2+15x+2=0
Ans: x=-0.17 or x=-0.67
Q8285) Solve 8x^2+6x+1=0
Ans: x=-0.25 or x=-0.5
Q8286) Solve 11x^2+36x+9=0
Ans: x=-0.27 or x=-3
Q8287) Solve 3x^2+16x+5=0
Ans: x=-0.33 or x=-5
Q8288) Solve 12x^2+50x+42=0
Ans: x=-1.17 or x=-3
Q8289) Solve 15x^2+37x+18=0
Ans: x=-0.67 or x=-1.8
Q8290) Solve 24x^2+34x+11=0
Ans: x=-0.5 or x=-0.92
Q8291) Solve 42x^2+43x+10=0
Ans: x=-0.36 or x=-0.67
Q8292) Solve 3x^2+38x+24=0
Ans: x=-0.67 or x=-12
Q8293) Solve 10x^2+11x+1=0
Ans: x=-0.1 or x=-1
Q8294) Solve 11x^2+12x+1=0
Ans: x=-0.09 or x=-1
Q8295) Solve 10x^2+41x+4=0
Ans: x=-0.1 or x=-4
Q8296) Solve 15x^2+48x+36=0
Ans: x=-1.2 or x=-2
Q8297) Solve 18x^2+31x+6=0
Ans: x=-0.22 or x=-1.5
Q8298) Solve 27x^2+35x+8=0
Ans: x=-0.3 or x=-1
Q8299) Solve 10x^2+31x+3=0
Ans: x=-0.1 or x=-3
Q8300) Solve 10x^2+45x+50=0
Ans: x=-2 or x=-2.5
Q8301) Solve 16x^2+46x+28=0
Ans: x=-0.87 or x=-2
Q8302) Solve 6x^2+17x+7=0
Ans: x=-0.5 or x=-2.33
Q8303) Solve 11x^2+38x+27=0
Ans: x=-1 or x=-2.45
Q8304) Solve 5x^2+43x+38=0
Ans: x=-1 or x=-7.6
Q8305) Solve 5x^2+23x+24=0
Ans: x=-1.6 or x=-3
Q8306) Solve 25x^2+41x+16=0
Ans: x=-0.64 or x=-1
Q8307) Solve 27x^2+32x+5=0
Ans: x=-0.19 or x=-1
Q8308) Solve 20x^2+43x+21=0
Ans: x=-0.75 or x=-1.4
Q8309) Solve 7x^2+37x+46=0
Ans: x=-2 or x=-3.29
Q8310) Solve 25x^2+26x+1=0
Ans: x=-0.04 or x=-1
Q8311) Solve 30x^2+45x+15=0
Ans: x=-0.5 or x=-1
Q8312) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q8313) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q8314) Solve 7x^2+39x+20=0
Ans: x=-0.57 or x=-5
Q8315) Solve 16x^2+50x+39=0
Ans: x=-1.5 or x=-1.62
Q8316) Solve 22x^2+44x+22=0
Ans: x=-1 or x=-1
Q8317) Solve 11x^2+25x+14=0
Ans: x=-1 or x=-1.27
Q8318) Solve 3x^2+34x+40=0
Ans: x=-1.33 or x=-10
Q8319) Solve 18x^2+12x+2=0
Ans: x=-0.33 or x=-0.33
Q8320) Solve 18x^2+48x+24=0
Ans: x=-0.67 or x=-2
Q8321) Solve 35x^2+43x+8=0
Ans: x=-0.23 or x=-1
Q8322) Solve 6x^2+8x+2=0
Ans: x=-0.33 or x=-1
Q8323) Solve 28x^2+31x+3=0
Ans: x=-0.11 or x=-1
Q8324) Solve 36x^2+36x+5=0
Ans: x=-0.17 or x=-0.83
Q8325) Solve 4x^2+44x+21=0
Ans: x=-0.5 or x=-10.5
Q8326) Solve 2x^2+17x+35=0
Ans: x=-3.5 or x=-5
Q8327) Solve 8x^2+30x+13=0
Ans: x=-0.5 or x=-3.25
Q8328) Solve 7x^2+31x+34=0
Ans: x=-2 or x=-2.43
Q8329) Solve 26x^2+27x+7=0
Ans: x=-0.5 or x=-0.54
Q8330) Solve 30x^2+31x+1=0
Ans: x=-0.03 or x=-1
Q8331) Solve 10x^2+11x+3=0
Ans: x=-0.5 or x=-0.6
Q8332) Solve 6x^2+32x+32=0
Ans: x=-1.33 or x=-4
Q8333) Solve 10x^2+14x+4=0
Ans: x=-0.4 or x=-1
Q8334) Solve x^2+13x+40=0
Ans: x=-5 or x=-8
Q8335) Solve 20x^2+41x+20=0
Ans: x=-0.8 or x=-1.25
Q8336) Solve 50x^2+39x+7=0
Ans: x=-0.28 or x=-0.5
Q8337) Solve 37x^2+45x+8=0
Ans: x=-0.22 or x=-1
Q8338) Solve 22x^2+35x+3=0
Ans: x=-0.09 or x=-1.5
Q8339) Solve 22x^2+35x+13=0
Ans: x=-0.59 or x=-1
Q8340) Solve 4x^2+22x+24=0
Ans: x=-1.5 or x=-4
Q8341) Solve 42x^2+17x+1=0
Ans: x=-0.07 or x=-0.33
Q8342) Solve 22x^2+29x+7=0
Ans: x=-0.32 or x=-1
Q8343) Solve 20x^2+34x+12=0
Ans: x=-0.5 or x=-1.2
Q8344) Solve 3x^2+24x+21=0
Ans: x=-1 or x=-7
Q8345) Solve 9x^2+25x+16=0
Ans: x=-1 or x=-1.78
Q8346) Solve 18x^2+24x+6=0
Ans: x=-0.33 or x=-1
Q8347) Solve 14x^2+15x+1=0
Ans: x=-0.07 or x=-1
Q8348) Solve 11x^2+19x+8=0
Ans: x=-0.73 or x=-1
Q8349) Solve 19x^2+26x+7=0
Ans: x=-0.37 or x=-1
Q8350) Solve 18x^2+28x+10=0
Ans: x=-0.56 or x=-1
Q8351) Solve 3x^2+30x+48=0
Ans: x=-2 or x=-8
Q8352) Solve 36x^2+39x+10=0
Ans: x=-0.42 or x=-0.67
Q8353) Solve 21x^2+47x+26=0
Ans: x=-1 or x=-1.24
Q8354) Solve 7x^2+36x+45=0
Ans: x=-2.14 or x=-3
Q8355) Solve 11x^2+18x+7=0
Ans: x=-0.64 or x=-1
Q8356) Solve 2x^2+9x+7=0
Ans: x=-1 or x=-3.5
Q8357) Solve 15x^2+19x+6=0
Ans: x=-0.6 or x=-0.67
Q8358) Solve 40x^2+34x+7=0
Ans: x=-0.35 or x=-0.5
Q8359) Solve 10x^2+39x+27=0
Ans: x=-0.9 or x=-3
Q8360) Solve 7x^2+40x+33=0
Ans: x=-1 or x=-4.71
Q8361) Solve 5x^2+32x+27=0
Ans: x=-1 or x=-5.4
Q8362) Solve 26x^2+47x+12=0
Ans: x=-0.31 or x=-1.5
Q8363) Solve 35x^2+38x+8=0
Ans: x=-0.29 or x=-0.8
Q8364) Solve 30x^2+43x+13=0
Ans: x=-0.43 or x=-1
Q8365) Solve 17x^2+31x+14=0
Ans: x=-0.82 or x=-1
Q8366) Solve 2x^2+29x+27=0
Ans: x=-1 or x=-13.5
Q8367) Solve 46x^2+35x+6=0
Ans: x=-0.26 or x=-0.5
Q8368) Solve 4x^2+29x+45=0
Ans: x=-2.25 or x=-5
Q8369) Solve 36x^2+45x+14=0
Ans: x=-0.58 or x=-0.67
Q8370) Solve 7x^2+33x+26=0
Ans: x=-1 or x=-3.71
Q8371) Solve 50x^2+45x+10=0
Ans: x=-0.4 or x=-0.5
Q8372) Solve 2x^2+43x+41=0
Ans: x=-1 or x=-20.5
Q8373) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q8374) Solve 30x^2+48x+18=0
Ans: x=-0.6 or x=-1
Q8375) Solve 4x^2+26x+12=0
Ans: x=-0.5 or x=-6
Q8376) Solve 7x^2+36x+45=0
Ans: x=-2.14 or x=-3
Q8377) Solve 4x^2+11x+6=0
Ans: x=-0.75 or x=-2
Q8378) Solve 44x^2+46x+12=0
Ans: x=-0.5 or x=-0.55
Q8379) Solve 26x^2+43x+6=0
Ans: x=-0.15 or x=-1.5
Q8380) Solve 4x^2+24x+20=0
Ans: x=-1 or x=-5
Q8381) Solve 24x^2+38x+3=0
Ans: x=-0.08 or x=-1.5
Q8382) Solve 7x^2+32x+25=0
Ans: x=-1 or x=-3.57
Q8383) Solve 10x^2+33x+27=0
Ans: x=-1.5 or x=-1.8
Q8384) Solve 5x^2+30x+40=0
Ans: x=-2 or x=-4
Q8385) Solve 6x^2+28x+30=0
Ans: x=-1.67 or x=-3
Q8386) Solve 20x^2+28x+8=0
Ans: x=-0.4 or x=-1
Q8387) Solve 11x^2+37x+26=0
Ans: x=-1 or x=-2.36
Q8388) Solve 2x^2+27x+13=0
Ans: x=-0.5 or x=-13
Q8389) Solve 22x^2+45x+18=0
Ans: x=-0.55 or x=-1.5
Q8390) Solve 5x^2+29x+36=0
Ans: x=-1.8 or x=-4
Q8391) Solve 45x^2+37x+6=0
Ans: x=-0.22 or x=-0.6
Q8392) Solve x^2+42x+41=0
Ans: x=-1 or x=-41
Q8393) Solve 12x^2+33x+18=0
Ans: x=-0.75 or x=-2
Q8394) Solve x^2+10x+9=0
Ans: x=-1 or x=-9
Q8395) Solve 6x^2+37x+31=0
Ans: x=-1 or x=-5.17
Q8396) Solve 14x^2+29x+15=0
Ans: x=-1 or x=-1.07
Q8397) Solve 39x^2+46x+11=0
Ans: x=-0.33 or x=-0.85
Q8398) Solve 12x^2+38x+30=0
Ans: x=-1.5 or x=-1.67
Q8399) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q8400) Solve 45x^2+34x+5=0
Ans: x=-0.2 or x=-0.56
Q8401) Solve 32x^2+47x+15=0
Ans: x=-0.47 or x=-1
Q8402) Solve 20x^2+46x+26=0
Ans: x=-1 or x=-1.3
Q8403) Solve 7x^2+34x+24=0
Ans: x=-0.86 or x=-4
Q8404) Solve 4x^2+4x+1=0
Ans: x=-0.5 or x=-0.5
Q8405) Solve 10x^2+39x+35=0
Ans: x=-1.4 or x=-2.5
Q8406) Solve 12x^2+35x+8=0
Ans: x=-0.25 or x=-2.67
Q8407) Solve 39x^2+46x+11=0
Ans: x=-0.33 or x=-0.85
Q8408) Solve 8x^2+41x+50=0
Ans: x=-2 or x=-3.12
Q8409) Solve 14x^2+45x+36=0
Ans: x=-1.5 or x=-1.71
Q8410) Solve 12x^2+31x+9=0
Ans: x=-0.33 or x=-2.25
Q8411) Solve 13x^2+45x+38=0
Ans: x=-1.46 or x=-2
Q8412) Solve 27x^2+34x+7=0
Ans: x=-0.26 or x=-1
Q8413) Solve 18x^2+47x+19=0
Ans: x=-0.5 or x=-2.11
Q8414) Solve 4x^2+17x+13=0
Ans: x=-1 or x=-3.25
Q8415) Solve 35x^2+43x+12=0
Ans: x=-0.43 or x=-0.8
Q8416) Solve 6x^2+21x+15=0
Ans: x=-1 or x=-2.5
Q8417) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q8418) Solve 28x^2+40x+12=0
Ans: x=-0.43 or x=-1
Q8419) Solve 32x^2+28x+5=0
Ans: x=-0.25 or x=-0.62
Q8420) Solve 7x^2+33x+38=0
Ans: x=-2 or x=-2.71
Q8421) Solve 23x^2+44x+21=0
Ans: x=-0.91 or x=-1
Q8422) Solve 13x^2+21x+8=0
Ans: x=-0.62 or x=-1
Q8423) Solve 10x^2+45x+20=0
Ans: x=-0.5 or x=-4
Q8424) Solve 2x^2+4x+2=0
Ans: x=-1 or x=-1
Q8425) Solve 30x^2+44x+16=0
Ans: x=-0.67 or x=-0.8
Q8426) Solve 48x^2+50x+2=0
Ans: x=-0.04 or x=-1
Q8427) Solve 3x^2+8x+4=0
Ans: x=-0.67 or x=-2
Q8428) Solve 9x^2+45x+36=0
Ans: x=-1 or x=-4
Q8429) Solve 10x^2+26x+16=0
Ans: x=-1 or x=-1.6
Q8430) Solve 24x^2+46x+17=0
Ans: x=-0.5 or x=-1.42
Q8431) Solve 17x^2+38x+21=0
Ans: x=-1 or x=-1.24
Q8432) Solve 17x^2+20x+3=0
Ans: x=-0.18 or x=-1
Q8433) Solve 39x^2+38x+8=0
Ans: x=-0.31 or x=-0.67
Q8434) Solve 2x^2+8x+6=0
Ans: x=-1 or x=-3
Q8435) Solve 18x^2+24x+6=0
Ans: x=-0.33 or x=-1
Q8436) Solve 15x^2+17x+4=0
Ans: x=-0.33 or x=-0.8
Q8437) Solve 8x^2+50x+42=0
Ans: x=-1 or x=-5.25
Q8438) Solve 4x^2+20x+21=0
Ans: x=-1.5 or x=-3.5
Q8439) Solve 14x^2+44x+32=0
Ans: x=-1.14 or x=-2
Q8440) Solve 2x^2+29x+50=0
Ans: x=-2 or x=-12.5
Q8441) Solve 2x^2+14x+12=0
Ans: x=-1 or x=-6
Q8442) Solve 36x^2+43x+7=0
Ans: x=-0.19 or x=-1
Q8443) Solve 14x^2+33x+10=0
Ans: x=-0.36 or x=-2
Q8444) Solve 32x^2+36x+4=0
Ans: x=-0.12 or x=-1
Q8445) Solve 37x^2+49x+12=0
Ans: x=-0.32 or x=-1
Q8446) Solve 2x^2+5x+2=0
Ans: x=-0.5 or x=-2
Q8447) Solve x^2+14x+33=0
Ans: x=-3 or x=-11
Q8448) Solve 42x^2+43x+6=0
Ans: x=-0.17 or x=-0.86
Q8449) Solve 24x^2+38x+15=0
Ans: x=-0.75 or x=-0.83
Q8450) Solve 5x^2+27x+22=0
Ans: x=-1 or x=-4.4
Q8451) Solve x^2+19x+18=0
Ans: x=-1 or x=-18
Q8452) Solve 10x^2+29x+10=0
Ans: x=-0.4 or x=-2.5
Q8453) Solve 2x^2+28x+48=0
Ans: x=-2 or x=-12
Q8454) Solve 19x^2+42x+23=0
Ans: x=-1 or x=-1.21
Q8455) Solve 48x^2+50x+13=0
Ans: x=-0.5 or x=-0.54
Q8456) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q8457) Solve x^2+14x+33=0
Ans: x=-3 or x=-11
Q8458) Solve 22x^2+39x+9=0
Ans: x=-0.27 or x=-1.5
Q8459) Solve 38x^2+39x+1=0
Ans: x=-0.03 or x=-1
Q8460) Solve 16x^2+37x+10=0
Ans: x=-0.31 or x=-2
Q8461) Solve 12x^2+24x+9=0
Ans: x=-0.5 or x=-1.5
Q8462) Solve 32x^2+41x+9=0
Ans: x=-0.28 or x=-1
Q8463) Solve 6x^2+47x+22=0
Ans: x=-0.5 or x=-7.33
Q8464) Solve 23x^2+48x+25=0
Ans: x=-1 or x=-1.09
Q8465) Solve 22x^2+50x+28=0
Ans: x=-1 or x=-1.27
Q8466) Solve 32x^2+41x+9=0
Ans: x=-0.28 or x=-1
Q8467) Solve 10x^2+33x+27=0
Ans: x=-1.5 or x=-1.8
Q8468) Solve 17x^2+35x+2=0
Ans: x=-0.06 or x=-2
Q8469) Solve 22x^2+40x+18=0
Ans: x=-0.82 or x=-1
Q8470) Solve x^2+12x+35=0
Ans: x=-5 or x=-7
Q8471) Solve 21x^2+49x+14=0
Ans: x=-0.33 or x=-2
Q8472) Solve x^2+16x+28=0
Ans: x=-2 or x=-14
Q8473) Solve 33x^2+26x+5=0
Ans: x=-0.33 or x=-0.45
Q8474) Solve 5x^2+42x+37=0
Ans: x=-1 or x=-7.4
Q8475) Solve 12x^2+8x+1=0
Ans: x=-0.17 or x=-0.5
Q8476) Solve 5x^2+24x+28=0
Ans: x=-2 or x=-2.8
Q8477) Solve 14x^2+15x+4=0
Ans: x=-0.5 or x=-0.57
Q8478) Solve 10x^2+24x+14=0
Ans: x=-1 or x=-1.4
Q8479) Solve 12x^2+36x+24=0
Ans: x=-1 or x=-2
Q8480) Solve 35x^2+38x+8=0
Ans: x=-0.29 or x=-0.8
Q8481) Solve 23x^2+40x+17=0
Ans: x=-0.74 or x=-1
Q8482) Solve 24x^2+50x+4=0
Ans: x=-0.08 or x=-2
Q8483) Solve 46x^2+49x+3=0
Ans: x=-0.07 or x=-1
Q8484) Solve 21x^2+38x+5=0
Ans: x=-0.14 or x=-1.67
Q8485) Solve 16x^2+29x+13=0
Ans: x=-0.81 or x=-1
Q8486) Solve 16x^2+12x+2=0
Ans: x=-0.25 or x=-0.5
Q8487) Solve 3x^2+15x+12=0
Ans: x=-1 or x=-4
Q8488) Solve 8x^2+33x+34=0
Ans: x=-2 or x=-2.12
Q8489) Solve 9x^2+33x+24=0
Ans: x=-1 or x=-2.67
Q8490) Solve 13x^2+46x+33=0
Ans: x=-1 or x=-2.54
Q8491) Solve 9x^2+40x+44=0
Ans: x=-2 or x=-2.44
Q8492) Solve 39x^2+40x+9=0
Ans: x=-0.33 or x=-0.69
Q8493) Solve 14x^2+25x+11=0
Ans: x=-0.79 or x=-1
Q8494) Solve 17x^2+26x+9=0
Ans: x=-0.53 or x=-1
Q8495) Solve 27x^2+41x+14=0
Ans: x=-0.52 or x=-1
Q8496) Solve 39x^2+44x+12=0
Ans: x=-0.46 or x=-0.67
Q8497) Solve 4x^2+45x+11=0
Ans: x=-0.25 or x=-11
Q8498) Solve 8x^2+38x+45=0
Ans: x=-2.25 or x=-2.5
Q8499) Solve 17x^2+18x+1=0
Ans: x=-0.06 or x=-1
Q8500) Solve 18x^2+45x+28=0
Ans: x=-1.17 or x=-1.33
Q8501) Solve 25x^2+25x+4=0
Ans: x=-0.2 or x=-0.8
Q8502) Solve 7x^2+23x+16=0
Ans: x=-1 or x=-2.29
Q8503) Solve 2x^2+17x+36=0
Ans: x=-4 or x=-4.5
Q8504) Solve 6x^2+31x+5=0
Ans: x=-0.17 or x=-5
Q8505) Solve x^2+13x+30=0
Ans: x=-3 or x=-10
Q8506) Solve 16x^2+40x+16=0
Ans: x=-0.5 or x=-2
Q8507) Solve 36x^2+37x+7=0
Ans: x=-0.25 or x=-0.78
Q8508) Solve 21x^2+31x+10=0
Ans: x=-0.48 or x=-1
Q8509) Solve 16x^2+50x+6=0
Ans: x=-0.12 or x=-3
Q8510) Solve 12x^2+44x+35=0
Ans: x=-1.17 or x=-2.5
Q8511) Solve 48x^2+48x+12=0
Ans: x=-0.5 or x=-0.5
Q8512) Solve 3x^2+19x+20=0
Ans: x=-1.33 or x=-5
Q8513) Solve 42x^2+43x+11=0
Ans: x=-0.5 or x=-0.52
Q8514) Solve 6x^2+43x+7=0
Ans: x=-0.17 or x=-7
Q8515) Solve 6x^2+50x+16=0
Ans: x=-0.33 or x=-8
Q8516) Solve 49x^2+14x+1=0
Ans: x=-0.14 or x=-0.14
Q8517) Solve 4x^2+14x+6=0
Ans: x=-0.5 or x=-3
Q8518) Solve 19x^2+50x+24=0
Ans: x=-0.63 or x=-2
Q8519) Solve 6x^2+35x+29=0
Ans: x=-1 or x=-4.83
Q8520) Solve 42x^2+35x+7=0
Ans: x=-0.33 or x=-0.5
Q8521) Solve 8x^2+33x+34=0
Ans: x=-2 or x=-2.12
Q8522) Solve 35x^2+47x+6=0
Ans: x=-0.14 or x=-1.2
Q8523) Solve 28x^2+15x+2=0
Ans: x=-0.25 or x=-0.29
Q8524) Solve 8x^2+35x+12=0
Ans: x=-0.37 or x=-4
Q8525) Solve 13x^2+16x+3=0
Ans: x=-0.23 or x=-1
Q8526) Solve 36x^2+45x+14=0
Ans: x=-0.58 or x=-0.67
Q8527) Solve 2x^2+12x+10=0
Ans: x=-1 or x=-5
Q8528) Solve 18x^2+45x+28=0
Ans: x=-1.17 or x=-1.33
Q8529) Solve 19x^2+31x+12=0
Ans: x=-0.63 or x=-1
Q8530) Solve 2x^2+33x+16=0
Ans: x=-0.5 or x=-16
Q8531) Solve 40x^2+39x+9=0
Ans: x=-0.37 or x=-0.6
Q8532) Solve 8x^2+33x+4=0
Ans: x=-0.12 or x=-4
Q8533) Solve 12x^2+16x+4=0
Ans: x=-0.33 or x=-1
Q8534) Solve 7x^2+29x+4=0
Ans: x=-0.14 or x=-4
Q8535) Solve 9x^2+26x+16=0
Ans: x=-0.89 or x=-2
Q8536) Solve 12x^2+28x+15=0
Ans: x=-0.83 or x=-1.5
Q8537) Solve 4x^2+23x+30=0
Ans: x=-2 or x=-3.75
Q8538) Solve 12x^2+49x+49=0
Ans: x=-1.75 or x=-2.33
Q8539) Solve 19x^2+41x+6=0
Ans: x=-0.16 or x=-2
Q8540) Solve 6x^2+19x+13=0
Ans: x=-1 or x=-2.17
Q8541) Solve 14x^2+43x+33=0
Ans: x=-1.5 or x=-1.57
Q8542) Solve 25x^2+50x+25=0
Ans: x=-1 or x=-1
Q8543) Solve 7x^2+33x+36=0
Ans: x=-1.71 or x=-3
Q8544) Solve 4x^2+21x+20=0
Ans: x=-1.25 or x=-4
Q8545) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q8546) Solve 8x^2+34x+36=0
Ans: x=-2 or x=-2.25
Q8547) Solve 21x^2+37x+16=0
Ans: x=-0.76 or x=-1
Q8548) Solve 45x^2+41x+4=0
Ans: x=-0.11 or x=-0.8
Q8549) Solve 6x^2+49x+30=0
Ans: x=-0.67 or x=-7.5
Q8550) Solve 2x^2+21x+27=0
Ans: x=-1.5 or x=-9
Q8551) Solve 29x^2+35x+6=0
Ans: x=-0.21 or x=-1
Q8552) Solve 20x^2+41x+9=0
Ans: x=-0.25 or x=-1.8
Q8553) Solve 10x^2+49x+18=0
Ans: x=-0.4 or x=-4.5
Q8554) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q8555) Solve 3x^2+37x+44=0
Ans: x=-1.33 or x=-11
Q8556) Solve 12x^2+26x+12=0
Ans: x=-0.67 or x=-1.5
Q8557) Solve 35x^2+37x+2=0
Ans: x=-0.06 or x=-1
Q8558) Solve 27x^2+30x+7=0
Ans: x=-0.33 or x=-0.78
Q8559) Solve 4x^2+15x+11=0
Ans: x=-1 or x=-2.75
Q8560) Solve 7x^2+35x+28=0
Ans: x=-1 or x=-4
Q8561) Solve 22x^2+17x+3=0
Ans: x=-0.27 or x=-0.5
Q8562) Solve 7x^2+43x+40=0
Ans: x=-1.14 or x=-5
Q8563) Solve 5x^2+10x+5=0
Ans: x=-1 or x=-1
Q8564) Solve 29x^2+39x+10=0
Ans: x=-0.34 or x=-1
Q8565) Solve 27x^2+46x+19=0
Ans: x=-0.7 or x=-1
Q8566) Solve 9x^2+19x+10=0
Ans: x=-1 or x=-1.11
Q8567) Solve 3x^2+18x+27=0
Ans: x=-3 or x=-3
Q8568) Solve 2x^2+20x+18=0
Ans: x=-1 or x=-9
Q8569) Solve 22x^2+23x+6=0
Ans: x=-0.5 or x=-0.55
Q8570) Solve 50x^2+37x+6=0
Ans: x=-0.24 or x=-0.5
Q8571) Solve 42x^2+31x+4=0
Ans: x=-0.17 or x=-0.57
Q8572) Solve 3x^2+36x+33=0
Ans: x=-1 or x=-11
Q8573) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q8574) Solve 14x^2+48x+40=0
Ans: x=-1.43 or x=-2
Q8575) Solve 14x^2+44x+32=0
Ans: x=-1.14 or x=-2
Q8576) Solve 4x^2+24x+35=0
Ans: x=-2.5 or x=-3.5
Q8577) Solve 36x^2+30x+4=0
Ans: x=-0.17 or x=-0.67
Q8578) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q8579) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q8580) Solve 24x^2+44x+12=0
Ans: x=-0.33 or x=-1.5
Q8581) Solve 19x^2+21x+2=0
Ans: x=-0.11 or x=-1
Q8582) Solve 31x^2+41x+10=0
Ans: x=-0.32 or x=-1
Q8583) Solve 11x^2+49x+38=0
Ans: x=-1 or x=-3.45
Q8584) Solve 6x^2+21x+18=0
Ans: x=-1.5 or x=-2
Q8585) Solve 10x^2+16x+6=0
Ans: x=-0.6 or x=-1
Q8586) Solve 22x^2+37x+15=0
Ans: x=-0.68 or x=-1
Q8587) Solve 3x^2+11x+10=0
Ans: x=-1.67 or x=-2
Q8588) Solve 21x^2+44x+15=0
Ans: x=-0.43 or x=-1.67
Q8589) Solve 8x^2+50x+12=0
Ans: x=-0.25 or x=-6
Q8590) Solve 9x^2+28x+3=0
Ans: x=-0.11 or x=-3
Q8591) Solve 44x^2+37x+3=0
Ans: x=-0.09 or x=-0.75
Q8592) Solve 10x^2+49x+18=0
Ans: x=-0.4 or x=-4.5
Q8593) Solve 6x^2+20x+6=0
Ans: x=-0.33 or x=-3
Q8594) Solve 3x^2+19x+16=0
Ans: x=-1 or x=-5.33
Q8595) Solve 4x^2+36x+45=0
Ans: x=-1.5 or x=-7.5
Q8596) Solve 10x^2+45x+20=0
Ans: x=-0.5 or x=-4
Q8597) Solve 27x^2+37x+10=0
Ans: x=-0.37 or x=-1
Q8598) Solve 20x^2+32x+12=0
Ans: x=-0.6 or x=-1
Q8599) Solve 26x^2+29x+3=0
Ans: x=-0.12 or x=-1
Q8600) Solve 14x^2+22x+8=0
Ans: x=-0.57 or x=-1
Q8601) Solve 5x^2+47x+18=0
Ans: x=-0.4 or x=-9
Q8602) Solve 16x^2+20x+6=0
Ans: x=-0.5 or x=-0.75
Q8603) Solve 5x^2+22x+17=0
Ans: x=-1 or x=-3.4
Q8604) Solve 6x^2+41x+44=0
Ans: x=-1.33 or x=-5.5
Q8605) Solve 2x^2+12x+18=0
Ans: x=-3 or x=-3
Q8606) Solve 15x^2+47x+14=0
Ans: x=-0.33 or x=-2.8
Q8607) Solve 5x^2+50x+45=0
Ans: x=-1 or x=-9
Q8608) Solve 28x^2+44x+15=0
Ans: x=-0.5 or x=-1.07
Q8609) Solve 22x^2+27x+8=0
Ans: x=-0.5 or x=-0.73
Q8610) Solve 17x^2+35x+2=0
Ans: x=-0.06 or x=-2
Q8611) Solve 9x^2+14x+5=0
Ans: x=-0.56 or x=-1
Q8612) Solve 2x^2+35x+33=0
Ans: x=-1 or x=-16.5
Q8613) Solve 18x^2+33x+9=0
Ans: x=-0.33 or x=-1.5
Q8614) Solve 24x^2+11x+1=0
Ans: x=-0.12 or x=-0.33
Q8615) Solve 27x^2+33x+10=0
Ans: x=-0.56 or x=-0.67
Q8616) Solve 18x^2+36x+18=0
Ans: x=-1 or x=-1
Q8617) Solve 14x^2+32x+8=0
Ans: x=-0.29 or x=-2
Q8618) Solve 6x^2+49x+30=0
Ans: x=-0.67 or x=-7.5
Q8619) Solve 15x^2+32x+16=0
Ans: x=-0.8 or x=-1.33
Q8620) Solve 7x^2+30x+32=0
Ans: x=-2 or x=-2.29
Q8621) Solve 3x^2+34x+31=0
Ans: x=-1 or x=-10.33
Q8622) Solve 6x^2+25x+21=0
Ans: x=-1.17 or x=-3
Q8623) Solve 25x^2+39x+14=0
Ans: x=-0.56 or x=-1
Q8624) Solve 5x^2+43x+38=0
Ans: x=-1 or x=-7.6
Q8625) Solve 19x^2+50x+24=0
Ans: x=-0.63 or x=-2
Q8626) Solve 12x^2+37x+11=0
Ans: x=-0.33 or x=-2.75
Q8627) Solve 10x^2+18x+8=0
Ans: x=-0.8 or x=-1
Q8628) Solve 39x^2+22x+3=0
Ans: x=-0.23 or x=-0.33
Q8629) Solve 7x^2+40x+48=0
Ans: x=-1.71 or x=-4
Q8630) Solve 25x^2+26x+1=0
Ans: x=-0.04 or x=-1
Q8631) Solve 2x^2+49x+47=0
Ans: x=-1 or x=-23.5
Q8632) Solve 8x^2+24x+18=0
Ans: x=-1.5 or x=-1.5
Q8633) Solve 40x^2+34x+6=0
Ans: x=-0.25 or x=-0.6
Q8634) Solve 16x^2+34x+13=0
Ans: x=-0.5 or x=-1.62
Q8635) Solve 4x^2+32x+15=0
Ans: x=-0.5 or x=-7.5
Q8636) Solve 2x^2+12x+10=0
Ans: x=-1 or x=-5
Q8637) Solve 15x^2+35x+20=0
Ans: x=-1 or x=-1.33
Q8638) Solve 3x^2+23x+20=0
Ans: x=-1 or x=-6.67
Q8639) Solve 15x^2+49x+24=0
Ans: x=-0.6 or x=-2.67
Q8640) Solve 4x^2+24x+20=0
Ans: x=-1 or x=-5
Q8641) Solve 40x^2+46x+6=0
Ans: x=-0.15 or x=-1
Q8642) Solve 21x^2+19x+4=0
Ans: x=-0.33 or x=-0.57
Q8643) Solve 40x^2+23x+3=0
Ans: x=-0.2 or x=-0.37
Q8644) Solve 3x^2+25x+50=0
Ans: x=-3.33 or x=-5
Q8645) Solve 16x^2+42x+20=0
Ans: x=-0.62 or x=-2
Q8646) Solve 13x^2+44x+36=0
Ans: x=-1.38 or x=-2
Q8647) Solve 16x^2+43x+27=0
Ans: x=-1 or x=-1.69
Q8648) Solve 24x^2+34x+5=0
Ans: x=-0.17 or x=-1.25
Q8649) Solve x^2+31x+30=0
Ans: x=-1 or x=-30
Q8650) Solve x^2+37x+36=0
Ans: x=-1 or x=-36
Q8651) Solve 26x^2+43x+17=0
Ans: x=-0.65 or x=-1
Q8652) Solve 6x^2+43x+37=0
Ans: x=-1 or x=-6.17
Q8653) Solve 11x^2+25x+14=0
Ans: x=-1 or x=-1.27
Q8654) Solve 27x^2+18x+3=0
Ans: x=-0.33 or x=-0.33
Q8655) Solve 19x^2+50x+24=0
Ans: x=-0.63 or x=-2
Q8656) Solve 25x^2+48x+23=0
Ans: x=-0.92 or x=-1
Q8657) Solve x^2+5x+4=0
Ans: x=-1 or x=-4
Q8658) Solve 39x^2+35x+6=0
Ans: x=-0.23 or x=-0.67
Q8659) Solve 42x^2+32x+6=0
Ans: x=-0.33 or x=-0.43
Q8660) Solve 12x^2+46x+40=0
Ans: x=-1.33 or x=-2.5
Q8661) Solve 25x^2+26x+1=0
Ans: x=-0.04 or x=-1
Q8662) Solve 36x^2+45x+9=0
Ans: x=-0.25 or x=-1
Q8663) Solve 12x^2+39x+30=0
Ans: x=-1.25 or x=-2
Q8664) Solve 9x^2+18x+8=0
Ans: x=-0.67 or x=-1.33
Q8665) Solve 13x^2+18x+5=0
Ans: x=-0.38 or x=-1
Q8666) Solve 14x^2+33x+18=0
Ans: x=-0.86 or x=-1.5
Q8667) Solve 6x^2+16x+10=0
Ans: x=-1 or x=-1.67
Q8668) Solve 4x^2+7x+3=0
Ans: x=-0.75 or x=-1
Q8669) Solve 5x^2+46x+9=0
Ans: x=-0.2 or x=-9
Q8670) Solve 9x^2+43x+34=0
Ans: x=-1 or x=-3.78
Q8671) Solve 3x^2+28x+25=0
Ans: x=-1 or x=-8.33
Q8672) Solve 44x^2+45x+1=0
Ans: x=-0.02 or x=-1
Q8673) Solve 11x^2+36x+25=0
Ans: x=-1 or x=-2.27
Q8674) Solve 15x^2+31x+10=0
Ans: x=-0.4 or x=-1.67
Q8675) Solve 2x^2+15x+22=0
Ans: x=-2 or x=-5.5
Q8676) Solve 21x^2+32x+12=0
Ans: x=-0.67 or x=-0.86
Q8677) Solve 36x^2+45x+9=0
Ans: x=-0.25 or x=-1
Q8678) Solve x^2+25x+24=0
Ans: x=-1 or x=-24
Q8679) Solve 22x^2+40x+18=0
Ans: x=-0.82 or x=-1
Q8680) Solve 3x^2+48x+45=0
Ans: x=-1 or x=-15
Q8681) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q8682) Solve 10x^2+46x+48=0
Ans: x=-1.6 or x=-3
Q8683) Solve 11x^2+24x+4=0
Ans: x=-0.18 or x=-2
Q8684) Solve 22x^2+39x+17=0
Ans: x=-0.77 or x=-1
Q8685) Solve 5x^2+36x+7=0
Ans: x=-0.2 or x=-7
Q8686) Solve 21x^2+41x+18=0
Ans: x=-0.67 or x=-1.29
Q8687) Solve 2x^2+8x+8=0
Ans: x=-2 or x=-2
Q8688) Solve 14x^2+28x+14=0
Ans: x=-1 or x=-1
Q8689) Solve 7x^2+39x+44=0
Ans: x=-1.57 or x=-4
Q8690) Solve 15x^2+30x+15=0
Ans: x=-1 or x=-1
Q8691) Solve 36x^2+32x+7=0
Ans: x=-0.39 or x=-0.5
Q8692) Solve 5x^2+22x+21=0
Ans: x=-1.4 or x=-3
Q8693) Solve 6x^2+20x+14=0
Ans: x=-1 or x=-2.33
Q8694) Solve 30x^2+43x+11=0
Ans: x=-0.33 or x=-1.1
Q8695) Solve 24x^2+26x+5=0
Ans: x=-0.25 or x=-0.83
Q8696) Solve 49x^2+28x+3=0
Ans: x=-0.14 or x=-0.43
Q8697) Solve 36x^2+41x+8=0
Ans: x=-0.25 or x=-0.89
Q8698) Solve 15x^2+25x+10=0
Ans: x=-0.67 or x=-1
Q8699) Solve 15x^2+41x+26=0
Ans: x=-1 or x=-1.73
Q8700) Solve 6x^2+28x+22=0
Ans: x=-1 or x=-3.67
Q8701) Solve 20x^2+33x+10=0
Ans: x=-0.4 or x=-1.25
Q8702) Solve 12x^2+10x+2=0
Ans: x=-0.33 or x=-0.5
Q8703) Solve 18x^2+49x+10=0
Ans: x=-0.22 or x=-2.5
Q8704) Solve 18x^2+49x+10=0
Ans: x=-0.22 or x=-2.5
Q8705) Solve 10x^2+19x+7=0
Ans: x=-0.5 or x=-1.4
Q8706) Solve 8x^2+40x+50=0
Ans: x=-2.5 or x=-2.5
Q8707) Solve 24x^2+38x+14=0
Ans: x=-0.58 or x=-1
Q8708) Solve 28x^2+38x+12=0
Ans: x=-0.5 or x=-0.86
Q8709) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q8710) Solve 8x^2+16x+6=0
Ans: x=-0.5 or x=-1.5
Q8711) Solve 27x^2+48x+21=0
Ans: x=-0.78 or x=-1
Q8712) Solve 3x^2+34x+31=0
Ans: x=-1 or x=-10.33
Q8713) Solve 3x^2+29x+40=0
Ans: x=-1.67 or x=-8
Q8714) Solve 2x^2+11x+5=0
Ans: x=-0.5 or x=-5
Q8715) Solve 7x^2+46x+39=0
Ans: x=-1 or x=-5.57
Q8716) Solve 6x^2+35x+49=0
Ans: x=-2.33 or x=-3.5
Q8717) Solve x^2+8x+7=0
Ans: x=-1 or x=-7
Q8718) Solve 11x^2+18x+7=0
Ans: x=-0.64 or x=-1
Q8719) Solve 6x^2+43x+7=0
Ans: x=-0.17 or x=-7
Q8720) Solve 20x^2+39x+18=0
Ans: x=-0.75 or x=-1.2
Q8721) Solve 10x^2+43x+28=0
Ans: x=-0.8 or x=-3.5
Q8722) Solve 7x^2+18x+11=0
Ans: x=-1 or x=-1.57
Q8723) Solve 20x^2+39x+7=0
Ans: x=-0.2 or x=-1.75
Q8724) Solve 7x^2+21x+14=0
Ans: x=-1 or x=-2
Q8725) Solve 40x^2+38x+6=0
Ans: x=-0.2 or x=-0.75
Q8726) Solve 5x^2+42x+37=0
Ans: x=-1 or x=-7.4
Q8727) Solve 29x^2+41x+12=0
Ans: x=-0.41 or x=-1
Q8728) Solve 12x^2+38x+28=0
Ans: x=-1.17 or x=-2
Q8729) Solve 17x^2+45x+22=0
Ans: x=-0.65 or x=-2
Q8730) Solve 10x^2+39x+17=0
Ans: x=-0.5 or x=-3.4
Q8731) Solve 48x^2+32x+5=0
Ans: x=-0.25 or x=-0.42
Q8732) Solve 16x^2+36x+18=0
Ans: x=-0.75 or x=-1.5
Q8733) Solve 4x^2+10x+6=0
Ans: x=-1 or x=-1.5
Q8734) Solve 36x^2+45x+11=0
Ans: x=-0.33 or x=-0.92
Q8735) Solve 6x^2+31x+14=0
Ans: x=-0.5 or x=-4.67
Q8736) Solve 9x^2+41x+32=0
Ans: x=-1 or x=-3.56
Q8737) Solve 33x^2+34x+1=0
Ans: x=-0.03 or x=-1
Q8738) Solve 24x^2+46x+7=0
Ans: x=-0.17 or x=-1.75
Q8739) Solve 4x^2+21x+27=0
Ans: x=-2.25 or x=-3
Q8740) Solve 11x^2+38x+15=0
Ans: x=-0.45 or x=-3
Q8741) Solve 48x^2+41x+6=0
Ans: x=-0.19 or x=-0.67
Q8742) Solve 24x^2+22x+5=0
Ans: x=-0.42 or x=-0.5
Q8743) Solve 2x^2+21x+19=0
Ans: x=-1 or x=-9.5
Q8744) Solve 31x^2+34x+3=0
Ans: x=-0.1 or x=-1
Q8745) Solve 22x^2+24x+2=0
Ans: x=-0.09 or x=-1
Q8746) Solve 6x^2+46x+28=0
Ans: x=-0.67 or x=-7
Q8747) Solve 16x^2+48x+36=0
Ans: x=-1.5 or x=-1.5
Q8748) Solve x^2+26x+48=0
Ans: x=-2 or x=-24
Q8749) Solve 3x^2+11x+6=0
Ans: x=-0.67 or x=-3
Q8750) Solve 24x^2+40x+16=0
Ans: x=-0.67 or x=-1
Q8751) Solve 18x^2+24x+6=0
Ans: x=-0.33 or x=-1
Q8752) Solve 18x^2+49x+31=0
Ans: x=-1 or x=-1.72
Q8753) Solve 4x^2+18x+14=0
Ans: x=-1 or x=-3.5
Q8754) Solve 24x^2+50x+24=0
Ans: x=-0.75 or x=-1.33
Q8755) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q8756) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q8757) Solve 3x^2+33x+30=0
Ans: x=-1 or x=-10
Q8758) Solve 4x^2+31x+42=0
Ans: x=-1.75 or x=-6
Q8759) Solve 2x^2+7x+5=0
Ans: x=-1 or x=-2.5
Q8760) Solve 20x^2+37x+8=0
Ans: x=-0.25 or x=-1.6
Q8761) Solve 6x^2+44x+14=0
Ans: x=-0.33 or x=-7
Q8762) Solve 6x^2+35x+11=0
Ans: x=-0.33 or x=-5.5
Q8763) Solve 2x^2+21x+40=0
Ans: x=-2.5 or x=-8
Q8764) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q8765) Solve 7x^2+34x+39=0
Ans: x=-1.86 or x=-3
Q8766) Solve 33x^2+38x+9=0
Ans: x=-0.33 or x=-0.82
Q8767) Solve 15x^2+25x+10=0
Ans: x=-0.67 or x=-1
Q8768) Solve 5x^2+38x+48=0
Ans: x=-1.6 or x=-6
Q8769) Solve x^2+17x+30=0
Ans: x=-2 or x=-15
Q8770) Solve x^2+5x+6=0
Ans: x=-2 or x=-3
Q8771) Solve 6x^2+23x+20=0
Ans: x=-1.33 or x=-2.5
Q8772) Solve 31x^2+50x+19=0
Ans: x=-0.61 or x=-1
Q8773) Solve 34x^2+47x+13=0
Ans: x=-0.38 or x=-1
Q8774) Solve 3x^2+31x+28=0
Ans: x=-1 or x=-9.33
Q8775) Solve 6x^2+34x+28=0
Ans: x=-1 or x=-4.67
Q8776) Solve 22x^2+25x+3=0
Ans: x=-0.14 or x=-1
Q8777) Solve 5x^2+47x+42=0
Ans: x=-1 or x=-8.4
Q8778) Solve 9x^2+43x+50=0
Ans: x=-2 or x=-2.78
Q8779) Solve 12x^2+15x+3=0
Ans: x=-0.25 or x=-1
Q8780) Solve 11x^2+39x+28=0
Ans: x=-1 or x=-2.55
Q8781) Solve 43x^2+44x+1=0
Ans: x=-0.02 or x=-1
Q8782) Solve 9x^2+21x+10=0
Ans: x=-0.67 or x=-1.67
Q8783) Solve 27x^2+48x+13=0
Ans: x=-0.33 or x=-1.44
Q8784) Solve 30x^2+31x+5=0
Ans: x=-0.2 or x=-0.83
Q8785) Solve 11x^2+44x+33=0
Ans: x=-1 or x=-3
Q8786) Solve 4x^2+32x+39=0
Ans: x=-1.5 or x=-6.5
Q8787) Solve 3x^2+19x+28=0
Ans: x=-2.33 or x=-4
Q8788) Solve 5x^2+14x+8=0
Ans: x=-0.8 or x=-2
Q8789) Solve 30x^2+13x+1=0
Ans: x=-0.1 or x=-0.33
Q8790) Solve 15x^2+43x+30=0
Ans: x=-1.2 or x=-1.67
Q8791) Solve 15x^2+34x+8=0
Ans: x=-0.27 or x=-2
Q8792) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q8793) Solve 34x^2+29x+6=0
Ans: x=-0.35 or x=-0.5
Q8794) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q8795) Solve 10x^2+31x+3=0
Ans: x=-0.1 or x=-3
Q8796) Solve 33x^2+37x+4=0
Ans: x=-0.12 or x=-1
Q8797) Solve 3x^2+17x+20=0
Ans: x=-1.67 or x=-4
Q8798) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q8799) Solve 10x^2+44x+48=0
Ans: x=-2 or x=-2.4
Q8800) Solve 14x^2+34x+20=0
Ans: x=-1 or x=-1.43
Q8801) Solve 13x^2+40x+27=0
Ans: x=-1 or x=-2.08
Q8802) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q8803) Solve 9x^2+43x+50=0
Ans: x=-2 or x=-2.78
Q8804) Solve 4x^2+8x+4=0
Ans: x=-1 or x=-1
Q8805) Solve 11x^2+41x+30=0
Ans: x=-1 or x=-2.73
Q8806) Solve 8x^2+16x+8=0
Ans: x=-1 or x=-1
Q8807) Solve 26x^2+15x+1=0
Ans: x=-0.08 or x=-0.5
Q8808) Solve 2x^2+9x+7=0
Ans: x=-1 or x=-3.5
Q8809) Solve 8x^2+43x+44=0
Ans: x=-1.37 or x=-4
Q8810) Solve 50x^2+15x+1=0
Ans: x=-0.1 or x=-0.2
Q8811) Solve 7x^2+43x+40=0
Ans: x=-1.14 or x=-5
Q8812) Solve 12x^2+35x+25=0
Ans: x=-1.25 or x=-1.67
Q8813) Solve 13x^2+19x+6=0
Ans: x=-0.46 or x=-1
Q8814) Solve 4x^2+21x+27=0
Ans: x=-2.25 or x=-3
Q8815) Solve 28x^2+41x+15=0
Ans: x=-0.71 or x=-0.75
Q8816) Solve 2x^2+17x+36=0
Ans: x=-4 or x=-4.5
Q8817) Solve 3x^2+14x+16=0
Ans: x=-2 or x=-2.67
Q8818) Solve 30x^2+34x+8=0
Ans: x=-0.33 or x=-0.8
Q8819) Solve 17x^2+32x+15=0
Ans: x=-0.88 or x=-1
Q8820) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q8821) Solve 7x^2+24x+17=0
Ans: x=-1 or x=-2.43
Q8822) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q8823) Solve 30x^2+31x+5=0
Ans: x=-0.2 or x=-0.83
Q8824) Solve 5x^2+21x+18=0
Ans: x=-1.2 or x=-3
Q8825) Solve 2x^2+45x+22=0
Ans: x=-0.5 or x=-22
Q8826) Solve 25x^2+45x+20=0
Ans: x=-0.8 or x=-1
Q8827) Solve 7x^2+41x+30=0
Ans: x=-0.86 or x=-5
Q8828) Solve 6x^2+31x+35=0
Ans: x=-1.67 or x=-3.5
Q8829) Solve 14x^2+30x+4=0
Ans: x=-0.14 or x=-2
Q8830) Solve 35x^2+36x+1=0
Ans: x=-0.03 or x=-1
Q8831) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q8832) Solve 48x^2+50x+13=0
Ans: x=-0.5 or x=-0.54
Q8833) Solve 21x^2+50x+25=0
Ans: x=-0.71 or x=-1.67
Q8834) Solve 12x^2+43x+38=0
Ans: x=-1.58 or x=-2
Q8835) Solve 9x^2+40x+31=0
Ans: x=-1 or x=-3.44
Q8836) Solve 50x^2+25x+2=0
Ans: x=-0.1 or x=-0.4
Q8837) Solve 26x^2+41x+15=0
Ans: x=-0.58 or x=-1
Q8838) Solve 4x^2+36x+17=0
Ans: x=-0.5 or x=-8.5
Q8839) Solve 12x^2+38x+28=0
Ans: x=-1.17 or x=-2
Q8840) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q8841) Solve 9x^2+22x+13=0
Ans: x=-1 or x=-1.44
Q8842) Solve 18x^2+45x+18=0
Ans: x=-0.5 or x=-2
Q8843) Solve 9x^2+45x+36=0
Ans: x=-1 or x=-4
Q8844) Solve 16x^2+23x+7=0
Ans: x=-0.44 or x=-1
Q8845) Solve 22x^2+46x+24=0
Ans: x=-1 or x=-1.09
Q8846) Solve 41x^2+42x+1=0
Ans: x=-0.02 or x=-1
Q8847) Solve 32x^2+28x+5=0
Ans: x=-0.25 or x=-0.62
Q8848) Solve 30x^2+50x+20=0
Ans: x=-0.67 or x=-1
Q8849) Solve 17x^2+27x+10=0
Ans: x=-0.59 or x=-1
Q8850) Solve 8x^2+38x+9=0
Ans: x=-0.25 or x=-4.5
Q8851) Solve 15x^2+42x+27=0
Ans: x=-1 or x=-1.8
Q8852) Solve 22x^2+45x+18=0
Ans: x=-0.55 or x=-1.5
Q8853) Solve 30x^2+26x+4=0
Ans: x=-0.2 or x=-0.67
Q8854) Solve 22x^2+47x+6=0
Ans: x=-0.14 or x=-2
Q8855) Solve 8x^2+27x+22=0
Ans: x=-1.37 or x=-2
Q8856) Solve 12x^2+34x+14=0
Ans: x=-0.5 or x=-2.33
Q8857) Solve 13x^2+45x+32=0
Ans: x=-1 or x=-2.46
Q8858) Solve 22x^2+46x+24=0
Ans: x=-1 or x=-1.09
Q8859) Solve 18x^2+39x+21=0
Ans: x=-1 or x=-1.17
Q8860) Solve 15x^2+38x+23=0
Ans: x=-1 or x=-1.53
Q8861) Solve 21x^2+28x+7=0
Ans: x=-0.33 or x=-1
Q8862) Solve 9x^2+31x+12=0
Ans: x=-0.44 or x=-3
Q8863) Solve 12x^2+31x+14=0
Ans: x=-0.58 or x=-2
Q8864) Solve 36x^2+34x+8=0
Ans: x=-0.44 or x=-0.5
Q8865) Solve 24x^2+47x+20=0
Ans: x=-0.62 or x=-1.33
Q8866) Solve 2x^2+27x+46=0
Ans: x=-2 or x=-11.5
Q8867) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q8868) Solve 34x^2+39x+5=0
Ans: x=-0.15 or x=-1
Q8869) Solve 42x^2+44x+2=0
Ans: x=-0.05 or x=-1
Q8870) Solve 6x^2+35x+25=0
Ans: x=-0.83 or x=-5
Q8871) Solve 41x^2+42x+1=0
Ans: x=-0.02 or x=-1
Q8872) Solve 20x^2+12x+1=0
Ans: x=-0.1 or x=-0.5
Q8873) Solve 28x^2+44x+15=0
Ans: x=-0.5 or x=-1.07
Q8874) Solve 44x^2+47x+3=0
Ans: x=-0.07 or x=-1
Q8875) Solve 10x^2+36x+18=0
Ans: x=-0.6 or x=-3
Q8876) Solve 4x^2+24x+27=0
Ans: x=-1.5 or x=-4.5
Q8877) Solve 30x^2+16x+2=0
Ans: x=-0.2 or x=-0.33
Q8878) Solve 24x^2+11x+1=0
Ans: x=-0.12 or x=-0.33
Q8879) Solve 7x^2+37x+46=0
Ans: x=-2 or x=-3.29
Q8880) Solve 2x^2+17x+33=0
Ans: x=-3 or x=-5.5
Q8881) Solve 4x^2+19x+12=0
Ans: x=-0.75 or x=-4
Q8882) Solve 4x^2+11x+7=0
Ans: x=-1 or x=-1.75
Q8883) Solve 18x^2+31x+11=0
Ans: x=-0.5 or x=-1.22
Q8884) Solve 2x^2+21x+27=0
Ans: x=-1.5 or x=-9
Q8885) Solve 35x^2+37x+2=0
Ans: x=-0.06 or x=-1
Q8886) Solve 4x^2+44x+40=0
Ans: x=-1 or x=-10
Q8887) Solve 11x^2+17x+6=0
Ans: x=-0.55 or x=-1
Q8888) Solve 27x^2+42x+16=0
Ans: x=-0.67 or x=-0.89
Q8889) Solve x^2+14x+24=0
Ans: x=-2 or x=-12
Q8890) Solve 6x^2+17x+10=0
Ans: x=-0.83 or x=-2
Q8891) Solve 48x^2+28x+2=0
Ans: x=-0.08 or x=-0.5
Q8892) Solve 45x^2+47x+12=0
Ans: x=-0.44 or x=-0.6
Q8893) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q8894) Solve 20x^2+24x+7=0
Ans: x=-0.5 or x=-0.7
Q8895) Solve 28x^2+29x+1=0
Ans: x=-0.04 or x=-1
Q8896) Solve 2x^2+12x+10=0
Ans: x=-1 or x=-5
Q8897) Solve 22x^2+46x+24=0
Ans: x=-1 or x=-1.09
Q8898) Solve 29x^2+43x+14=0
Ans: x=-0.48 or x=-1
Q8899) Solve 12x^2+32x+21=0
Ans: x=-1.17 or x=-1.5
Q8900) Solve 9x^2+47x+10=0
Ans: x=-0.22 or x=-5
Q8901) Solve 6x^2+29x+30=0
Ans: x=-1.5 or x=-3.33
Q8902) Solve 7x^2+43x+6=0
Ans: x=-0.14 or x=-6
Q8903) Solve 49x^2+49x+12=0
Ans: x=-0.43 or x=-0.57
Q8904) Solve x^2+23x+42=0
Ans: x=-2 or x=-21
Q8905) Solve 2x^2+7x+5=0
Ans: x=-1 or x=-2.5
Q8906) Solve 19x^2+28x+9=0
Ans: x=-0.47 or x=-1
Q8907) Solve 26x^2+39x+13=0
Ans: x=-0.5 or x=-1
Q8908) Solve 4x^2+27x+35=0
Ans: x=-1.75 or x=-5
Q8909) Solve 3x^2+45x+42=0
Ans: x=-1 or x=-14
Q8910) Solve 14x^2+24x+10=0
Ans: x=-0.71 or x=-1
Q8911) Solve 24x^2+16x+2=0
Ans: x=-0.17 or x=-0.5
Q8912) Solve 11x^2+37x+30=0
Ans: x=-1.36 or x=-2
Q8913) Solve 14x^2+46x+32=0
Ans: x=-1 or x=-2.29
Q8914) Solve 10x^2+19x+9=0
Ans: x=-0.9 or x=-1
Q8915) Solve 31x^2+47x+16=0
Ans: x=-0.52 or x=-1
Q8916) Solve 33x^2+36x+3=0
Ans: x=-0.09 or x=-1
Q8917) Solve 15x^2+24x+9=0
Ans: x=-0.6 or x=-1
Q8918) Solve 8x^2+45x+25=0
Ans: x=-0.62 or x=-5
Q8919) Solve 3x^2+20x+28=0
Ans: x=-2 or x=-4.67
Q8920) Solve 7x^2+31x+24=0
Ans: x=-1 or x=-3.43
Q8921) Solve x^2+10x+16=0
Ans: x=-2 or x=-8
Q8922) Solve 9x^2+32x+15=0
Ans: x=-0.56 or x=-3
Q8923) Solve x^2+12x+27=0
Ans: x=-3 or x=-9
Q8924) Solve 11x^2+23x+12=0
Ans: x=-1 or x=-1.09
Q8925) Solve 17x^2+27x+10=0
Ans: x=-0.59 or x=-1
Q8926) Solve 16x^2+40x+16=0
Ans: x=-0.5 or x=-2
Q8927) Solve 15x^2+37x+18=0
Ans: x=-0.67 or x=-1.8
Q8928) Solve 25x^2+43x+18=0
Ans: x=-0.72 or x=-1
Q8929) Solve 21x^2+26x+5=0
Ans: x=-0.24 or x=-1
Q8930) Solve 6x^2+33x+27=0
Ans: x=-1 or x=-4.5
Q8931) Solve 14x^2+35x+14=0
Ans: x=-0.5 or x=-2
Q8932) Solve 12x^2+44x+32=0
Ans: x=-1 or x=-2.67
Q8933) Solve 10x^2+31x+13=0
Ans: x=-0.5 or x=-2.6
Q8934) Solve 14x^2+30x+16=0
Ans: x=-1 or x=-1.14
Q8935) Solve x^2+7x+12=0
Ans: x=-3 or x=-4
Q8936) Solve 11x^2+37x+30=0
Ans: x=-1.36 or x=-2
Q8937) Solve 6x^2+49x+49=0
Ans: x=-1.17 or x=-7
Q8938) Solve 12x^2+44x+24=0
Ans: x=-0.67 or x=-3
Q8939) Solve 17x^2+44x+27=0
Ans: x=-1 or x=-1.59
Q8940) Solve 14x^2+47x+15=0
Ans: x=-0.36 or x=-3
Q8941) Solve 13x^2+46x+40=0
Ans: x=-1.54 or x=-2
Q8942) Solve 16x^2+26x+10=0
Ans: x=-0.62 or x=-1
Q8943) Solve 30x^2+42x+12=0
Ans: x=-0.4 or x=-1
Q8944) Solve 30x^2+47x+16=0
Ans: x=-0.5 or x=-1.07
Q8945) Solve 40x^2+49x+9=0
Ans: x=-0.22 or x=-1
Q8946) Solve 21x^2+46x+24=0
Ans: x=-0.86 or x=-1.33
Q8947) Solve 44x^2+45x+9=0
Ans: x=-0.27 or x=-0.75
Q8948) Solve 20x^2+49x+9=0
Ans: x=-0.2 or x=-2.25
Q8949) Solve 7x^2+34x+27=0
Ans: x=-1 or x=-3.86
Q8950) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q8951) Solve 8x^2+34x+30=0
Ans: x=-1.25 or x=-3
Q8952) Solve 11x^2+20x+9=0
Ans: x=-0.82 or x=-1
Q8953) Solve 12x^2+40x+25=0
Ans: x=-0.83 or x=-2.5
Q8954) Solve 15x^2+37x+18=0
Ans: x=-0.67 or x=-1.8
Q8955) Solve 17x^2+48x+31=0
Ans: x=-1 or x=-1.82
Q8956) Solve 45x^2+39x+8=0
Ans: x=-0.33 or x=-0.53
Q8957) Solve 2x^2+35x+48=0
Ans: x=-1.5 or x=-16
Q8958) Solve 28x^2+42x+14=0
Ans: x=-0.5 or x=-1
Q8959) Solve 4x^2+25x+36=0
Ans: x=-2.25 or x=-4
Q8960) Solve 4x^2+42x+38=0
Ans: x=-1 or x=-9.5
Q8961) Solve 34x^2+45x+14=0
Ans: x=-0.5 or x=-0.82
Q8962) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q8963) Solve 5x^2+26x+32=0
Ans: x=-2 or x=-3.2
Q8964) Solve 4x^2+33x+29=0
Ans: x=-1 or x=-7.25
Q8965) Solve 18x^2+48x+14=0
Ans: x=-0.33 or x=-2.33
Q8966) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q8967) Solve 23x^2+43x+20=0
Ans: x=-0.87 or x=-1
Q8968) Solve 36x^2+47x+15=0
Ans: x=-0.56 or x=-0.75
Q8969) Solve 31x^2+35x+4=0
Ans: x=-0.13 or x=-1
Q8970) Solve 14x^2+30x+4=0
Ans: x=-0.14 or x=-2
Q8971) Solve 2x^2+47x+23=0
Ans: x=-0.5 or x=-23
Q8972) Solve 21x^2+48x+12=0
Ans: x=-0.29 or x=-2
Q8973) Solve 48x^2+50x+2=0
Ans: x=-0.04 or x=-1
Q8974) Solve 9x^2+19x+10=0
Ans: x=-1 or x=-1.11
Q8975) Solve 7x^2+36x+29=0
Ans: x=-1 or x=-4.14
Q8976) Solve 22x^2+43x+21=0
Ans: x=-0.95 or x=-1
Q8977) Solve 35x^2+39x+10=0
Ans: x=-0.4 or x=-0.71
Q8978) Solve 19x^2+48x+29=0
Ans: x=-1 or x=-1.53
Q8979) Solve 3x^2+29x+26=0
Ans: x=-1 or x=-8.67
Q8980) Solve 10x^2+15x+5=0
Ans: x=-0.5 or x=-1
Q8981) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q8982) Solve 32x^2+36x+7=0
Ans: x=-0.25 or x=-0.87
Q8983) Solve 8x^2+50x+12=0
Ans: x=-0.25 or x=-6
Q8984) Solve 5x^2+34x+29=0
Ans: x=-1 or x=-5.8
Q8985) Solve 14x^2+46x+12=0
Ans: x=-0.29 or x=-3
Q8986) Solve 30x^2+31x+8=0
Ans: x=-0.5 or x=-0.53
Q8987) Solve 19x^2+45x+26=0
Ans: x=-1 or x=-1.37
Q8988) Solve 22x^2+47x+18=0
Ans: x=-0.5 or x=-1.64
Q8989) Solve x^2+13x+40=0
Ans: x=-5 or x=-8
Q8990) Solve 32x^2+36x+4=0
Ans: x=-0.12 or x=-1
Q8991) Solve 6x^2+13x+2=0
Ans: x=-0.17 or x=-2
Q8992) Solve 46x^2+25x+1=0
Ans: x=-0.04 or x=-0.5
Q8993) Solve 32x^2+28x+5=0
Ans: x=-0.25 or x=-0.62
Q8994) Solve 9x^2+28x+19=0
Ans: x=-1 or x=-2.11
Q8995) Solve 3x^2+25x+28=0
Ans: x=-1.33 or x=-7
Q8996) Solve 2x^2+49x+24=0
Ans: x=-0.5 or x=-24
Q8997) Solve 10x^2+27x+18=0
Ans: x=-1.2 or x=-1.5
Q8998) Solve 3x^2+22x+32=0
Ans: x=-2 or x=-5.33
Q8999) Solve 4x^2+37x+40=0
Ans: x=-1.25 or x=-8
Q9000) Solve 20x^2+31x+11=0
Ans: x=-0.55 or x=-1
Q9001) Solve 31x^2+40x+9=0
Ans: x=-0.29 or x=-1
Q9002) Solve 27x^2+12x+1=0
Ans: x=-0.11 or x=-0.33
Q9003) Solve 24x^2+40x+6=0
Ans: x=-0.17 or x=-1.5
Q9004) Solve 12x^2+38x+26=0
Ans: x=-1 or x=-2.17
Q9005) Solve 18x^2+49x+26=0
Ans: x=-0.72 or x=-2
Q9006) Solve 5x^2+41x+42=0
Ans: x=-1.2 or x=-7
Q9007) Solve 42x^2+13x+1=0
Ans: x=-0.14 or x=-0.17
Q9008) Solve 15x^2+47x+28=0
Ans: x=-0.8 or x=-2.33
Q9009) Solve 4x^2+24x+20=0
Ans: x=-1 or x=-5
Q9010) Solve 6x^2+29x+35=0
Ans: x=-2.33 or x=-2.5
Q9011) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q9012) Solve 40x^2+44x+4=0
Ans: x=-0.1 or x=-1
Q9013) Solve 6x^2+34x+28=0
Ans: x=-1 or x=-4.67
Q9014) Solve 38x^2+41x+11=0
Ans: x=-0.5 or x=-0.58
Q9015) Solve 10x^2+29x+19=0
Ans: x=-1 or x=-1.9
Q9016) Solve 26x^2+40x+14=0
Ans: x=-0.54 or x=-1
Q9017) Solve 9x^2+31x+12=0
Ans: x=-0.44 or x=-3
Q9018) Solve 24x^2+40x+6=0
Ans: x=-0.17 or x=-1.5
Q9019) Solve 4x^2+45x+50=0
Ans: x=-1.25 or x=-10
Q9020) Solve 9x^2+43x+34=0
Ans: x=-1 or x=-3.78
Q9021) Solve 14x^2+43x+20=0
Ans: x=-0.57 or x=-2.5
Q9022) Solve 36x^2+47x+11=0
Ans: x=-0.31 or x=-1
Q9023) Solve 15x^2+38x+7=0
Ans: x=-0.2 or x=-2.33
Q9024) Solve 10x^2+7x+1=0
Ans: x=-0.2 or x=-0.5
Q9025) Solve 2x^2+15x+18=0
Ans: x=-1.5 or x=-6
Q9026) Solve 30x^2+29x+4=0
Ans: x=-0.17 or x=-0.8
Q9027) Solve 38x^2+40x+2=0
Ans: x=-0.05 or x=-1
Q9028) Solve 5x^2+36x+31=0
Ans: x=-1 or x=-6.2
Q9029) Solve 15x^2+46x+31=0
Ans: x=-1 or x=-2.07
Q9030) Solve x^2+11x+24=0
Ans: x=-3 or x=-8
Q9031) Solve 16x^2+48x+32=0
Ans: x=-1 or x=-2
Q9032) Solve 4x^2+32x+48=0
Ans: x=-2 or x=-6
Q9033) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q9034) Solve 3x^2+8x+4=0
Ans: x=-0.67 or x=-2
Q9035) Solve 16x^2+28x+10=0
Ans: x=-0.5 or x=-1.25
Q9036) Solve 16x^2+45x+29=0
Ans: x=-1 or x=-1.81
Q9037) Solve 4x^2+14x+10=0
Ans: x=-1 or x=-2.5
Q9038) Solve 13x^2+50x+48=0
Ans: x=-1.85 or x=-2
Q9039) Solve 2x^2+20x+42=0
Ans: x=-3 or x=-7
Q9040) Solve 36x^2+13x+1=0
Ans: x=-0.11 or x=-0.25
Q9041) Solve 13x^2+46x+21=0
Ans: x=-0.54 or x=-3
Q9042) Solve 9x^2+29x+20=0
Ans: x=-1 or x=-2.22
Q9043) Solve 32x^2+49x+17=0
Ans: x=-0.53 or x=-1
Q9044) Solve 8x^2+27x+22=0
Ans: x=-1.37 or x=-2
Q9045) Solve x^2+12x+11=0
Ans: x=-1 or x=-11
Q9046) Solve 39x^2+35x+6=0
Ans: x=-0.23 or x=-0.67
Q9047) Solve 33x^2+49x+18=0
Ans: x=-0.67 or x=-0.82
Q9048) Solve 16x^2+46x+33=0
Ans: x=-1.37 or x=-1.5
Q9049) Solve 28x^2+46x+6=0
Ans: x=-0.14 or x=-1.5
Q9050) Solve 20x^2+43x+21=0
Ans: x=-0.75 or x=-1.4
Q9051) Solve 20x^2+42x+4=0
Ans: x=-0.1 or x=-2
Q9052) Solve 4x^2+6x+2=0
Ans: x=-0.5 or x=-1
Q9053) Solve 26x^2+33x+7=0
Ans: x=-0.27 or x=-1
Q9054) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q9055) Solve 28x^2+43x+9=0
Ans: x=-0.25 or x=-1.29
Q9056) Solve 8x^2+37x+20=0
Ans: x=-0.62 or x=-4
Q9057) Solve x^2+40x+39=0
Ans: x=-1 or x=-39
Q9058) Solve 39x^2+44x+12=0
Ans: x=-0.46 or x=-0.67
Q9059) Solve 8x^2+48x+40=0
Ans: x=-1 or x=-5
Q9060) Solve 8x^2+23x+15=0
Ans: x=-1 or x=-1.87
Q9061) Solve 14x^2+43x+30=0
Ans: x=-1.07 or x=-2
Q9062) Solve 36x^2+49x+10=0
Ans: x=-0.25 or x=-1.11
Q9063) Solve 20x^2+40x+15=0
Ans: x=-0.5 or x=-1.5
Q9064) Solve 12x^2+43x+38=0
Ans: x=-1.58 or x=-2
Q9065) Solve 10x^2+33x+26=0
Ans: x=-1.3 or x=-2
Q9066) Solve 7x^2+47x+40=0
Ans: x=-1 or x=-5.71
Q9067) Solve 40x^2+22x+1=0
Ans: x=-0.05 or x=-0.5
Q9068) Solve 16x^2+50x+25=0
Ans: x=-0.62 or x=-2.5
Q9069) Solve 29x^2+47x+18=0
Ans: x=-0.62 or x=-1
Q9070) Solve 6x^2+40x+34=0
Ans: x=-1 or x=-5.67
Q9071) Solve 13x^2+27x+14=0
Ans: x=-1 or x=-1.08
Q9072) Solve 36x^2+49x+10=0
Ans: x=-0.25 or x=-1.11
Q9073) Solve 9x^2+25x+16=0
Ans: x=-1 or x=-1.78
Q9074) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q9075) Solve 2x^2+22x+48=0
Ans: x=-3 or x=-8
Q9076) Solve 18x^2+49x+10=0
Ans: x=-0.22 or x=-2.5
Q9077) Solve 6x^2+41x+48=0
Ans: x=-1.5 or x=-5.33
Q9078) Solve 5x^2+17x+14=0
Ans: x=-1.4 or x=-2
Q9079) Solve 8x^2+44x+20=0
Ans: x=-0.5 or x=-5
Q9080) Solve 15x^2+43x+26=0
Ans: x=-0.87 or x=-2
Q9081) Solve 15x^2+24x+9=0
Ans: x=-0.6 or x=-1
Q9082) Solve 12x^2+35x+8=0
Ans: x=-0.25 or x=-2.67
Q9083) Solve x^2+13x+36=0
Ans: x=-4 or x=-9
Q9084) Solve 16x^2+21x+5=0
Ans: x=-0.31 or x=-1
Q9085) Solve 10x^2+39x+27=0
Ans: x=-0.9 or x=-3
Q9086) Solve 24x^2+47x+23=0
Ans: x=-0.96 or x=-1
Q9087) Solve x^2+39x+38=0
Ans: x=-1 or x=-38
Q9088) Solve 17x^2+43x+18=0
Ans: x=-0.53 or x=-2
Q9089) Solve 39x^2+42x+3=0
Ans: x=-0.08 or x=-1
Q9090) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q9091) Solve 7x^2+10x+3=0
Ans: x=-0.43 or x=-1
Q9092) Solve 22x^2+28x+6=0
Ans: x=-0.27 or x=-1
Q9093) Solve 3x^2+14x+15=0
Ans: x=-1.67 or x=-3
Q9094) Solve 36x^2+40x+11=0
Ans: x=-0.5 or x=-0.61
Q9095) Solve 26x^2+29x+8=0
Ans: x=-0.5 or x=-0.62
Q9096) Solve 5x^2+28x+39=0
Ans: x=-2.6 or x=-3
Q9097) Solve 7x^2+34x+27=0
Ans: x=-1 or x=-3.86
Q9098) Solve 19x^2+34x+15=0
Ans: x=-0.79 or x=-1
Q9099) Solve 22x^2+31x+9=0
Ans: x=-0.41 or x=-1
Q9100) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Q9101) Solve 7x^2+31x+34=0
Ans: x=-2 or x=-2.43
Q9102) Solve 16x^2+8x+1=0
Ans: x=-0.25 or x=-0.25
Q9103) Solve 12x^2+43x+10=0
Ans: x=-0.25 or x=-3.33
Q9104) Solve 10x^2+29x+21=0
Ans: x=-1.4 or x=-1.5
Q9105) Solve 20x^2+44x+21=0
Ans: x=-0.7 or x=-1.5
Q9106) Solve 12x^2+36x+27=0
Ans: x=-1.5 or x=-1.5
Q9107) Solve 10x^2+50x+40=0
Ans: x=-1 or x=-4
Q9108) Solve 2x^2+43x+21=0
Ans: x=-0.5 or x=-21
Q9109) Solve 18x^2+9x+1=0
Ans: x=-0.17 or x=-0.33
Q9110) Solve 8x^2+40x+48=0
Ans: x=-2 or x=-3
Q9111) Solve 9x^2+36x+36=0
Ans: x=-2 or x=-2
Q9112) Solve 10x^2+29x+18=0
Ans: x=-0.9 or x=-2
Q9113) Solve 21x^2+49x+14=0
Ans: x=-0.33 or x=-2
Q9114) Solve 22x^2+47x+25=0
Ans: x=-1 or x=-1.14
Q9115) Solve 2x^2+21x+27=0
Ans: x=-1.5 or x=-9
Q9116) Solve 5x^2+27x+22=0
Ans: x=-1 or x=-4.4
Q9117) Solve 3x^2+28x+25=0
Ans: x=-1 or x=-8.33
Q9118) Solve 15x^2+41x+14=0
Ans: x=-0.4 or x=-2.33
Q9119) Solve 8x^2+38x+44=0
Ans: x=-2 or x=-2.75
Q9120) Solve 13x^2+40x+3=0
Ans: x=-0.08 or x=-3
Q9121) Solve 3x^2+29x+46=0
Ans: x=-2 or x=-7.67
Q9122) Solve 2x^2+15x+28=0
Ans: x=-3.5 or x=-4
Q9123) Solve 2x^2+19x+39=0
Ans: x=-3 or x=-6.5
Q9124) Solve x^2+10x+9=0
Ans: x=-1 or x=-9
Q9125) Solve 5x^2+34x+24=0
Ans: x=-0.8 or x=-6
Q9126) Solve 4x^2+46x+22=0
Ans: x=-0.5 or x=-11
Q9127) Solve x^2+31x+30=0
Ans: x=-1 or x=-30
Q9128) Solve 2x^2+19x+24=0
Ans: x=-1.5 or x=-8
Q9129) Solve 28x^2+28x+7=0
Ans: x=-0.5 or x=-0.5
Q9130) Solve 8x^2+26x+18=0
Ans: x=-1 or x=-2.25
Q9131) Solve 18x^2+27x+7=0
Ans: x=-0.33 or x=-1.17
Q9132) Solve 9x^2+13x+4=0
Ans: x=-0.44 or x=-1
Q9133) Solve 40x^2+31x+6=0
Ans: x=-0.37 or x=-0.4
Q9134) Solve 15x^2+27x+12=0
Ans: x=-0.8 or x=-1
Q9135) Solve x^2+36x+35=0
Ans: x=-1 or x=-35
Q9136) Solve 15x^2+37x+14=0
Ans: x=-0.47 or x=-2
Q9137) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q9138) Solve 20x^2+36x+13=0
Ans: x=-0.5 or x=-1.3
Q9139) Solve 9x^2+30x+25=0
Ans: x=-1.67 or x=-1.67
Q9140) Solve 27x^2+36x+12=0
Ans: x=-0.67 or x=-0.67
Q9141) Solve 13x^2+44x+36=0
Ans: x=-1.38 or x=-2
Q9142) Solve 18x^2+43x+17=0
Ans: x=-0.5 or x=-1.89
Q9143) Solve 5x^2+20x+20=0
Ans: x=-2 or x=-2
Q9144) Solve 29x^2+45x+16=0
Ans: x=-0.55 or x=-1
Q9145) Solve 15x^2+48x+9=0
Ans: x=-0.2 or x=-3
Q9146) Solve 28x^2+19x+3=0
Ans: x=-0.25 or x=-0.43
Q9147) Solve 50x^2+37x+6=0
Ans: x=-0.24 or x=-0.5
Q9148) Solve 25x^2+35x+12=0
Ans: x=-0.6 or x=-0.8
Q9149) Solve 3x^2+29x+18=0
Ans: x=-0.67 or x=-9
Q9150) Solve 10x^2+23x+9=0
Ans: x=-0.5 or x=-1.8
Q9151) Solve x^2+5x+4=0
Ans: x=-1 or x=-4
Q9152) Solve 44x^2+38x+8=0
Ans: x=-0.36 or x=-0.5
Q9153) Solve 4x^2+40x+36=0
Ans: x=-1 or x=-9
Q9154) Solve 17x^2+40x+23=0
Ans: x=-1 or x=-1.35
Q9155) Solve 10x^2+47x+48=0
Ans: x=-1.5 or x=-3.2
Q9156) Solve 42x^2+41x+10=0
Ans: x=-0.48 or x=-0.5
Q9157) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Q9158) Solve 7x^2+12x+5=0
Ans: x=-0.71 or x=-1
Q9159) Solve 2x^2+17x+35=0
Ans: x=-3.5 or x=-5
Q9160) Solve 48x^2+46x+10=0
Ans: x=-0.33 or x=-0.62
Q9161) Solve 31x^2+41x+10=0
Ans: x=-0.32 or x=-1
Q9162) Solve 24x^2+42x+15=0
Ans: x=-0.5 or x=-1.25
Q9163) Solve 28x^2+44x+16=0
Ans: x=-0.57 or x=-1
Q9164) Solve 15x^2+46x+32=0
Ans: x=-1.07 or x=-2
Q9165) Solve 5x^2+42x+16=0
Ans: x=-0.4 or x=-8
Q9166) Solve 6x^2+25x+26=0
Ans: x=-2 or x=-2.17
Q9167) Solve 8x^2+42x+27=0
Ans: x=-0.75 or x=-4.5
Q9168) Solve 32x^2+40x+8=0
Ans: x=-0.25 or x=-1
Q9169) Solve 12x^2+7x+1=0
Ans: x=-0.25 or x=-0.33
Q9170) Solve 12x^2+44x+35=0
Ans: x=-1.17 or x=-2.5
Q9171) Solve 6x^2+15x+9=0
Ans: x=-1 or x=-1.5
Q9172) Solve 9x^2+29x+6=0
Ans: x=-0.22 or x=-3
Q9173) Solve 44x^2+46x+12=0
Ans: x=-0.5 or x=-0.55
Q9174) Solve 3x^2+18x+24=0
Ans: x=-2 or x=-4
Q9175) Solve 12x^2+28x+11=0
Ans: x=-0.5 or x=-1.83
Q9176) Solve 11x^2+30x+16=0
Ans: x=-0.73 or x=-2
Q9177) Solve 27x^2+42x+15=0
Ans: x=-0.56 or x=-1
Q9178) Solve 29x^2+42x+13=0
Ans: x=-0.45 or x=-1
Q9179) Solve 36x^2+43x+12=0
Ans: x=-0.44 or x=-0.75
Q9180) Solve 6x^2+17x+5=0
Ans: x=-0.33 or x=-2.5
Q9181) Solve 9x^2+47x+10=0
Ans: x=-0.22 or x=-5
Q9182) Solve 30x^2+44x+14=0
Ans: x=-0.47 or x=-1
Q9183) Solve 12x^2+46x+30=0
Ans: x=-0.83 or x=-3
Q9184) Solve 8x^2+6x+1=0
Ans: x=-0.25 or x=-0.5
Q9185) Solve 6x^2+37x+50=0
Ans: x=-2 or x=-4.17
Q9186) Solve 19x^2+48x+20=0
Ans: x=-0.53 or x=-2
Q9187) Solve 20x^2+50x+30=0
Ans: x=-1 or x=-1.5
Q9188) Solve 4x^2+20x+21=0
Ans: x=-1.5 or x=-3.5
Q9189) Solve 22x^2+30x+8=0
Ans: x=-0.36 or x=-1
Q9190) Solve 21x^2+44x+20=0
Ans: x=-0.67 or x=-1.43
Q9191) Solve 8x^2+34x+8=0
Ans: x=-0.25 or x=-4
Q9192) Solve 18x^2+48x+14=0
Ans: x=-0.33 or x=-2.33
Q9193) Solve 12x^2+44x+19=0
Ans: x=-0.5 or x=-3.17
Q9194) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q9195) Solve 10x^2+23x+9=0
Ans: x=-0.5 or x=-1.8
Q9196) Solve 28x^2+24x+5=0
Ans: x=-0.36 or x=-0.5
Q9197) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q9198) Solve 20x^2+22x+6=0
Ans: x=-0.5 or x=-0.6
Q9199) Solve 16x^2+40x+16=0
Ans: x=-0.5 or x=-2
Q9200) Solve 16x^2+17x+1=0
Ans: x=-0.06 or x=-1
Q9201) Solve 7x^2+31x+12=0
Ans: x=-0.43 or x=-4
Q9202) Solve 16x^2+49x+3=0
Ans: x=-0.06 or x=-3
Q9203) Solve 22x^2+39x+14=0
Ans: x=-0.5 or x=-1.27
Q9204) Solve 27x^2+49x+22=0
Ans: x=-0.81 or x=-1
Q9205) Solve 48x^2+14x+1=0
Ans: x=-0.12 or x=-0.17
Q9206) Solve 14x^2+45x+25=0
Ans: x=-0.71 or x=-2.5
Q9207) Solve 2x^2+43x+41=0
Ans: x=-1 or x=-20.5
Q9208) Solve 5x^2+34x+29=0
Ans: x=-1 or x=-5.8
Q9209) Solve 2x^2+15x+22=0
Ans: x=-2 or x=-5.5
Q9210) Solve 16x^2+36x+20=0
Ans: x=-1 or x=-1.25
Q9211) Solve 45x^2+30x+5=0
Ans: x=-0.33 or x=-0.33
Q9212) Solve 12x^2+37x+11=0
Ans: x=-0.33 or x=-2.75
Q9213) Solve 44x^2+50x+14=0
Ans: x=-0.5 or x=-0.64
Q9214) Solve 2x^2+20x+18=0
Ans: x=-1 or x=-9
Q9215) Solve 24x^2+45x+21=0
Ans: x=-0.87 or x=-1
Q9216) Solve 19x^2+21x+2=0
Ans: x=-0.11 or x=-1
Q9217) Solve 24x^2+49x+25=0
Ans: x=-1 or x=-1.04
Q9218) Solve x^2+37x+36=0
Ans: x=-1 or x=-36
Q9219) Solve 7x^2+19x+10=0
Ans: x=-0.71 or x=-2
Q9220) Solve 12x^2+49x+37=0
Ans: x=-1 or x=-3.08
Q9221) Solve 36x^2+39x+3=0
Ans: x=-0.08 or x=-1
Q9222) Solve 35x^2+44x+9=0
Ans: x=-0.26 or x=-1
Q9223) Solve 17x^2+25x+8=0
Ans: x=-0.47 or x=-1
Q9224) Solve 3x^2+19x+16=0
Ans: x=-1 or x=-5.33
Q9225) Solve x^2+8x+12=0
Ans: x=-2 or x=-6
Q9226) Solve 33x^2+45x+12=0
Ans: x=-0.36 or x=-1
Q9227) Solve 22x^2+31x+9=0
Ans: x=-0.41 or x=-1
Q9228) Solve 23x^2+33x+10=0
Ans: x=-0.43 or x=-1
Q9229) Solve 16x^2+27x+11=0
Ans: x=-0.69 or x=-1
Q9230) Solve 21x^2+37x+10=0
Ans: x=-0.33 or x=-1.43
Q9231) Solve 4x^2+17x+13=0
Ans: x=-1 or x=-3.25
Q9232) Solve 11x^2+36x+28=0
Ans: x=-1.27 or x=-2
Q9233) Solve 14x^2+47x+20=0
Ans: x=-0.5 or x=-2.86
Q9234) Solve 15x^2+39x+24=0
Ans: x=-1 or x=-1.6
Q9235) Solve 3x^2+29x+26=0
Ans: x=-1 or x=-8.67
Q9236) Solve 12x^2+37x+3=0
Ans: x=-0.08 or x=-3
Q9237) Solve 20x^2+25x+5=0
Ans: x=-0.25 or x=-1
Q9238) Solve 4x^2+40x+36=0
Ans: x=-1 or x=-9
Q9239) Solve 32x^2+32x+8=0
Ans: x=-0.5 or x=-0.5
Q9240) Solve 4x^2+27x+38=0
Ans: x=-2 or x=-4.75
Q9241) Solve 9x^2+24x+15=0
Ans: x=-1 or x=-1.67
Q9242) Solve 12x^2+40x+33=0
Ans: x=-1.5 or x=-1.83
Q9243) Solve 24x^2+29x+5=0
Ans: x=-0.21 or x=-1
Q9244) Solve 25x^2+15x+2=0
Ans: x=-0.2 or x=-0.4
Q9245) Solve 9x^2+14x+5=0
Ans: x=-0.56 or x=-1
Q9246) Solve 11x^2+12x+1=0
Ans: x=-0.09 or x=-1
Q9247) Solve 42x^2+41x+10=0
Ans: x=-0.48 or x=-0.5
Q9248) Solve 13x^2+39x+26=0
Ans: x=-1 or x=-2
Q9249) Solve 4x^2+27x+18=0
Ans: x=-0.75 or x=-6
Q9250) Solve 14x^2+17x+5=0
Ans: x=-0.5 or x=-0.71
Q9251) Solve 7x^2+31x+24=0
Ans: x=-1 or x=-3.43
Q9252) Solve 20x^2+42x+22=0
Ans: x=-1 or x=-1.1
Q9253) Solve 10x^2+45x+45=0
Ans: x=-1.5 or x=-3
Q9254) Solve 3x^2+17x+20=0
Ans: x=-1.67 or x=-4
Q9255) Solve 35x^2+49x+14=0
Ans: x=-0.4 or x=-1
Q9256) Solve 35x^2+43x+8=0
Ans: x=-0.23 or x=-1
Q9257) Solve 24x^2+37x+13=0
Ans: x=-0.54 or x=-1
Q9258) Solve 25x^2+26x+1=0
Ans: x=-0.04 or x=-1
Q9259) Solve 16x^2+44x+30=0
Ans: x=-1.25 or x=-1.5
Q9260) Solve 14x^2+43x+18=0
Ans: x=-0.5 or x=-2.57
Q9261) Solve 2x^2+27x+13=0
Ans: x=-0.5 or x=-13
Q9262) Solve 12x^2+49x+37=0
Ans: x=-1 or x=-3.08
Q9263) Solve 30x^2+35x+5=0
Ans: x=-0.17 or x=-1
Q9264) Solve 3x^2+43x+14=0
Ans: x=-0.33 or x=-14
Q9265) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Q9266) Solve 24x^2+30x+6=0
Ans: x=-0.25 or x=-1
Q9267) Solve 24x^2+36x+12=0
Ans: x=-0.5 or x=-1
Q9268) Solve 12x^2+43x+36=0
Ans: x=-1.33 or x=-2.25
Q9269) Solve 44x^2+46x+2=0
Ans: x=-0.05 or x=-1
Q9270) Solve 12x^2+29x+10=0
Ans: x=-0.42 or x=-2
Q9271) Solve 4x^2+36x+32=0
Ans: x=-1 or x=-8
Q9272) Solve 48x^2+40x+7=0
Ans: x=-0.25 or x=-0.58
Q9273) Solve 9x^2+39x+36=0
Ans: x=-1.33 or x=-3
Q9274) Solve 40x^2+46x+12=0
Ans: x=-0.4 or x=-0.75
Q9275) Solve 6x^2+17x+5=0
Ans: x=-0.33 or x=-2.5
Q9276) Solve 12x^2+29x+15=0
Ans: x=-0.75 or x=-1.67
Q9277) Solve 48x^2+20x+2=0
Ans: x=-0.17 or x=-0.25
Q9278) Solve 9x^2+11x+2=0
Ans: x=-0.22 or x=-1
Q9279) Solve 34x^2+23x+3=0
Ans: x=-0.18 or x=-0.5
Q9280) Solve 16x^2+46x+19=0
Ans: x=-0.5 or x=-2.37
Q9281) Solve x^2+36x+35=0
Ans: x=-1 or x=-35
Q9282) Solve 14x^2+31x+6=0
Ans: x=-0.21 or x=-2
Q9283) Solve 17x^2+41x+14=0
Ans: x=-0.41 or x=-2
Q9284) Solve 15x^2+18x+3=0
Ans: x=-0.2 or x=-1
Q9285) Solve 45x^2+43x+10=0
Ans: x=-0.4 or x=-0.56
Q9286) Solve 18x^2+11x+1=0
Ans: x=-0.11 or x=-0.5
Q9287) Solve 6x^2+37x+42=0
Ans: x=-1.5 or x=-4.67
Q9288) Solve 20x^2+44x+17=0
Ans: x=-0.5 or x=-1.7
Q9289) Solve 42x^2+26x+4=0
Ans: x=-0.29 or x=-0.33
Q9290) Solve 50x^2+35x+3=0
Ans: x=-0.1 or x=-0.6
Q9291) Solve 35x^2+31x+6=0
Ans: x=-0.29 or x=-0.6
Q9292) Solve 26x^2+43x+15=0
Ans: x=-0.5 or x=-1.15
Q9293) Solve 22x^2+41x+12=0
Ans: x=-0.36 or x=-1.5
Q9294) Solve 22x^2+50x+12=0
Ans: x=-0.27 or x=-2
Q9295) Solve 2x^2+17x+15=0
Ans: x=-1 or x=-7.5
Q9296) Solve 13x^2+41x+30=0
Ans: x=-1.15 or x=-2
Q9297) Solve 11x^2+42x+31=0
Ans: x=-1 or x=-2.82
Q9298) Solve 18x^2+28x+10=0
Ans: x=-0.56 or x=-1
Q9299) Solve 14x^2+29x+2=0
Ans: x=-0.07 or x=-2
Q9300) Solve 6x^2+35x+46=0
Ans: x=-2 or x=-3.83
Q9301) Solve 40x^2+39x+9=0
Ans: x=-0.37 or x=-0.6
Q9302) Solve 2x^2+29x+27=0
Ans: x=-1 or x=-13.5
Q9303) Solve 16x^2+50x+21=0
Ans: x=-0.5 or x=-2.62
Q9304) Solve 9x^2+48x+15=0
Ans: x=-0.33 or x=-5
Q9305) Solve 7x^2+32x+33=0
Ans: x=-1.57 or x=-3
Q9306) Solve 21x^2+23x+6=0
Ans: x=-0.43 or x=-0.67
Q9307) Solve 6x^2+28x+16=0
Ans: x=-0.67 or x=-4
Q9308) Solve 30x^2+29x+7=0
Ans: x=-0.47 or x=-0.5
Q9309) Solve 10x^2+22x+12=0
Ans: x=-1 or x=-1.2
Q9310) Solve 4x^2+15x+11=0
Ans: x=-1 or x=-2.75
Q9311) Solve 6x^2+26x+8=0
Ans: x=-0.33 or x=-4
Q9312) Solve 6x^2+32x+40=0
Ans: x=-2 or x=-3.33
Q9313) Solve 23x^2+39x+16=0
Ans: x=-0.7 or x=-1
Q9314) Solve 12x^2+41x+15=0
Ans: x=-0.42 or x=-3
Q9315) Solve 5x^2+12x+7=0
Ans: x=-1 or x=-1.4
Q9316) Solve 24x^2+43x+19=0
Ans: x=-0.79 or x=-1
Q9317) Solve 25x^2+38x+13=0
Ans: x=-0.52 or x=-1
Q9318) Solve 38x^2+47x+14=0
Ans: x=-0.5 or x=-0.74
Q9319) Solve 15x^2+43x+30=0
Ans: x=-1.2 or x=-1.67
Q9320) Solve x^2+12x+35=0
Ans: x=-5 or x=-7
Q9321) Solve 25x^2+33x+8=0
Ans: x=-0.32 or x=-1
Q9322) Solve 18x^2+41x+23=0
Ans: x=-1 or x=-1.28
Q9323) Solve 28x^2+27x+5=0
Ans: x=-0.25 or x=-0.71
Q9324) Solve 16x^2+38x+15=0
Ans: x=-0.5 or x=-1.87
Q9325) Solve 42x^2+47x+11=0
Ans: x=-0.33 or x=-0.79
Q9326) Solve 22x^2+30x+8=0
Ans: x=-0.36 or x=-1
Q9327) Solve 5x^2+27x+22=0
Ans: x=-1 or x=-4.4
Q9328) Solve 10x^2+33x+14=0
Ans: x=-0.5 or x=-2.8
Q9329) Solve 16x^2+41x+18=0
Ans: x=-0.56 or x=-2
Q9330) Solve 12x^2+34x+20=0
Ans: x=-0.83 or x=-2
Q9331) Solve 4x^2+8x+4=0
Ans: x=-1 or x=-1
Q9332) Solve 13x^2+24x+11=0
Ans: x=-0.85 or x=-1
Q9333) Solve 22x^2+47x+6=0
Ans: x=-0.14 or x=-2
Q9334) Solve 16x^2+8x+1=0
Ans: x=-0.25 or x=-0.25
Q9335) Solve 3x^2+35x+32=0
Ans: x=-1 or x=-10.67
Q9336) Solve 17x^2+39x+10=0
Ans: x=-0.29 or x=-2
Q9337) Solve 44x^2+19x+2=0
Ans: x=-0.18 or x=-0.25
Q9338) Solve 6x^2+19x+15=0
Ans: x=-1.5 or x=-1.67
Q9339) Solve 36x^2+35x+6=0
Ans: x=-0.22 or x=-0.75
Q9340) Solve 32x^2+48x+18=0
Ans: x=-0.75 or x=-0.75
Q9341) Solve x^2+9x+20=0
Ans: x=-4 or x=-5
Q9342) Solve 16x^2+48x+36=0
Ans: x=-1.5 or x=-1.5
Q9343) Solve 35x^2+12x+1=0
Ans: x=-0.14 or x=-0.2
Q9344) Solve 2x^2+19x+24=0
Ans: x=-1.5 or x=-8
Q9345) Solve 15x^2+31x+16=0
Ans: x=-1 or x=-1.07
Q9346) Solve 3x^2+18x+15=0
Ans: x=-1 or x=-5
Q9347) Solve x^2+34x+33=0
Ans: x=-1 or x=-33
Q9348) Solve 31x^2+34x+3=0
Ans: x=-0.1 or x=-1
Q9349) Solve 11x^2+35x+26=0
Ans: x=-1.18 or x=-2
Q9350) Solve 36x^2+41x+5=0
Ans: x=-0.14 or x=-1
Q9351) Solve 4x^2+23x+33=0
Ans: x=-2.75 or x=-3
Q9352) Solve 21x^2+46x+24=0
Ans: x=-0.86 or x=-1.33
Q9353) Solve 6x^2+13x+6=0
Ans: x=-0.67 or x=-1.5
Q9354) Solve 4x^2+17x+4=0
Ans: x=-0.25 or x=-4
Q9355) Solve 14x^2+41x+15=0
Ans: x=-0.43 or x=-2.5
Q9356) Solve 24x^2+23x+5=0
Ans: x=-0.33 or x=-0.62
Q9357) Solve 32x^2+50x+18=0
Ans: x=-0.56 or x=-1
Q9358) Solve 23x^2+50x+27=0
Ans: x=-1 or x=-1.17
Q9359) Solve 7x^2+40x+48=0
Ans: x=-1.71 or x=-4
Q9360) Solve 35x^2+47x+8=0
Ans: x=-0.2 or x=-1.14
Q9361) Solve 49x^2+14x+1=0
Ans: x=-0.14 or x=-0.14
Q9362) Solve 17x^2+41x+24=0
Ans: x=-1 or x=-1.41
Q9363) Solve 10x^2+33x+20=0
Ans: x=-0.8 or x=-2.5
Q9364) Solve 17x^2+36x+4=0
Ans: x=-0.12 or x=-2
Q9365) Solve 35x^2+27x+4=0
Ans: x=-0.2 or x=-0.57
Q9366) Solve 26x^2+27x+1=0
Ans: x=-0.04 or x=-1
Q9367) Solve 18x^2+28x+10=0
Ans: x=-0.56 or x=-1
Q9368) Solve 3x^2+19x+20=0
Ans: x=-1.33 or x=-5
Q9369) Solve 24x^2+36x+12=0
Ans: x=-0.5 or x=-1
Q9370) Solve 20x^2+30x+10=0
Ans: x=-0.5 or x=-1
Q9371) Solve 26x^2+48x+22=0
Ans: x=-0.85 or x=-1
Q9372) Solve 6x^2+21x+15=0
Ans: x=-1 or x=-2.5
Q9373) Solve 16x^2+50x+6=0
Ans: x=-0.12 or x=-3
Q9374) Solve x^2+23x+22=0
Ans: x=-1 or x=-22
Q9375) Solve 22x^2+28x+6=0
Ans: x=-0.27 or x=-1
Q9376) Solve 15x^2+29x+8=0
Ans: x=-0.33 or x=-1.6
Q9377) Solve 36x^2+31x+3=0
Ans: x=-0.11 or x=-0.75
Q9378) Solve 3x^2+37x+34=0
Ans: x=-1 or x=-11.33
Q9379) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q9380) Solve 2x^2+18x+36=0
Ans: x=-3 or x=-6
Q9381) Solve 14x^2+39x+27=0
Ans: x=-1.29 or x=-1.5
Q9382) Solve 8x^2+47x+39=0
Ans: x=-1 or x=-4.87
Q9383) Solve 37x^2+40x+3=0
Ans: x=-0.08 or x=-1
Q9384) Solve 8x^2+34x+15=0
Ans: x=-0.5 or x=-3.75
Q9385) Solve 10x^2+30x+20=0
Ans: x=-1 or x=-2
Q9386) Solve 3x^2+23x+40=0
Ans: x=-2.67 or x=-5
Q9387) Solve 4x^2+20x+9=0
Ans: x=-0.5 or x=-4.5
Q9388) Solve 49x^2+28x+3=0
Ans: x=-0.14 or x=-0.43
Q9389) Solve 9x^2+20x+11=0
Ans: x=-1 or x=-1.22
Q9390) Solve 28x^2+34x+6=0
Ans: x=-0.21 or x=-1
Q9391) Solve 14x^2+44x+30=0
Ans: x=-1 or x=-2.14
Q9392) Solve 16x^2+20x+4=0
Ans: x=-0.25 or x=-1
Q9393) Solve 6x^2+29x+33=0
Ans: x=-1.83 or x=-3
Q9394) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q9395) Solve 7x^2+21x+14=0
Ans: x=-1 or x=-2
Q9396) Solve 13x^2+20x+7=0
Ans: x=-0.54 or x=-1
Q9397) Solve 3x^2+43x+14=0
Ans: x=-0.33 or x=-14
Q9398) Solve 32x^2+45x+13=0
Ans: x=-0.41 or x=-1
Q9399) Solve 12x^2+37x+28=0
Ans: x=-1.33 or x=-1.75
Q9400) Solve 21x^2+26x+8=0
Ans: x=-0.57 or x=-0.67
Q9401) Solve 44x^2+32x+5=0
Ans: x=-0.23 or x=-0.5
Q9402) Solve 4x^2+21x+5=0
Ans: x=-0.25 or x=-5
Q9403) Solve 15x^2+32x+4=0
Ans: x=-0.13 or x=-2
Q9404) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q9405) Solve 15x^2+47x+34=0
Ans: x=-1.13 or x=-2
Q9406) Solve 20x^2+12x+1=0
Ans: x=-0.1 or x=-0.5
Q9407) Solve 8x^2+41x+5=0
Ans: x=-0.12 or x=-5
Q9408) Solve 27x^2+39x+14=0
Ans: x=-0.67 or x=-0.78
Q9409) Solve 12x^2+45x+33=0
Ans: x=-1 or x=-2.75
Q9410) Solve 5x^2+23x+24=0
Ans: x=-1.6 or x=-3
Q9411) Solve 19x^2+43x+10=0
Ans: x=-0.26 or x=-2
Q9412) Solve x^2+9x+20=0
Ans: x=-4 or x=-5
Q9413) Solve 3x^2+30x+27=0
Ans: x=-1 or x=-9
Q9414) Solve 34x^2+40x+6=0
Ans: x=-0.18 or x=-1
Q9415) Solve 2x^2+21x+34=0
Ans: x=-2 or x=-8.5
Q9416) Solve 20x^2+38x+12=0
Ans: x=-0.4 or x=-1.5
Q9417) Solve 12x^2+21x+9=0
Ans: x=-0.75 or x=-1
Q9418) Solve 22x^2+45x+2=0
Ans: x=-0.05 or x=-2
Q9419) Solve 29x^2+34x+5=0
Ans: x=-0.17 or x=-1
Q9420) Solve 10x^2+46x+36=0
Ans: x=-1 or x=-3.6
Q9421) Solve 21x^2+50x+25=0
Ans: x=-0.71 or x=-1.67
Q9422) Solve 39x^2+32x+4=0
Ans: x=-0.15 or x=-0.67
Q9423) Solve 17x^2+36x+19=0
Ans: x=-1 or x=-1.12
Q9424) Solve x^2+46x+45=0
Ans: x=-1 or x=-45
Q9425) Solve 10x^2+33x+9=0
Ans: x=-0.3 or x=-3
Q9426) Solve 15x^2+43x+22=0
Ans: x=-0.67 or x=-2.2
Q9427) Solve 39x^2+19x+2=0
Ans: x=-0.15 or x=-0.33
Q9428) Solve 15x^2+32x+17=0
Ans: x=-1 or x=-1.13
Q9429) Solve 7x^2+23x+16=0
Ans: x=-1 or x=-2.29
Q9430) Solve x^2+15x+14=0
Ans: x=-1 or x=-14
Q9431) Solve 44x^2+42x+10=0
Ans: x=-0.45 or x=-0.5
Q9432) Solve 2x^2+27x+25=0
Ans: x=-1 or x=-12.5
Q9433) Solve 21x^2+40x+16=0
Ans: x=-0.57 or x=-1.33
Q9434) Solve 7x^2+30x+32=0
Ans: x=-2 or x=-2.29
Q9435) Solve 25x^2+15x+2=0
Ans: x=-0.2 or x=-0.4
Q9436) Solve 6x^2+49x+23=0
Ans: x=-0.5 or x=-7.67
Q9437) Solve 18x^2+21x+6=0
Ans: x=-0.5 or x=-0.67
Q9438) Solve 7x^2+36x+32=0
Ans: x=-1.14 or x=-4
Q9439) Solve 21x^2+46x+25=0
Ans: x=-1 or x=-1.19
Q9440) Solve 4x^2+15x+9=0
Ans: x=-0.75 or x=-3
Q9441) Solve 12x^2+49x+30=0
Ans: x=-0.75 or x=-3.33
Q9442) Solve 32x^2+38x+11=0
Ans: x=-0.5 or x=-0.69
Q9443) Solve 15x^2+32x+4=0
Ans: x=-0.13 or x=-2
Q9444) Solve 9x^2+45x+36=0
Ans: x=-1 or x=-4
Q9445) Solve x^2+4x+3=0
Ans: x=-1 or x=-3
Q9446) Solve 24x^2+34x+7=0
Ans: x=-0.25 or x=-1.17
Q9447) Solve 8x^2+22x+5=0
Ans: x=-0.25 or x=-2.5
Q9448) Solve 10x^2+35x+30=0
Ans: x=-1.5 or x=-2
Q9449) Solve 50x^2+33x+4=0
Ans: x=-0.16 or x=-0.5
Q9450) Solve 10x^2+47x+48=0
Ans: x=-1.5 or x=-3.2
Q9451) Solve 29x^2+33x+4=0
Ans: x=-0.14 or x=-1
Q9452) Solve 9x^2+31x+22=0
Ans: x=-1 or x=-2.44
Q9453) Solve 25x^2+49x+24=0
Ans: x=-0.96 or x=-1
Q9454) Solve 18x^2+23x+5=0
Ans: x=-0.28 or x=-1
Q9455) Solve 45x^2+18x+1=0
Ans: x=-0.07 or x=-0.33
Q9456) Solve 28x^2+30x+2=0
Ans: x=-0.07 or x=-1
Q9457) Solve x^2+34x+33=0
Ans: x=-1 or x=-33
Q9458) Solve 22x^2+28x+6=0
Ans: x=-0.27 or x=-1
Q9459) Solve 13x^2+35x+18=0
Ans: x=-0.69 or x=-2
Q9460) Solve 34x^2+37x+10=0
Ans: x=-0.5 or x=-0.59
Q9461) Solve 29x^2+30x+1=0
Ans: x=-0.03 or x=-1
Q9462) Solve 4x^2+23x+30=0
Ans: x=-2 or x=-3.75
Q9463) Solve 8x^2+32x+32=0
Ans: x=-2 or x=-2
Q9464) Solve x^2+19x+48=0
Ans: x=-3 or x=-16
Q9465) Solve 24x^2+27x+3=0
Ans: x=-0.12 or x=-1
Q9466) Solve 2x^2+49x+47=0
Ans: x=-1 or x=-23.5
Q9467) Solve 14x^2+30x+16=0
Ans: x=-1 or x=-1.14
Q9468) Solve 5x^2+21x+16=0
Ans: x=-1 or x=-3.2
Q9469) Solve 40x^2+22x+3=0
Ans: x=-0.25 or x=-0.3
Q9470) Solve 25x^2+48x+23=0
Ans: x=-0.92 or x=-1
Q9471) Solve 28x^2+16x+1=0
Ans: x=-0.07 or x=-0.5
Q9472) Solve 24x^2+11x+1=0
Ans: x=-0.12 or x=-0.33
Q9473) Solve 5x^2+29x+38=0
Ans: x=-2 or x=-3.8
Q9474) Solve 24x^2+26x+2=0
Ans: x=-0.08 or x=-1
Q9475) Solve 7x^2+43x+40=0
Ans: x=-1.14 or x=-5
Q9476) Solve 21x^2+35x+14=0
Ans: x=-0.67 or x=-1
Q9477) Solve 2x^2+41x+20=0
Ans: x=-0.5 or x=-20
Q9478) Solve 50x^2+40x+6=0
Ans: x=-0.2 or x=-0.6
Q9479) Solve 35x^2+45x+10=0
Ans: x=-0.29 or x=-1
Q9480) Solve 14x^2+48x+34=0
Ans: x=-1 or x=-2.43
Q9481) Solve 18x^2+48x+24=0
Ans: x=-0.67 or x=-2
Q9482) Solve 3x^2+46x+15=0
Ans: x=-0.33 or x=-15
Q9483) Solve 25x^2+49x+24=0
Ans: x=-0.96 or x=-1
Q9484) Solve 4x^2+6x+2=0
Ans: x=-0.5 or x=-1
Q9485) Solve 2x^2+12x+10=0
Ans: x=-1 or x=-5
Q9486) Solve 18x^2+37x+14=0
Ans: x=-0.5 or x=-1.56
Q9487) Solve 26x^2+29x+8=0
Ans: x=-0.5 or x=-0.62
Q9488) Solve 36x^2+37x+1=0
Ans: x=-0.03 or x=-1
Q9489) Solve 10x^2+38x+28=0
Ans: x=-1 or x=-2.8
Q9490) Solve 42x^2+43x+6=0
Ans: x=-0.17 or x=-0.86
Q9491) Solve 12x^2+32x+5=0
Ans: x=-0.17 or x=-2.5
Q9492) Solve 27x^2+47x+20=0
Ans: x=-0.74 or x=-1
Q9493) Solve 2x^2+9x+4=0
Ans: x=-0.5 or x=-4
Q9494) Solve 7x^2+30x+8=0
Ans: x=-0.29 or x=-4
Q9495) Solve 48x^2+28x+4=0
Ans: x=-0.25 or x=-0.33
Q9496) Solve 15x^2+46x+35=0
Ans: x=-1.4 or x=-1.67
Q9497) Solve 17x^2+42x+25=0
Ans: x=-1 or x=-1.47
Q9498) Solve 4x^2+27x+18=0
Ans: x=-0.75 or x=-6
Q9499) Solve 40x^2+14x+1=0
Ans: x=-0.1 or x=-0.25
Q9500) Solve 20x^2+36x+13=0
Ans: x=-0.5 or x=-1.3
Q9501) Solve 42x^2+43x+1=0
Ans: x=-0.02 or x=-1
Q9502) Solve 20x^2+40x+20=0
Ans: x=-1 or x=-1
Q9503) Solve 18x^2+9x+1=0
Ans: x=-0.17 or x=-0.33
Q9504) Solve 8x^2+38x+44=0
Ans: x=-2 or x=-2.75
Q9505) Solve 25x^2+25x+6=0
Ans: x=-0.4 or x=-0.6
Q9506) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q9507) Solve 15x^2+20x+5=0
Ans: x=-0.33 or x=-1
Q9508) Solve 27x^2+39x+4=0
Ans: x=-0.11 or x=-1.33
Q9509) Solve 28x^2+30x+8=0
Ans: x=-0.5 or x=-0.57
Q9510) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q9511) Solve 24x^2+11x+1=0
Ans: x=-0.12 or x=-0.33
Q9512) Solve 14x^2+9x+1=0
Ans: x=-0.14 or x=-0.5
Q9513) Solve 12x^2+32x+5=0
Ans: x=-0.17 or x=-2.5
Q9514) Solve 11x^2+28x+17=0
Ans: x=-1 or x=-1.55
Q9515) Solve 16x^2+26x+3=0
Ans: x=-0.12 or x=-1.5
Q9516) Solve 28x^2+40x+13=0
Ans: x=-0.5 or x=-0.93
Q9517) Solve 18x^2+35x+17=0
Ans: x=-0.94 or x=-1
Q9518) Solve 18x^2+27x+9=0
Ans: x=-0.5 or x=-1
Q9519) Solve 16x^2+35x+6=0
Ans: x=-0.19 or x=-2
Q9520) Solve 19x^2+49x+22=0
Ans: x=-0.58 or x=-2
Q9521) Solve 5x^2+33x+18=0
Ans: x=-0.6 or x=-6
Q9522) Solve 2x^2+37x+35=0
Ans: x=-1 or x=-17.5
Q9523) Solve 8x^2+36x+16=0
Ans: x=-0.5 or x=-4
Q9524) Solve 33x^2+43x+14=0
Ans: x=-0.64 or x=-0.67
Q9525) Solve 44x^2+44x+11=0
Ans: x=-0.5 or x=-0.5
Q9526) Solve 11x^2+27x+16=0
Ans: x=-1 or x=-1.45
Q9527) Solve 31x^2+43x+12=0
Ans: x=-0.39 or x=-1
Q9528) Solve 25x^2+34x+9=0
Ans: x=-0.36 or x=-1
Q9529) Solve 5x^2+33x+18=0
Ans: x=-0.6 or x=-6
Q9530) Solve 24x^2+30x+9=0
Ans: x=-0.5 or x=-0.75
Q9531) Solve 8x^2+44x+48=0
Ans: x=-1.5 or x=-4
Q9532) Solve 10x^2+40x+30=0
Ans: x=-1 or x=-3
Q9533) Solve 11x^2+35x+26=0
Ans: x=-1.18 or x=-2
Q9534) Solve 13x^2+27x+2=0
Ans: x=-0.08 or x=-2
Q9535) Solve 5x^2+22x+24=0
Ans: x=-2 or x=-2.4
Q9536) Solve 24x^2+26x+5=0
Ans: x=-0.25 or x=-0.83
Q9537) Solve 30x^2+39x+12=0
Ans: x=-0.5 or x=-0.8
Q9538) Solve 20x^2+46x+12=0
Ans: x=-0.3 or x=-2
Q9539) Solve 2x^2+40x+38=0
Ans: x=-1 or x=-19
Q9540) Solve 19x^2+23x+4=0
Ans: x=-0.21 or x=-1
Q9541) Solve 10x^2+19x+7=0
Ans: x=-0.5 or x=-1.4
Q9542) Solve 44x^2+24x+1=0
Ans: x=-0.05 or x=-0.5
Q9543) Solve 17x^2+21x+4=0
Ans: x=-0.24 or x=-1
Q9544) Solve 23x^2+33x+10=0
Ans: x=-0.43 or x=-1
Q9545) Solve 24x^2+22x+3=0
Ans: x=-0.17 or x=-0.75
Q9546) Solve 18x^2+39x+15=0
Ans: x=-0.5 or x=-1.67
Q9547) Solve 6x^2+27x+30=0
Ans: x=-2 or x=-2.5
Q9548) Solve 42x^2+45x+3=0
Ans: x=-0.07 or x=-1
Q9549) Solve 4x^2+5x+1=0
Ans: x=-0.25 or x=-1
Q9550) Solve 6x^2+18x+12=0
Ans: x=-1 or x=-2
Q9551) Solve 6x^2+43x+7=0
Ans: x=-0.17 or x=-7
Q9552) Solve 9x^2+41x+46=0
Ans: x=-2 or x=-2.56
Q9553) Solve 8x^2+41x+50=0
Ans: x=-2 or x=-3.12
Q9554) Solve 15x^2+28x+12=0
Ans: x=-0.67 or x=-1.2
Q9555) Solve 28x^2+15x+2=0
Ans: x=-0.25 or x=-0.29
Q9556) Solve 18x^2+29x+10=0
Ans: x=-0.5 or x=-1.11
Q9557) Solve 30x^2+47x+7=0
Ans: x=-0.17 or x=-1.4
Q9558) Solve 6x^2+32x+10=0
Ans: x=-0.33 or x=-5
Q9559) Solve 15x^2+43x+8=0
Ans: x=-0.2 or x=-2.67
Q9560) Solve 9x^2+24x+7=0
Ans: x=-0.33 or x=-2.33
Q9561) Solve 18x^2+35x+13=0
Ans: x=-0.5 or x=-1.44
Q9562) Solve 8x^2+44x+36=0
Ans: x=-1 or x=-4.5
Q9563) Solve 4x^2+21x+20=0
Ans: x=-1.25 or x=-4
Q9564) Solve 8x^2+32x+30=0
Ans: x=-1.5 or x=-2.5
Q9565) Solve 30x^2+39x+12=0
Ans: x=-0.5 or x=-0.8
Q9566) Solve 22x^2+27x+5=0
Ans: x=-0.23 or x=-1
Q9567) Solve 4x^2+32x+15=0
Ans: x=-0.5 or x=-7.5
Q9568) Solve x^2+15x+44=0
Ans: x=-4 or x=-11
Q9569) Solve 8x^2+26x+20=0
Ans: x=-1.25 or x=-2
Q9570) Solve 3x^2+7x+2=0
Ans: x=-0.33 or x=-2
Q9571) Solve 17x^2+45x+22=0
Ans: x=-0.65 or x=-2
Q9572) Solve 49x^2+42x+8=0
Ans: x=-0.29 or x=-0.57
Q9573) Solve 21x^2+41x+20=0
Ans: x=-0.95 or x=-1
Q9574) Solve 15x^2+11x+2=0
Ans: x=-0.33 or x=-0.4
Q9575) Solve 8x^2+26x+15=0
Ans: x=-0.75 or x=-2.5
Q9576) Solve 50x^2+15x+1=0
Ans: x=-0.1 or x=-0.2
Q9577) Solve 8x^2+43x+44=0
Ans: x=-1.37 or x=-4
Q9578) Solve 36x^2+36x+8=0
Ans: x=-0.33 or x=-0.67
Q9579) Solve 4x^2+24x+20=0
Ans: x=-1 or x=-5
Q9580) Solve 10x^2+43x+46=0
Ans: x=-2 or x=-2.3
Q9581) Solve 11x^2+34x+24=0
Ans: x=-1.09 or x=-2
Q9582) Solve 17x^2+32x+15=0
Ans: x=-0.88 or x=-1
Q9583) Solve 12x^2+29x+17=0
Ans: x=-1 or x=-1.42
Q9584) Solve 44x^2+46x+12=0
Ans: x=-0.5 or x=-0.55
Q9585) Solve 13x^2+43x+34=0
Ans: x=-1.31 or x=-2
Q9586) Solve 6x^2+21x+9=0
Ans: x=-0.5 or x=-3
Q9587) Solve x^2+11x+28=0
Ans: x=-4 or x=-7
Q9588) Solve 36x^2+33x+6=0
Ans: x=-0.25 or x=-0.67
Q9589) Solve 4x^2+15x+14=0
Ans: x=-1.75 or x=-2
Q9590) Solve 22x^2+49x+10=0
Ans: x=-0.23 or x=-2
Q9591) Solve 4x^2+11x+7=0
Ans: x=-1 or x=-1.75
Q9592) Solve 9x^2+17x+8=0
Ans: x=-0.89 or x=-1
Q9593) Solve 28x^2+30x+2=0
Ans: x=-0.07 or x=-1
Q9594) Solve 34x^2+31x+7=0
Ans: x=-0.41 or x=-0.5
Q9595) Solve 20x^2+28x+8=0
Ans: x=-0.4 or x=-1
Q9596) Solve 8x^2+25x+17=0
Ans: x=-1 or x=-2.12
Q9597) Solve 32x^2+49x+17=0
Ans: x=-0.53 or x=-1
Q9598) Solve 36x^2+24x+3=0
Ans: x=-0.17 or x=-0.5
Q9599) Solve 6x^2+31x+28=0
Ans: x=-1.17 or x=-4
Q9600) Solve 38x^2+47x+9=0
Ans: x=-0.24 or x=-1
Q9601) Solve 26x^2+30x+4=0
Ans: x=-0.15 or x=-1
Q9602) Solve 4x^2+43x+39=0
Ans: x=-1 or x=-9.75
Q9603) Solve 39x^2+47x+8=0
Ans: x=-0.21 or x=-1
Q9604) Solve 12x^2+32x+13=0
Ans: x=-0.5 or x=-2.17
Q9605) Solve 18x^2+36x+10=0
Ans: x=-0.33 or x=-1.67
Q9606) Solve 36x^2+42x+10=0
Ans: x=-0.33 or x=-0.83
Q9607) Solve 32x^2+24x+4=0
Ans: x=-0.25 or x=-0.5
Q9608) Solve 50x^2+30x+4=0
Ans: x=-0.2 or x=-0.4
Q9609) Solve 3x^2+26x+23=0
Ans: x=-1 or x=-7.67
Q9610) Solve 16x^2+32x+16=0
Ans: x=-1 or x=-1
Q9611) Solve 7x^2+45x+38=0
Ans: x=-1 or x=-5.43
Q9612) Solve 35x^2+41x+6=0
Ans: x=-0.17 or x=-1
Q9613) Solve 2x^2+16x+30=0
Ans: x=-3 or x=-5
Q9614) Solve 7x^2+23x+18=0
Ans: x=-1.29 or x=-2
Q9615) Solve 8x^2+40x+48=0
Ans: x=-2 or x=-3
Q9616) Solve 13x^2+29x+6=0
Ans: x=-0.23 or x=-2
Q9617) Solve 9x^2+44x+32=0
Ans: x=-0.89 or x=-4
Q9618) Solve 14x^2+15x+1=0
Ans: x=-0.07 or x=-1
Q9619) Solve 11x^2+39x+34=0
Ans: x=-1.55 or x=-2
Q9620) Solve 22x^2+31x+10=0
Ans: x=-0.5 or x=-0.91
Q9621) Solve 43x^2+47x+4=0
Ans: x=-0.09 or x=-1
Q9622) Solve 12x^2+34x+24=0
Ans: x=-1.33 or x=-1.5
Q9623) Solve 32x^2+12x+1=0
Ans: x=-0.12 or x=-0.25
Q9624) Solve 4x^2+41x+45=0
Ans: x=-1.25 or x=-9
Q9625) Solve 30x^2+34x+8=0
Ans: x=-0.33 or x=-0.8
Q9626) Solve 19x^2+29x+10=0
Ans: x=-0.53 or x=-1
Q9627) Solve 10x^2+46x+24=0
Ans: x=-0.6 or x=-4
Q9628) Solve 11x^2+25x+6=0
Ans: x=-0.27 or x=-2
Q9629) Solve 3x^2+14x+15=0
Ans: x=-1.67 or x=-3
Q9630) Solve 45x^2+36x+7=0
Ans: x=-0.33 or x=-0.47
Q9631) Solve 14x^2+29x+12=0
Ans: x=-0.57 or x=-1.5
Q9632) Solve 50x^2+47x+11=0
Ans: x=-0.44 or x=-0.5
Q9633) Solve 38x^2+23x+2=0
Ans: x=-0.11 or x=-0.5
Q9634) Solve 3x^2+18x+24=0
Ans: x=-2 or x=-4
Q9635) Solve 18x^2+21x+6=0
Ans: x=-0.5 or x=-0.67
Q9636) Solve 4x^2+29x+25=0
Ans: x=-1 or x=-6.25
Q9637) Solve 22x^2+30x+8=0
Ans: x=-0.36 or x=-1
Q9638) Solve 18x^2+43x+24=0
Ans: x=-0.89 or x=-1.5
Q9639) Solve x^2+15x+36=0
Ans: x=-3 or x=-12
Q9640) Solve 2x^2+27x+13=0
Ans: x=-0.5 or x=-13
Q9641) Solve 12x^2+34x+24=0
Ans: x=-1.33 or x=-1.5
Q9642) Solve 2x^2+21x+40=0
Ans: x=-2.5 or x=-8
Q9643) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q9644) Solve 30x^2+28x+6=0
Ans: x=-0.33 or x=-0.6
Q9645) Solve 5x^2+22x+21=0
Ans: x=-1.4 or x=-3
Q9646) Solve 2x^2+23x+30=0
Ans: x=-1.5 or x=-10
Q9647) Solve 16x^2+36x+14=0
Ans: x=-0.5 or x=-1.75
Q9648) Solve 6x^2+35x+50=0
Ans: x=-2.5 or x=-3.33
Q9649) Solve 10x^2+9x+2=0
Ans: x=-0.4 or x=-0.5
Q9650) Solve 15x^2+28x+5=0
Ans: x=-0.2 or x=-1.67
Q9651) Solve 3x^2+9x+6=0
Ans: x=-1 or x=-2
Q9652) Solve 18x^2+15x+3=0
Ans: x=-0.33 or x=-0.5
Q9653) Solve 9x^2+36x+32=0
Ans: x=-1.33 or x=-2.67
Q9654) Solve 18x^2+25x+8=0
Ans: x=-0.5 or x=-0.89
Q9655) Solve 12x^2+46x+14=0
Ans: x=-0.33 or x=-3.5
Q9656) Solve x^2+35x+34=0
Ans: x=-1 or x=-34
Q9657) Solve 18x^2+49x+33=0
Ans: x=-1.22 or x=-1.5
Q9658) Solve 24x^2+29x+7=0
Ans: x=-0.33 or x=-0.87
Q9659) Solve 20x^2+31x+11=0
Ans: x=-0.55 or x=-1
Q9660) Solve x^2+15x+44=0
Ans: x=-4 or x=-11
Q9661) Solve 16x^2+23x+7=0
Ans: x=-0.44 or x=-1
Q9662) Solve 28x^2+50x+18=0
Ans: x=-0.5 or x=-1.29
Q9663) Solve 3x^2+34x+31=0
Ans: x=-1 or x=-10.33
Q9664) Solve 12x^2+40x+12=0
Ans: x=-0.33 or x=-3
Q9665) Solve 7x^2+15x+8=0
Ans: x=-1 or x=-1.14
Q9666) Solve 12x^2+26x+4=0
Ans: x=-0.17 or x=-2
Q9667) Solve 42x^2+37x+5=0
Ans: x=-0.17 or x=-0.71
Q9668) Solve 18x^2+27x+7=0
Ans: x=-0.33 or x=-1.17
Q9669) Solve 8x^2+35x+12=0
Ans: x=-0.37 or x=-4
Q9670) Solve 16x^2+44x+28=0
Ans: x=-1 or x=-1.75
Q9671) Solve 5x^2+42x+49=0
Ans: x=-1.4 or x=-7
Q9672) Solve 10x^2+32x+22=0
Ans: x=-1 or x=-2.2
Q9673) Solve 20x^2+20x+5=0
Ans: x=-0.5 or x=-0.5
Q9674) Solve 25x^2+40x+7=0
Ans: x=-0.2 or x=-1.4
Q9675) Solve 33x^2+46x+13=0
Ans: x=-0.39 or x=-1
Q9676) Solve 24x^2+50x+11=0
Ans: x=-0.25 or x=-1.83
Q9677) Solve 32x^2+30x+7=0
Ans: x=-0.44 or x=-0.5
Q9678) Solve 46x^2+29x+3=0
Ans: x=-0.13 or x=-0.5
Q9679) Solve 23x^2+27x+4=0
Ans: x=-0.17 or x=-1
Q9680) Solve 39x^2+42x+3=0
Ans: x=-0.08 or x=-1
Q9681) Solve 35x^2+12x+1=0
Ans: x=-0.14 or x=-0.2
Q9682) Solve 25x^2+25x+4=0
Ans: x=-0.2 or x=-0.8
Q9683) Solve 11x^2+25x+14=0
Ans: x=-1 or x=-1.27
Q9684) Solve 14x^2+50x+44=0
Ans: x=-1.57 or x=-2
Q9685) Solve 8x^2+41x+33=0
Ans: x=-1 or x=-4.12
Q9686) Solve 12x^2+17x+6=0
Ans: x=-0.67 or x=-0.75
Q9687) Solve 13x^2+43x+34=0
Ans: x=-1.31 or x=-2
Q9688) Solve 8x^2+46x+30=0
Ans: x=-0.75 or x=-5
Q9689) Solve 4x^2+25x+39=0
Ans: x=-3 or x=-3.25
Q9690) Solve 5x^2+28x+23=0
Ans: x=-1 or x=-4.6
Q9691) Solve 2x^2+38x+36=0
Ans: x=-1 or x=-18
Q9692) Solve 5x^2+8x+3=0
Ans: x=-0.6 or x=-1
Q9693) Solve 20x^2+21x+4=0
Ans: x=-0.25 or x=-0.8
Q9694) Solve 8x^2+27x+9=0
Ans: x=-0.37 or x=-3
Q9695) Solve 19x^2+38x+19=0
Ans: x=-1 or x=-1
Q9696) Solve 18x^2+49x+10=0
Ans: x=-0.22 or x=-2.5
Q9697) Solve 44x^2+46x+2=0
Ans: x=-0.05 or x=-1
Q9698) Solve 20x^2+23x+6=0
Ans: x=-0.4 or x=-0.75
Q9699) Solve 20x^2+47x+14=0
Ans: x=-0.35 or x=-2
Q9700) Solve 21x^2+31x+8=0
Ans: x=-0.33 or x=-1.14
Q9701) Solve x^2+14x+48=0
Ans: x=-6 or x=-8
Q9702) Solve 4x^2+25x+25=0
Ans: x=-1.25 or x=-5
Q9703) Solve 11x^2+22x+11=0
Ans: x=-1 or x=-1
Q9704) Solve 8x^2+50x+50=0
Ans: x=-1.25 or x=-5
Q9705) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q9706) Solve 3x^2+29x+46=0
Ans: x=-2 or x=-7.67
Q9707) Solve 21x^2+50x+24=0
Ans: x=-0.67 or x=-1.71
Q9708) Solve 22x^2+36x+14=0
Ans: x=-0.64 or x=-1
Q9709) Solve 16x^2+33x+2=0
Ans: x=-0.06 or x=-2
Q9710) Solve 27x^2+43x+16=0
Ans: x=-0.59 or x=-1
Q9711) Solve 14x^2+29x+11=0
Ans: x=-0.5 or x=-1.57
Q9712) Solve 5x^2+39x+28=0
Ans: x=-0.8 or x=-7
Q9713) Solve 10x^2+46x+36=0
Ans: x=-1 or x=-3.6
Q9714) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q9715) Solve 21x^2+45x+6=0
Ans: x=-0.14 or x=-2
Q9716) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q9717) Solve 16x^2+24x+8=0
Ans: x=-0.5 or x=-1
Q9718) Solve 20x^2+19x+3=0
Ans: x=-0.2 or x=-0.75
Q9719) Solve 11x^2+42x+31=0
Ans: x=-1 or x=-2.82
Q9720) Solve 32x^2+39x+7=0
Ans: x=-0.22 or x=-1
Q9721) Solve 12x^2+39x+9=0
Ans: x=-0.25 or x=-3
Q9722) Solve 12x^2+43x+31=0
Ans: x=-1 or x=-2.58
Q9723) Solve 18x^2+27x+9=0
Ans: x=-0.5 or x=-1
Q9724) Solve 15x^2+37x+22=0
Ans: x=-1 or x=-1.47
Q9725) Solve 10x^2+19x+7=0
Ans: x=-0.5 or x=-1.4
Q9726) Solve x^2+12x+11=0
Ans: x=-1 or x=-11
Q9727) Solve 5x^2+25x+20=0
Ans: x=-1 or x=-4
Q9728) Solve 5x^2+37x+42=0
Ans: x=-1.4 or x=-6
Q9729) Solve 2x^2+19x+9=0
Ans: x=-0.5 or x=-9
Q9730) Solve 12x^2+44x+32=0
Ans: x=-1 or x=-2.67
Q9731) Solve 7x^2+48x+36=0
Ans: x=-0.86 or x=-6
Q9732) Solve 33x^2+40x+12=0
Ans: x=-0.55 or x=-0.67
Q9733) Solve 18x^2+42x+20=0
Ans: x=-0.67 or x=-1.67
Q9734) Solve 49x^2+14x+1=0
Ans: x=-0.14 or x=-0.14
Q9735) Solve 46x^2+31x+4=0
Ans: x=-0.17 or x=-0.5
Q9736) Solve 8x^2+34x+26=0
Ans: x=-1 or x=-3.25
Q9737) Solve x^2+8x+16=0
Ans: x=-4 or x=-4
Q9738) Solve 3x^2+22x+39=0
Ans: x=-3 or x=-4.33
Q9739) Solve 6x^2+43x+20=0
Ans: x=-0.5 or x=-6.67
Q9740) Solve 10x^2+11x+3=0
Ans: x=-0.5 or x=-0.6
Q9741) Solve 8x^2+17x+2=0
Ans: x=-0.12 or x=-2
Q9742) Solve 12x^2+29x+10=0
Ans: x=-0.42 or x=-2
Q9743) Solve 26x^2+27x+1=0
Ans: x=-0.04 or x=-1
Q9744) Solve 2x^2+15x+7=0
Ans: x=-0.5 or x=-7
Q9745) Solve 12x^2+41x+15=0
Ans: x=-0.42 or x=-3
Q9746) Solve 24x^2+47x+13=0
Ans: x=-0.33 or x=-1.62
Q9747) Solve 2x^2+12x+16=0
Ans: x=-2 or x=-4
Q9748) Solve 3x^2+43x+14=0
Ans: x=-0.33 or x=-14
Q9749) Solve 4x^2+13x+3=0
Ans: x=-0.25 or x=-3
Q9750) Solve 4x^2+41x+37=0
Ans: x=-1 or x=-9.25
Q9751) Solve 21x^2+13x+2=0
Ans: x=-0.29 or x=-0.33
Q9752) Solve x^2+19x+48=0
Ans: x=-3 or x=-16
Q9753) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q9754) Solve 10x^2+30x+20=0
Ans: x=-1 or x=-2
Q9755) Solve 6x^2+37x+17=0
Ans: x=-0.5 or x=-5.67
Q9756) Solve 35x^2+37x+6=0
Ans: x=-0.2 or x=-0.86
Q9757) Solve 14x^2+47x+30=0
Ans: x=-0.86 or x=-2.5
Q9758) Solve 16x^2+46x+19=0
Ans: x=-0.5 or x=-2.37
Q9759) Solve 17x^2+19x+2=0
Ans: x=-0.12 or x=-1
Q9760) Solve 19x^2+42x+8=0
Ans: x=-0.21 or x=-2
Q9761) Solve 38x^2+25x+3=0
Ans: x=-0.16 or x=-0.5
Q9762) Solve 15x^2+32x+4=0
Ans: x=-0.13 or x=-2
Q9763) Solve 6x^2+16x+8=0
Ans: x=-0.67 or x=-2
Q9764) Solve 16x^2+49x+33=0
Ans: x=-1 or x=-2.06
Q9765) Solve 50x^2+33x+4=0
Ans: x=-0.16 or x=-0.5
Q9766) Solve 6x^2+34x+40=0
Ans: x=-1.67 or x=-4
Q9767) Solve 28x^2+45x+18=0
Ans: x=-0.75 or x=-0.86
Q9768) Solve 20x^2+43x+14=0
Ans: x=-0.4 or x=-1.75
Q9769) Solve 12x^2+37x+26=0
Ans: x=-1.08 or x=-2
Q9770) Solve 20x^2+50x+30=0
Ans: x=-1 or x=-1.5
Q9771) Solve 21x^2+31x+4=0
Ans: x=-0.14 or x=-1.33
Q9772) Solve 28x^2+50x+22=0
Ans: x=-0.79 or x=-1
Q9773) Solve 32x^2+44x+14=0
Ans: x=-0.5 or x=-0.87
Q9774) Solve 20x^2+47x+21=0
Ans: x=-0.6 or x=-1.75
Q9775) Solve 9x^2+45x+36=0
Ans: x=-1 or x=-4
Q9776) Solve 8x^2+29x+26=0
Ans: x=-1.62 or x=-2
Q9777) Solve 12x^2+30x+18=0
Ans: x=-1 or x=-1.5
Q9778) Solve 16x^2+14x+3=0
Ans: x=-0.37 or x=-0.5
Q9779) Solve 14x^2+49x+42=0
Ans: x=-1.5 or x=-2
Q9780) Solve 29x^2+46x+17=0
Ans: x=-0.59 or x=-1
Q9781) Solve 4x^2+50x+46=0
Ans: x=-1 or x=-11.5
Q9782) Solve 24x^2+50x+19=0
Ans: x=-0.5 or x=-1.58
Q9783) Solve 21x^2+32x+11=0
Ans: x=-0.52 or x=-1
Q9784) Solve 42x^2+44x+2=0
Ans: x=-0.05 or x=-1
Q9785) Solve 21x^2+40x+11=0
Ans: x=-0.33 or x=-1.57
Q9786) Solve 14x^2+49x+42=0
Ans: x=-1.5 or x=-2
Q9787) Solve 13x^2+32x+19=0
Ans: x=-1 or x=-1.46
Q9788) Solve 23x^2+48x+4=0
Ans: x=-0.09 or x=-2
Q9789) Solve 9x^2+9x+2=0
Ans: x=-0.33 or x=-0.67
Q9790) Solve 4x^2+28x+13=0
Ans: x=-0.5 or x=-6.5
Q9791) Solve 33x^2+50x+17=0
Ans: x=-0.52 or x=-1
Q9792) Solve 11x^2+48x+16=0
Ans: x=-0.36 or x=-4
Q9793) Solve 12x^2+32x+5=0
Ans: x=-0.17 or x=-2.5
Q9794) Solve 2x^2+20x+50=0
Ans: x=-5 or x=-5
Q9795) Solve 17x^2+43x+18=0
Ans: x=-0.53 or x=-2
Q9796) Solve 10x^2+49x+39=0
Ans: x=-1 or x=-3.9
Q9797) Solve 35x^2+38x+3=0
Ans: x=-0.09 or x=-1
Q9798) Solve 11x^2+34x+24=0
Ans: x=-1.09 or x=-2
Q9799) Solve 42x^2+20x+2=0
Ans: x=-0.14 or x=-0.33
Q9800) Solve 30x^2+41x+14=0
Ans: x=-0.67 or x=-0.7
Q9801) Solve 6x^2+31x+40=0
Ans: x=-2.5 or x=-2.67
Q9802) Solve 39x^2+40x+9=0
Ans: x=-0.33 or x=-0.69
Q9803) Solve 50x^2+35x+6=0
Ans: x=-0.3 or x=-0.4
Q9804) Solve 5x^2+32x+48=0
Ans: x=-2.4 or x=-4
Q9805) Solve 36x^2+38x+2=0
Ans: x=-0.06 or x=-1
Q9806) Solve 30x^2+43x+15=0
Ans: x=-0.6 or x=-0.83
Q9807) Solve 4x^2+21x+26=0
Ans: x=-2 or x=-3.25
Q9808) Solve 5x^2+41x+42=0
Ans: x=-1.2 or x=-7
Q9809) Solve 17x^2+25x+8=0
Ans: x=-0.47 or x=-1
Q9810) Solve 31x^2+42x+11=0
Ans: x=-0.35 or x=-1
Q9811) Solve 2x^2+17x+36=0
Ans: x=-4 or x=-4.5
Q9812) Solve 17x^2+37x+20=0
Ans: x=-1 or x=-1.18
Q9813) Solve 39x^2+43x+10=0
Ans: x=-0.33 or x=-0.77
Q9814) Solve 9x^2+37x+4=0
Ans: x=-0.11 or x=-4
Q9815) Solve 4x^2+31x+27=0
Ans: x=-1 or x=-6.75
Q9816) Solve 18x^2+47x+5=0
Ans: x=-0.11 or x=-2.5
Q9817) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q9818) Solve 2x^2+5x+2=0
Ans: x=-0.5 or x=-2
Q9819) Solve 3x^2+22x+24=0
Ans: x=-1.33 or x=-6
Q9820) Solve 9x^2+43x+28=0
Ans: x=-0.78 or x=-4
Q9821) Solve 7x^2+26x+15=0
Ans: x=-0.71 or x=-3
Q9822) Solve 8x^2+46x+38=0
Ans: x=-1 or x=-4.75
Q9823) Solve 2x^2+23x+11=0
Ans: x=-0.5 or x=-11
Q9824) Solve 10x^2+33x+27=0
Ans: x=-1.5 or x=-1.8
Q9825) Solve 17x^2+37x+20=0
Ans: x=-1 or x=-1.18
Q9826) Solve 40x^2+39x+9=0
Ans: x=-0.37 or x=-0.6
Q9827) Solve 18x^2+42x+24=0
Ans: x=-1 or x=-1.33
Q9828) Solve 14x^2+41x+17=0
Ans: x=-0.5 or x=-2.43
Q9829) Solve 20x^2+27x+9=0
Ans: x=-0.6 or x=-0.75
Q9830) Solve 6x^2+25x+26=0
Ans: x=-2 or x=-2.17
Q9831) Solve 9x^2+27x+8=0
Ans: x=-0.33 or x=-2.67
Q9832) Solve 11x^2+40x+21=0
Ans: x=-0.64 or x=-3
Q9833) Solve 8x^2+45x+37=0
Ans: x=-1 or x=-4.62
Q9834) Solve 2x^2+24x+40=0
Ans: x=-2 or x=-10
Q9835) Solve 28x^2+37x+9=0
Ans: x=-0.32 or x=-1
Q9836) Solve 9x^2+48x+15=0
Ans: x=-0.33 or x=-5
Q9837) Solve 18x^2+29x+3=0
Ans: x=-0.11 or x=-1.5
Q9838) Solve x^2+21x+20=0
Ans: x=-1 or x=-20
Q9839) Solve 20x^2+48x+28=0
Ans: x=-1 or x=-1.4
Q9840) Solve 40x^2+42x+8=0
Ans: x=-0.25 or x=-0.8
Q9841) Solve 28x^2+28x+7=0
Ans: x=-0.5 or x=-0.5
Q9842) Solve 7x^2+30x+27=0
Ans: x=-1.29 or x=-3
Q9843) Solve 30x^2+35x+10=0
Ans: x=-0.5 or x=-0.67
Q9844) Solve 24x^2+22x+5=0
Ans: x=-0.42 or x=-0.5
Q9845) Solve 2x^2+21x+49=0
Ans: x=-3.5 or x=-7
Q9846) Solve 12x^2+22x+8=0
Ans: x=-0.5 or x=-1.33
Q9847) Solve 8x^2+19x+11=0
Ans: x=-1 or x=-1.37
Q9848) Solve 22x^2+49x+10=0
Ans: x=-0.23 or x=-2
Q9849) Solve 8x^2+18x+10=0
Ans: x=-1 or x=-1.25
Q9850) Solve 25x^2+50x+21=0
Ans: x=-0.6 or x=-1.4
Q9851) Solve 19x^2+47x+18=0
Ans: x=-0.47 or x=-2
Q9852) Solve 21x^2+42x+21=0
Ans: x=-1 or x=-1
Q9853) Solve 22x^2+39x+17=0
Ans: x=-0.77 or x=-1
Q9854) Solve 44x^2+26x+2=0
Ans: x=-0.09 or x=-0.5
Q9855) Solve 13x^2+20x+7=0
Ans: x=-0.54 or x=-1
Q9856) Solve 20x^2+42x+18=0
Ans: x=-0.6 or x=-1.5
Q9857) Solve 32x^2+50x+17=0
Ans: x=-0.5 or x=-1.06
Q9858) Solve 21x^2+40x+16=0
Ans: x=-0.57 or x=-1.33
Q9859) Solve 34x^2+46x+12=0
Ans: x=-0.35 or x=-1
Q9860) Solve 24x^2+16x+2=0
Ans: x=-0.17 or x=-0.5
Q9861) Solve 12x^2+36x+27=0
Ans: x=-1.5 or x=-1.5
Q9862) Solve 4x^2+9x+2=0
Ans: x=-0.25 or x=-2
Q9863) Solve 36x^2+38x+2=0
Ans: x=-0.06 or x=-1
Q9864) Solve 3x^2+27x+24=0
Ans: x=-1 or x=-8
Q9865) Solve 10x^2+45x+35=0
Ans: x=-1 or x=-3.5
Q9866) Solve 36x^2+48x+16=0
Ans: x=-0.67 or x=-0.67
Q9867) Solve 48x^2+30x+3=0
Ans: x=-0.12 or x=-0.5
Q9868) Solve 7x^2+50x+48=0
Ans: x=-1.14 or x=-6
Q9869) Solve 4x^2+27x+38=0
Ans: x=-2 or x=-4.75
Q9870) Solve 48x^2+50x+12=0
Ans: x=-0.37 or x=-0.67
Q9871) Solve 5x^2+39x+28=0
Ans: x=-0.8 or x=-7
Q9872) Solve 22x^2+23x+1=0
Ans: x=-0.05 or x=-1
Q9873) Solve 44x^2+49x+12=0
Ans: x=-0.36 or x=-0.75
Q9874) Solve 6x^2+25x+24=0
Ans: x=-1.5 or x=-2.67
Q9875) Solve 22x^2+21x+5=0
Ans: x=-0.45 or x=-0.5
Q9876) Solve 25x^2+38x+13=0
Ans: x=-0.52 or x=-1
Q9877) Solve 48x^2+16x+1=0
Ans: x=-0.08 or x=-0.25
Q9878) Solve 7x^2+42x+35=0
Ans: x=-1 or x=-5
Q9879) Solve 23x^2+47x+24=0
Ans: x=-1 or x=-1.04
Q9880) Solve 25x^2+32x+7=0
Ans: x=-0.28 or x=-1
Q9881) Solve 21x^2+48x+27=0
Ans: x=-1 or x=-1.29
Q9882) Solve 2x^2+13x+11=0
Ans: x=-1 or x=-5.5
Q9883) Solve 42x^2+49x+7=0
Ans: x=-0.17 or x=-1
Q9884) Solve 33x^2+45x+12=0
Ans: x=-0.36 or x=-1
Q9885) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q9886) Solve 13x^2+38x+24=0
Ans: x=-0.92 or x=-2
Q9887) Solve 6x^2+11x+5=0
Ans: x=-0.83 or x=-1
Q9888) Solve 6x^2+13x+5=0
Ans: x=-0.5 or x=-1.67
Q9889) Solve 7x^2+37x+30=0
Ans: x=-1 or x=-4.29
Q9890) Solve 6x^2+23x+21=0
Ans: x=-1.5 or x=-2.33
Q9891) Solve 3x^2+15x+18=0
Ans: x=-2 or x=-3
Q9892) Solve 6x^2+35x+36=0
Ans: x=-1.33 or x=-4.5
Q9893) Solve 22x^2+45x+2=0
Ans: x=-0.05 or x=-2
Q9894) Solve 27x^2+44x+17=0
Ans: x=-0.63 or x=-1
Q9895) Solve 5x^2+23x+26=0
Ans: x=-2 or x=-2.6
Q9896) Solve 3x^2+16x+21=0
Ans: x=-2.33 or x=-3
Q9897) Solve 36x^2+12x+1=0
Ans: x=-0.17 or x=-0.17
Q9898) Solve 6x^2+36x+30=0
Ans: x=-1 or x=-5
Q9899) Solve 5x^2+29x+20=0
Ans: x=-0.8 or x=-5
Q9900) Solve 40x^2+47x+7=0
Ans: x=-0.17 or x=-1
Q9901) Solve 22x^2+47x+6=0
Ans: x=-0.14 or x=-2
Q9902) Solve 18x^2+25x+8=0
Ans: x=-0.5 or x=-0.89
Q9903) Solve 16x^2+40x+16=0
Ans: x=-0.5 or x=-2
Q9904) Solve 34x^2+42x+8=0
Ans: x=-0.24 or x=-1
Q9905) Solve 44x^2+35x+6=0
Ans: x=-0.25 or x=-0.55
Q9906) Solve 3x^2+37x+44=0
Ans: x=-1.33 or x=-11
Q9907) Solve 20x^2+35x+15=0
Ans: x=-0.75 or x=-1
Q9908) Solve 40x^2+43x+7=0
Ans: x=-0.2 or x=-0.87
Q9909) Solve 40x^2+48x+8=0
Ans: x=-0.2 or x=-1
Q9910) Solve 18x^2+26x+8=0
Ans: x=-0.44 or x=-1
Q9911) Solve 2x^2+28x+26=0
Ans: x=-1 or x=-13
Q9912) Solve 9x^2+45x+26=0
Ans: x=-0.67 or x=-4.33
Q9913) Solve 29x^2+49x+20=0
Ans: x=-0.69 or x=-1
Q9914) Solve 41x^2+43x+2=0
Ans: x=-0.05 or x=-1
Q9915) Solve x^2+4x+4=0
Ans: x=-2 or x=-2
Q9916) Solve 38x^2+49x+11=0
Ans: x=-0.29 or x=-1
Q9917) Solve 8x^2+46x+45=0
Ans: x=-1.25 or x=-4.5
Q9918) Solve 12x^2+47x+45=0
Ans: x=-1.67 or x=-2.25
Q9919) Solve 10x^2+11x+3=0
Ans: x=-0.5 or x=-0.6
Q9920) Solve 37x^2+49x+12=0
Ans: x=-0.32 or x=-1
Q9921) Solve x^2+19x+34=0
Ans: x=-2 or x=-17
Q9922) Solve 30x^2+41x+7=0
Ans: x=-0.2 or x=-1.17
Q9923) Solve 8x^2+40x+50=0
Ans: x=-2.5 or x=-2.5
Q9924) Solve 27x^2+18x+3=0
Ans: x=-0.33 or x=-0.33
Q9925) Solve 14x^2+34x+12=0
Ans: x=-0.43 or x=-2
Q9926) Solve 6x^2+49x+43=0
Ans: x=-1 or x=-7.17
Q9927) Solve 3x^2+20x+28=0
Ans: x=-2 or x=-4.67
Q9928) Solve 9x^2+21x+12=0
Ans: x=-1 or x=-1.33
Q9929) Solve 18x^2+39x+15=0
Ans: x=-0.5 or x=-1.67
Q9930) Solve 14x^2+45x+9=0
Ans: x=-0.21 or x=-3
Q9931) Solve 5x^2+43x+24=0
Ans: x=-0.6 or x=-8
Q9932) Solve 4x^2+47x+33=0
Ans: x=-0.75 or x=-11
Q9933) Solve 12x^2+20x+8=0
Ans: x=-0.67 or x=-1
Q9934) Solve 15x^2+38x+16=0
Ans: x=-0.53 or x=-2
Q9935) Solve 25x^2+50x+21=0
Ans: x=-0.6 or x=-1.4
Q9936) Solve 36x^2+42x+12=0
Ans: x=-0.5 or x=-0.67
Q9937) Solve 12x^2+49x+44=0
Ans: x=-1.33 or x=-2.75
Q9938) Solve 7x^2+15x+8=0
Ans: x=-1 or x=-1.14
Q9939) Solve 16x^2+42x+26=0
Ans: x=-1 or x=-1.62
Q9940) Solve 12x^2+39x+30=0
Ans: x=-1.25 or x=-2
Q9941) Solve 9x^2+23x+10=0
Ans: x=-0.56 or x=-2
Q9942) Solve 12x^2+40x+12=0
Ans: x=-0.33 or x=-3
Q9943) Solve 35x^2+48x+13=0
Ans: x=-0.37 or x=-1
Q9944) Solve 28x^2+46x+16=0
Ans: x=-0.5 or x=-1.14
Q9945) Solve 30x^2+41x+13=0
Ans: x=-0.5 or x=-0.87
Q9946) Solve 6x^2+29x+23=0
Ans: x=-1 or x=-3.83
Q9947) Solve 22x^2+27x+5=0
Ans: x=-0.23 or x=-1
Q9948) Solve 26x^2+49x+18=0
Ans: x=-0.5 or x=-1.38
Q9949) Solve 2x^2+19x+44=0
Ans: x=-4 or x=-5.5
Q9950) Solve 2x^2+27x+25=0
Ans: x=-1 or x=-12.5
Q9951) Solve 4x^2+34x+42=0
Ans: x=-1.5 or x=-7
Q9952) Solve 9x^2+45x+36=0
Ans: x=-1 or x=-4
Q9953) Solve 5x^2+23x+18=0
Ans: x=-1 or x=-3.6
Q9954) Solve 18x^2+19x+1=0
Ans: x=-0.06 or x=-1
Q9955) Solve 25x^2+50x+25=0
Ans: x=-1 or x=-1
Q9956) Solve 20x^2+27x+7=0
Ans: x=-0.35 or x=-1
Q9957) Solve 20x^2+49x+30=0
Ans: x=-1.2 or x=-1.25
Q9958) Solve 6x^2+17x+7=0
Ans: x=-0.5 or x=-2.33
Q9959) Solve 16x^2+38x+12=0
Ans: x=-0.37 or x=-2
Q9960) Solve 25x^2+10x+1=0
Ans: x=-0.2 or x=-0.2
Q9961) Solve 17x^2+30x+13=0
Ans: x=-0.76 or x=-1
Q9962) Solve 45x^2+34x+5=0
Ans: x=-0.2 or x=-0.56
Q9963) Solve 12x^2+43x+21=0
Ans: x=-0.58 or x=-3
Q9964) Solve 12x^2+49x+39=0
Ans: x=-1.08 or x=-3
Q9965) Solve 5x^2+49x+36=0
Ans: x=-0.8 or x=-9
Q9966) Solve 12x^2+25x+13=0
Ans: x=-1 or x=-1.08
Q9967) Solve 6x^2+44x+48=0
Ans: x=-1.33 or x=-6
Q9968) Solve 18x^2+45x+27=0
Ans: x=-1 or x=-1.5
Q9969) Solve 24x^2+42x+18=0
Ans: x=-0.75 or x=-1
Q9970) Solve 3x^2+26x+40=0
Ans: x=-2 or x=-6.67
Q9971) Solve 28x^2+44x+3=0
Ans: x=-0.07 or x=-1.5
Q9972) Solve 31x^2+42x+11=0
Ans: x=-0.35 or x=-1
Q9973) Solve 35x^2+39x+4=0
Ans: x=-0.11 or x=-1
Q9974) Solve 24x^2+47x+23=0
Ans: x=-0.96 or x=-1
Q9975) Solve 29x^2+34x+5=0
Ans: x=-0.17 or x=-1
Q9976) Solve 28x^2+48x+20=0
Ans: x=-0.71 or x=-1
Q9977) Solve 40x^2+34x+6=0
Ans: x=-0.25 or x=-0.6
Q9978) Solve 23x^2+41x+18=0
Ans: x=-0.78 or x=-1
Q9979) Solve 2x^2+25x+42=0
Ans: x=-2 or x=-10.5
Q9980) Solve 12x^2+26x+10=0
Ans: x=-0.5 or x=-1.67
Q9981) Solve 20x^2+25x+5=0
Ans: x=-0.25 or x=-1
Q9982) Solve 9x^2+41x+20=0
Ans: x=-0.56 or x=-4
Q9983) Solve 14x^2+33x+13=0
Ans: x=-0.5 or x=-1.86
Q9984) Solve 39x^2+42x+3=0
Ans: x=-0.08 or x=-1
Q9985) Solve 11x^2+45x+36=0
Ans: x=-1.09 or x=-3
Q9986) Solve 4x^2+35x+49=0
Ans: x=-1.75 or x=-7
Q9987) Solve 42x^2+25x+3=0
Ans: x=-0.17 or x=-0.43
Q9988) Solve 3x^2+16x+20=0
Ans: x=-2 or x=-3.33
Q9989) Solve 3x^2+15x+18=0
Ans: x=-2 or x=-3
Q9990) Solve 6x^2+29x+9=0
Ans: x=-0.33 or x=-4.5
Q9991) Solve 6x^2+17x+12=0
Ans: x=-1.33 or x=-1.5
Q9992) Solve 9x^2+26x+17=0
Ans: x=-1 or x=-1.89
Q9993) Solve 45x^2+48x+3=0
Ans: x=-0.07 or x=-1
Q9994) Solve 15x^2+47x+36=0
Ans: x=-1.33 or x=-1.8
Q9995) Solve 38x^2+31x+6=0
Ans: x=-0.32 or x=-0.5
Q9996) Solve 10x^2+40x+40=0
Ans: x=-2 or x=-2
Q9997) Solve 14x^2+23x+8=0
Ans: x=-0.5 or x=-1.14
Q9998) Solve 20x^2+46x+24=0
Ans: x=-0.8 or x=-1.5
Q9999) Solve 46x^2+50x+4=0
Ans: x=-0.09 or x=-1
Q10000) Solve 6x^2+9x+3=0
Ans: x=-0.5 or x=-1
Advertisement
Sum of roots and Product of roots of Quadratic Equation
Given a quadratic equation $ax^2+bx+c=0$ with roots $\alpha$ and $\beta$, we have:
$\displaystyle\boxed{\alpha+\beta=\frac{-b}{a}}$
$\displaystyle\boxed{\alpha\beta=\frac{c}{a}}$
How do we prove this? It is actually due to the quadratic formula!
Recall that the quadratic formula gives the roots of the quadratic equation as: $\displaystyle\boxed{x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}$
Now, we can let
$\displaystyle \alpha=\frac{-b+\sqrt{b^2-4ac}}{2a}$
$\displaystyle \beta=\frac{-b-\sqrt{b^2-4ac}}{2a}$
Hence,
$\displaystyle \alpha+\beta=\frac{-2b}{2a}=\frac{-b}{a}$
$\begin{array}{rcl} \displaystyle \alpha\beta&=&\frac{-b+\sqrt{b^2-4ac}}{2a}\times\frac{-b-\sqrt{b^2-4ac}}{2a}\\ &=&\frac{b^2-(b^2-4ac)}{4a^2}\\ &=&\frac{4ac}{4a^2}\\ &=&\frac{c}{a}\end{array}$
In the above proof, we made use of the identity $(A+B)(A-B)=A^2-B^2$
The above formulas are also known as Vieta’s formulas (for quadratic). There we have it, this is how we prove the formula for the sum and product of roots!
Hwa Chong IP Sec 2 Maths Question – Equation of Parabola
Question:
Given that a parabola intersects the x-axis at x=-4 and x=2, and intersects the y-axis at y=-16, find the equation of the parabola.
Solution:
Sketch of graph:
Now, there is a fast and slow method to this question. The slower method is to let $y=ax^2+bx+c$, and solve 3 simultaneous equations.
The faster method is to let $y=k(x+4)(x-2)$.
Why? We know that x=4 is a root of the polynomial, so it has a factor of (x-4). Similarly, the polynomial has a factor of (x-2). The constant k (to be determined) is added to scale the graph, so that the graph will satisfy y=-16 when x=0.
So, we just substitute in y=-16, x=0 into our new equation.
$-16=k(4)(-2)$.
$-16=-8k$.
So $k=2$.
In conclusion, the equation of the parabola is $y=2(x+4)(x-2)$. |
Explaining Fractions to Kids
May 25, 2022
Welcome to Kids Academy’s Second Math Learning Bundle! This learning bundle is tailored to facilitate the teaching-learning process at home between you and your kids. The lessons here are arranged in a manner wherein the students will be able to use the preceding lesson as a bridge to the next lesson, which sustains increasing levels of challenge. Kids Academy, the creator of this bundle, has been engrossed in making your kids’ exploration of concepts worthwhile, fascinating, remarkable, and impactful. This bundle contains the following articles, which aid in introducing the topic, Fractions:
• What Are Fractions?;
• What Are Equivalent Fractions?; and
• Comparing Fractions.
The first article, What Are Fractions?, is set to familiarize kids with what fractions are so that they can use this knowledge as an anchor in dealing with fraction-related activities in real life. At the end of this article, they will also know what numerators and denominators are, and they will realize the importance of dealing with fractions and their parts. As an overview of the lesson, a fraction represents a specific part of a whole number and comprises two parts: the numerator which represents the parts taken or considered and the denominator which tells the total number of parts in the said whole. Here are some learning worksheets designed to aid your kids in learning the concept of fractions:
The second article, What Are Equivalent Fractions?, is created to show your children that there are fractions with the same value. This helps them distinguish when fractions are equal and when they are not. To give you a glimpse, equivalent or equal fractions, no matter how different their numerators and denominators are, can be determined by your kids, especially through visual representations. At the same time, these representations elevate the chances of your children to remember and apply what has been taught to them. Below are the sample math worksheets, which are brainstormed and produced by our very own, Kids Academy team:
The third article, Comparing Fractions, allows the kids to compare fractions to determine which has greater or lesser value. This is helpful especially when they are faced with scenarios where they are required to choose among things with different values. Take a look at the two worksheets below, which are carefully constructed to make your kids’ excitement even more alive.
In summary, this learning bundle comprises articles that strive to guide your kids in gaining their momentum in learning fractions, going through the challenges with smiles in their faces, and gradually obtaining the mastery of the lesson. From explaining what fractions are, to determining equivalent fractions, until arriving with the time to compare fractions, the children’s lessons are well-paced. With your compassionate heart and keen eyes toward your kids’ learning, they will be able to surpass the difficulty the concept of fractions poses. |
# Mathematical Curio
If you take any integer and rearrange the digits however you like, the difference between the original number and the rearranged number will always be divisible by 9.
For example:
number = 12345
rearranged = 43215
difference = 43215 - 12345 = 30870
difference / 9 = 3430
More generally, you can do the same with numbers in any base, and the resulting difference will be divisible by one less than the base.
For example:
number in base 8: o12345 (= 5349 in decimal)
rearrange = o43215 (= 18061 in decimal)
difference = o43215 - o12345 = o30650 (= 12712 in decimal)
difference / 7 = 1816 (decimal)
We can prove that this holds generally, not just is specially chosen examples.
Let $$b$$ be the base of the numbers we're considering, and let $$n$$ be the number of digits. Then the original number, $$x$$, is defined as
$$x = \sum_{i=0}^n a_i b^i$$
where $$a_i$$ are the digits of $$x$$.
When we rearrange the digits of $$x$$, let $$j(i)$$ denote the new position of the digit originally at position $$i$$. Then the rearranged number, $$y$$, is
$$y = \sum_{i=0}^n a_i b^{j(i)}$$
Then the difference is
$$x - y = \sum_{i=0}^n a_i [b^i - b^{j(i)}]$$
Each term in the series is a constant times $$b^n - b^m$$ for some $$n > m$$. [Note: if $$i < j(i)$$, then the constant is $$-a_i$$.]
$$b^n - b^m = b^m(b^{n-m} - 1)$$
And $$b-1$$ is a factor of $$b^N - 1$$ for any natural number $$N$$. Thus, $$x-y$$ is divisible by $$b-1$$, i.e. one less than the base of $$x$$ and $$y$$.
### Mathematical "Magic"
This is the basis of a mathematical magic trick (the origins of which I don't know). In it, the magician gives the following instructions to the audience:
1. Write down a four digit number
2. Underneath is, rearrange the digits of the number however you like, to form another four digit number.
3. Subtract the smaller number from the larger.
4. Circle a non-zero digit of the result.
5. Tell me the other digits of the result.
The magician will then tell the audience what the circled digit is.
This works because, as we established, the difference will always be divisible by nine. And thus the sum of the digits will also be a multiple of nine. So the magician needs only to sum up the other digits, and subtract it from the next bigger multiple of nine. (This is why it has to be a non-zero digit that is circled: otherwise you wouldn't be able to tell if it was a zero or a nine.) |
#### Need solution for RD Sharma maths class 12 chapter 11 Higher Order Derivatives exercise very short answer type question 7
\begin{aligned} &\text { The value of } \frac{d^{2} x}{d y^{2}} \text { is } \frac{-e^{x}}{\left(1+e^{x}\right)^{3}} \end{aligned}
Hint:
First differentiate given equation w.r.t to $x$
\begin{aligned} &\text { Use } \frac{d y}{d x} =\frac{1}{\frac{dy}{dx}} \end{aligned}
Given:
$y=x+e^{x}$
Explanation:
It is given that
$y=x+e^{x}$
Diff w. r .t be $x$
\begin{aligned} &\frac{d y}{d x}=1+e^{x} \\ &\therefore \frac{d y}{d x}=\frac{1}{\frac{d y}{d x}} \\ &\therefore \frac{d x}{d y}=\frac{1}{1+e^{x}} \end{aligned}
Diff w.r. to $y$
\begin{aligned} &\frac{d}{d y}\left(\frac{d x}{d y}\right)=\frac{d}{d y}\left(\frac{1}{1+e^{x}}\right) \\\\ &\frac{d^{2} x}{d y^{2}}=\frac{d}{d x}\left(\frac{1}{1+e^{x}}\right) \frac{d x}{d y} \\ &=\frac{-1}{\left(1+e^{x}\right)^{2}} e^{x}\left(\frac{1}{1+e^{x}}\right) \\ &=-\left(\frac{1}{1+e^{x}}\right)^{3} e^{x} \\ &\quad \\ &=\frac{-e^{x}}{\left(1+e^{x}\right)^{3}} \end{aligned}
\begin{aligned} Thus,\; \; \frac{d^{2}x}{dy^{2}}=\frac{-e^{x}}{\left(1+e^{x}\right)^{3}} \end{aligned} |
### Kissing Triangles
Determine the total shaded area of the 'kissing triangles'.
### Trice
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
# Hex
##### Stage: 3 Challenge Level:
Malcolm Findlay from Madras College in St Andrews, Scotland has solved the first part of this problem:
Izumi Tomioka, Carol Chow and Priscilla Luk from The Mount School in York solved the second part of the problem:
The original hexagon has sides of length 3 units and we need to work out $x$, the lengths of the sides of the smaller hexagons.
The hexagon below has been made from 6 of the shaded triangles above.
We need to find the length of one of the sides of this hexagon.
Splitting the shaded triangle into half gives us a right angled triangle such that: $$\cos30 = {1.5\over H}$$ $$H = {1.5\over \cos 30}$$ $$H = 1.73$$
Andrei Lazanu (aged 12) from School 205 in Bucharest, Romania, solved both parts of the problem.
This is how he tackled the second part:
To calculate the lengths of the sides of the smaller hexagons I used the following notations:
l for the length of side of the great hexagon
a for the length of side of the small hexagon
I used the following notation:
Triangle ACE is equilateral, because its sides are congruent.
So, angle EAC is $60$°.
Angle FAB is $120$°, since each angle of a regular hexagon is $120$°.
Triangle AEF is congruent with triangle ACB, having all sides congruent. They are also isosceles triangles.
This means that each of the angles FAE and CAB is $30$°.
Therefore angle EAB is $90$°.
Triangle AMN is also equilateral, because it has a $60$° angle (MAN) and AM = AN.
Triangle ANB is isosceles, so AN and NB are congruent.
Therefore, in the right angled triangle MAB,
MA = MN = NB = $a$
AB has length $l$
Applying the Pythagorean Theorem: $$l^2 = (a + a)^2 - a^2$$ $$l^2 = 4a^2 - a^2$$ $$l^2 = 3a^2$$ $$l = a \sqrt{3}$$
If the length of the side of the great hexagon is 3 units long $l = 3$
Therefore $a = \sqrt{3}$ units
An alternative way of calculating "a" takes into account the first part of the problem:
the area of the great hexagon is three times the area of the small one.
For a hexagon of side $l$, the area is calculated as 6 times the area of an equilateral triangle of side $l$,
this means that the area of the great hexagon is:
$6{l^2 \sqrt3\over 4} = {l^2 3\sqrt3\over2}$ $(1)$
The 3 smaller hexagons of side "a" have a total area of:
$3{a^2 3\sqrt3\over 2} = {a^2 9\sqrt3\over2}$ $(2)$
Since, $(1)$ and $(2)$ represent the same area, $3a^2 = l$
which is the same as we found above. |
# CBSE Class 10 Maths Chapter 8 Important Questions with Answers: Introduction to Trigonometry
CBSE Class 10 Maths Chapter 8 Important Questions: In this article, we will be discussing all the important questions with answers from Chapter 8 Introduction to Trigonometry.
CBSE Class 10 Maths Chapter 8 Important Questions with Answers: Introduction to Trigonometry
CBSE Class 10 Maths Chapter 8 Important Questions: In this article, we will be discussing all the different kinds of important questions from Chapter 8 Introduction to Trigonometry in CBSE Class 10 syllabus for the year 2022-23. Students preparing for CBSE Class 10 Mathematics Standard and Mathematics Basic examination must prepare these questions very well in order to get great scores.
These questions have been prepared by subject experts keeping in mind the latest CBSE Class 10 Maths Syllabus 2022-2023. Candidates should prepare these along with the NCERT textbook exercises and sample paper questions to ace in their examinations.
## CBSE Class 10 Maths Chapter 8 Important Questions
ALSO TRY TO SOLVE:
Related: CBSE Class 10 Maths Important Formulas for Last Minute Revision for Board Exam 2023
### MULTIPLE CHOICE QUESTIONS
Q1. If cos⍬ = ½ then cos⍬ -sec ⍬ is equal to
Q2. If sin ⍬ =x and sec⍬ = y, then tan ⍬ is equal to
Q3. If Cos A = ⅗, find the value of 9 + 9 tan2 A
Q4. If 0 ≤ A, B ≤ 900 such that Sin A = ½ and Cos B = ½ , A + B =
Q5. In a Δ ABC, right angled at B, the value of Sin (A +C) is
### VERY SHORT ANSWER QUESTIONS (1MARK each)
Q1. Find the value of (Sin 30 +Cos 30) – (Sin60 + Cos 60)
Q2. If Sin⍬ - Cos ⍬ = 0, find the value of ⍬
Q3. ΔABC is right angled at C, and AC = √3 BC, prove that ∠ ABC = 60o
Q4. If 2 Sin 3x= √3, then find the value of x
Q5. If Sin A + Sin2A=1 then find Cos2 A +Cos4 A
### SHORT ANSWER TYPE QUESTIONS (2MARK questions)
Q1. If tanβ=24/7, then the value of sinβ +cosβ is
Q2. If tan 3x = sin450 cos45o + sin30o then value of x is
Q3. In triangle ABC ,right angled triangled at B, AB =5cm and <ACB=30o,then the length of the side AC is
Q4. Given that the sinβ = √(3/2) and the cosα= 0,then the value of β - α is
Q5. In a triangle ABC i, right angled at C if <A=30o, AB=40 units find BC
### SHORT ANSWER TYPE QUESTIONS (3MARK questions)
Q1. In ∆ DEF, ∠E = 90°, DF – DE =2 cm and EF = 6 cm. Find cos D + sin D.
Q2. In the figure, ∠ACB = 90°, ∠BDC = 90°, CD = 4 cm, BD = 3 cm and AC = 12 cm. Find Cos A – Sin A
### CASE STUDY BASED QUESTION
Pragya noticed a tower built near her colony’s playground. She sees that it is being held by a wire, attached to the top of the tower. The wire makes an angle of 60°with the ground. Using these Information find the answers to the following questions
1. What is the measure of ∠CAB ?
(a) 15°
(b)25°
(c) 30°
(d) 45°
1. What is the sin ratio of ∠CAB ?
(a) 0
(b) 1/2
(c) 1/√2
(d) (√3)/2
1. What is the value of cos ∠ACB ?
(a) 0
(b) 1/2
(c) 1/√2
(d) (√3)/2
4. sin2∠CAB + cos2∠CAB =
(a) 0
(b) 1
(c) 2
(d) 3
5.What is the value of tan 90?
(a) 0
(b) 1
(c) 1/2
(d) not defined
## Answers of CBSE Class 10 Maths Chapter 8 Important Questions
Chapter 8 Introduction to Trigonometry is the first chapter in Unit 5 Trigonometry. This unit has a total weightage of 12 marks. Thus it is a very important chapter for the students planning to appear in CBSE Class 10 Math board examination 2022-23.
Related resources:
To check your preparation for CBSE Class 10 board examination, click on the link below:
All the best!
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0000002881 00000 n >> 0000001939 00000 n 6 0 R 6 0 obj << /Length 614 The Quotient Rule The quotient rule is used to take the derivative of two functions that are being divided. Differentiate x(x² + 1) let u = x and v = x² + 1 d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 . Understanding the Quotient Rule Let's say that you have y = u / v, where both u and v depend on x. Quotient rule is one of the techniques in derivative that is applied to differentiate rational functions. 400 549 300 300 333 576 453 250 333 300 310 500 750 750 750 444 ] The Product and Quotient Rules are covered in this section. << /FontName /TimesNewRomanPSMT /Info 2 0 R Example. The Quotient Rule is a method of differentiating two functions when one function is divided by the other.This a variation on the Product Rule, otherwise known as Leibniz's Law.Usually the upper function is designated the letter U, while the lower is given the letter V. 1 075 ' and v'q) = -1, find the derivative of Using the Quotient Rule with u(q) = 5q1/5, v(q) = 1 - 9, u'q) = sva Then, simplify the two terms in the numerator. (-1) [1-4] dp da = 1-9 + 94/5 (1 - 9)2 >> /Contents 11 0 R Use the quotient rule to answer each of the questions below. >> xref }$$The quotient rule states that the derivative of$${\displaystyle f(x)}$$is 2. This is another very useful formula: d (uv) = vdu + udv dx dx dx. /Filter /FlateDecode 500 500 500 500 500 500 500 500 500 500 278 278 564 564 564 444 To see why this is the case, we consider a situation involving functions with physical context. (a) y = u/v, if u = eax, and v = ebx (b) y = u/v, if u = x+1, and v = xâ1 Exercise 5. /FirstChar 0 /Outlines 1 0 R /ProcSet [/PDF /Text /ImageB /ImageC] >> Product rule: uâv+vâu Quotient Rule: (uâv-vâu)/v2 8. y = -2t2 + 6t - 3 u= v= uâ= vâ= 9. f(x) = (x + 1) (x2 - 3). The quotient rule is a formula for taking the derivative of a quotient of two functions. /Size 12 PRODUCT RULE. Let's look at the formula. The product rule tells us that if $$P$$ is a product of differentiable functions $$f$$ and $$g$$ according to the rule $$P(x) = f(x) g(x)\text{,}$$ then /Length 494 The Product and Quotient Rules are covered in this section. << 0000002096 00000 n << For example, if 11 y, 2 then y can be written as the quotient of two functions. stream stream let u = x and v = x² + 1d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 . 778 778 778 778 778 778 778 778 778 778 778 778 778 778 778 778 /Ascent 891 /Count 2 500 500 500 500 500 500 500 549 500 500 500 500 500 500 500 500 by M. Bourne. That is, if youâre given a formula for f (x), clearly label the formula you find for f' (x). 444 444 444 444 444 444 667 444 444 444 444 444 278 278 278 278 It follows from the limit definition of derivative and is given by⦠Remember the rule in the following way. /F15 If a function is a sum, product, or quotient of simpler functions, then we can use the sum, product, or quotient rules to differentiate it in terms of the simpler functions and their derivatives. We will accept this rule as true without a formal proof. endobj Let$${\displaystyle f(x)=g(x)/h(x),}$$where both$${\displaystyle g}$$and$${\displaystyle h}$$are differentiable and$${\displaystyle h(x)\neq 0. Letâs look at an example of how these two derivative r /StemV 0 endobj This is used when differentiating a product of two functions. 8 0 obj 2 0 obj We write this as y = u v where we identify u as cosx and v as x2. endobj /Type /Page /MediaBox [ 0 0 612 792 ] endobj . /Kids [ >> Then you want to find dy/dx, or d/dx (u / v). /Flags 34 /FontBBox [0 -216 2568 891] xڽUMo�0��W�(�c��l�e�v�i|�wjSE�El���Ӈ�| �{�,�����-��A�P��g�P�g��%ԕ 7�>+���徿��k���FH��37C|� �C����ژ���/�?Z�Z�����IK�ַЩ^�W)�47i�wz1�4{t���ii�ƪ << +u(x)v(x) to obtain So, the quotient rule for differentiation is the derivative of the first times the second minus the first times the derivative of the second over the second squared.'' (2) As an application of the Quotient Rule Integration by Parts formula, consider the %PDF-1.3 endobj /BaseFont /TimesNewRomanPSMT 0000001372 00000 n It is the most important topic of differentiation (a function that is broken down into small functions). Now what we're essentially going to do is reapply the product rule to do what many of your calculus books might call the quotient rule. [ Use the quotient rule to diï¬erentiate the following with The Product Rule. 722 722 722 722 722 722 889 667 611 611 611 611 333 333 333 333 �r\/J�"�-P��9N�j�r�bs�S�-j����rg�Q����br��ɓH�ɽz\�9[N��1;Po���H��b���"��O��������0�Nc�='��[_:����r�7�b���ns0^)�̟�q������w�o:��U}�/��M��s�>��/{D���)8H�]]��u�4VQ֕���o��j 1-9 94/5 p = v(9) u'q) - u(q) v'q) dp da = [vca)] 1.*)-(5925). d (u/v) = v(du/dx) - u(dv/dx) I have mixed feelings about the quotient rule. dx endobj 3 0 obj x + 1 c) Use the chain and product rules (and not the quotient rule) to show that the derivative âof u(x)(v(x)) 1 equals u (x)v(x) â u(x)v ⦠Quotient rule is one of the subtopics of differentiation in calculus. Always start with the bottom'' function and end with the bottom'' function squared. u= v= uâ= vâ= 10. f(x) = (2x + 5) /(2x) >> /LastChar 255 /Resources << << /FontDescriptor 8 0 R Letting u = g(x)and v = f (x)and observing that du = g (x)dxand dv = f (x)dx, we obtain a Quotient Rule Integration by Parts formula: dv u = v u + v u2 du. The quotient rule is a formal rule for differentiating problems where one function is divided by another. endobj The quotient rule states that for two functions, u and v, (See if you can use the product rule and the chain rule on y = uv -1 to derive this formula.) 11 0 obj Quotient rule: The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. /Count 0 333 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 Remember the rule in the following way. << endstream 10 0 R (x + 4)² (x + 4)². dx dx dx. Let y = uv be the product of the functions u and v. Find y â² (2) if u(2)= 3, u â² (2)= â4, v(2)= 1, and v â² (2)= 2 Example 6 Differentiating a Quotient Differentiate f x( )= x2 â1 x2 +1 Example 7 Second and Higher Order Derivatives Find the first four derivatives of y = x3 â 5x2 + 2 If u = 3x + 11 and v = 7x â 2, then u y. v To find the derivative of a function written as a quotient of two function, we can use the quotient rule. As part (b) of Example2.35 shows, it is not true in general that the derivative of a product of two functions is the product of the derivatives of those functions. 0000000069 00000 n Section 3: The Quotient Rule 10 Exercise 4. %���� MIT grad shows an easy way to use the Quotient Rule to differentiate rational functions and a shortcut to remember the formula. startxref 778 778 778 778 778 778 778 778 778 778 778 778 778 778 778 778 Copyright © 2004 - 2020 Revision World Networks Ltd. Subsection The Product Rule. /Filter /FlateDecode 7 0 obj /Widths 7 0 R The quotient rule is actually the product rule in disguise and is used when differentiating a fraction. /Font 5 0 R 921 722 667 667 722 611 556 722 722 333 389 722 611 889 722 722 Use the quotient rule to diï¬erentiate the functions below with respect to x (click on the green letters for the solutions). /Descent -216 556 722 667 556 611 722 722 944 722 722 611 333 278 333 469 500 >> /Pages 4 0 R �̎/JL$�DcY��2�tm�LK�bș��-�;,z�����)pgM�#���6�Bg�0���Ur�tMYE�N��9��:��9��\��#DP����p����أ����\�@=Ym��,!��k[��͉� /ID[<33ec5d477ae4164631e257d5171e8891><33ec5d477ae4164631e257d5171e8891>] trailer In this unit we will state and use the quotient rule. Explanation: Assuming that those who are reading have a minimum level in Maths, everyone knows perfectly that the quotient rule is color (blue) (((u (x))/ (v (x)))^'= (u^' (x)*v (x)-u (x)*v' (x))/ ((v (x))²)), where u (x) and v ⦠It is not necessary to algebraically simplify any of the derivatives you compute. Section 3: The Quotient Rule 10 Exercise 4. 0000003107 00000 n 0000000015 00000 n endobj (a) y = u/v, if u = eax, and v = ebx (b) y = u/v, if u = x+1, and v = xâ1 Exercise 5. /Parent 4 0 R This approach is much easier for more complicated compositions. endobj You can expand it that way if you want, or you can use the chain rule $$\frac d{dt}(t^2+2)^2=2(t^2+2)\frac d{dt}(t^2+2)=2(t^2+2)\cdot 2t$$ which is the same as you got another way. It is basically used in a differentiation problem where one function is divided by the other Quotient Rule: 3466 << 4 0 obj /Parent 4 0 R >> In other words, we always use the quotient rule to take the derivative of rational functions, but sometimes weâll need to apply chain rule as well when parts of that rational function require it. �T6P�9�A�MmK���U��N�2��hâ8�,ƌ�Ђad�}lF��T&Iͩ!: ����Tb)]�܆V��$�\)>o y��N㕑�29O�x�V��iIΡ0X�yN�Zb�%��2�H��"��N@��#���S��ET""A�6�P�y~�,�i�b�e5�;O� The quotient rule is a formal rule for differentiating problems where one function is divided by another. >> Say that an investor is regularly purchasing stock in a particular company. 250 333 500 500 500 500 200 500 333 760 276 500 564 333 760 500 Example 2.36. /ItalicAngle 0 /CapHeight 784 It follows from the limit definition of derivative and is given by . >> x + 4, Let u = x³ and v = (x + 4). 10 0 obj 6. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. d (uv) = vdu + udv (x + 4)(3x²) - x³(1) = 2x³ + 12x² endstream 0000003040 00000 n 722 722 722 722 722 722 722 564 722 722 722 722 722 722 556 500 250 333 408 500 500 833 778 180 333 333 500 564 250 333 250 278 1 0 obj << >> There are two ways to find that. The quotient rule states that if u and v are both functions of x and y, then: if y = u / v then dy / dx = ( v du / dx â u dv / dx ) / v 2 Example 2: Consider y = 1 â sin ( x ) . endobj 0000002127 00000 n Use the quotient rule to diï¬erentiate the following with This is the product rule. 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Derivatives of Products and Quotients. /Contents 9 0 R %%EOF. /Subtype /TrueType 0 12 /Resources << /Type /Pages /MediaBox [ 0 0 612 792 ] If you know it, it might make some operations a little bit faster, but it really comes straight out of the product rule. /Producer (BCL easyPDF 3.11.49) Use the quotient rule to diï¬erentiate the functions below with respect to x (click on the green letters for the solutions). Implicit diï¬erentiation Letâs say you want to ï¬nd y from an equation like y 3 + 3xy 2 = 8 d Instead of solving for y and then taking its derivative, just take of the whole dx 0000000000 65535 f 500 778 333 500 444 1000 500 500 333 1000 556 333 889 778 611 778 Always start with the âbottomâ function and end with the âbottomâ function squared. Then, the quotient rule can be used to find the derivative of U/V as shown below. The quotient rule states that given functions u and v such that #y = (u(x))/(v(x)), dy/dx = (u'(x)v(x) - u(x)v'(x))/(v^2(x))# By assigning u and v equal to the numerator and denominator, respectively, in the example function, we arrive at #dy/dx = [(-sin x)(1+sin x) - (1+cos x)(cos x)]/(1+sin x)^2#. , the quotient rule Let 's say that you have y = /... 5 ) / ( 2x ) 6 from the limit definition of derivative and is byâ¦! Uv ) = vdu + udv dx dx that an investor is regularly purchasing stock in a company! To x ( click on the green letters for the solutions ) shortcut remember! Follows from the limit definition of derivative and is given by⦠remember the rule in form! For example, if 11 y, 2 then y can be written as the quotient rule a. U/V as shown below every time that is broken down into small functions ) for more complicated compositions as quotient. 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How To
# How To Determine a Star's Radius
How do you measure something that’s light-years away? How can we comprehend the enormous size scales that dominate our universe when we’re nothing but little grains of sand on a floating rock? Amazingly, mathematics combined with observation has given us the tools to determine stellar radii, which in turn can tell us just how massive that little blip in the sky really is. Here’s how it works:
1. First, you need to know the luminosity and the surface temperature.
Without this step, you won’t get very far. Luminosity is a star’s intrinsic brightness. The surface temperature is fairly self explanatory. With these two pieces of knowledge, you can discover a lot about a distant star. We’ll use Betelgeuse as an example in this tutorial.
• LuminosityBetelgeuse = 4.6 x 1031 watts (you always want your luminosity units in watts)
• Surface TemperatureBetelgeuse = 3650 K (K is Kelvin, the standard temperature unit used in scientific calculations)
2. Rearrange the luminosity formula to solve for the radius.
The luminosity formula consists of three values that are all pieces of the puzzle: luminosity, surface area, and temperature of the star you’re solving the equation for. If you know two, you can figure out the third. Take a look:
L = 4πr2 x σT4
Breaking this down, L is the luminosity, 4πr2 is the surface area, and σT4 represents the power emitted per square meter [σ (sigma, which has a value of 5.7 x 10-8 watt/(m2 x K4) multiplied by the surface temperature to the fourth power].
We’re not looking for the luminosity, so solving for L would be pointless (and impossible if we don’t know the radius). So, we rearrange the formula using basic algebra to solve for r, resulting in:
3. Plug in your luminosity and temperature values and solve for r.
Now, it’s just a matter of plugging in the numbers and doing a little math. Using the figures we gave for Betelgeuse, plug in the luminosity and temperature.
The radius of Betelgeuse is about 590 billion meters. That’s four times the distance from the Earth to the Sun. Crazy, isn’t it?
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Sample Spaces Unit 6 > Lesson 3 of 12
Experiment 1: What is the probability of each outcome when a dime is tossed?
Outcomes: The outcomes of this experiment are head and tail.
Probabilities:
P(head) = 1 2 P(tail) = 1 2
Definition: The sample space of an experiment is the set of all possible outcomes of that experiment. The sample space of Experiment 1 is: {head, tail}
Experiment 2: A spinner has 4 equal sectors colored yellow, blue, green and red. What is the probability of landing on each color after spinning this spinner?
Sample Space: {yellow, blue, green, red}
Probabilities:
P(yellow) = 1 4 P(blue) = 1 4 P(green) = 1 4 P(red) = 1 4
Experiment 3: What is the probability of each outcome when a single 6-sided die is rolled?
Sample Space: {1, 2, 3, 4, 5, 6}
Probabilities:
P(1) = 1 6 P(2) = 1 6 P(3) = 1 6 P(4) = 1 6 P(5) = 1 6 P(6) = 1 6
Experiment 4: A glass jar contains 1 red, 3 green, 2 blue and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of each outcome?
Sample Space: {red, green, blue, yellow}
Probabilities:
P(red) = 1 10 P(green) = 3 10 P(blue) = 2 = 1 10 5 P(yellow) = 4 = 2 10 5
Summary: The sample space of an experiment is the set of all possible outcomes for that experiment. You may have noticed that for each of the experiments above, the sum of the probabilities of each outcome is 1. This is no coincidence. The sum of the probabilities of the distinct outcomes within a sample space is 1.
The sample space for choosing a single card at random from a deck of 52 playing cards is shown below. There are 52 possible outcomes in this sample space.
The probability of each outcome of this experiment is:
P(card) = 1 52
The sum of the probabilities of the distinct outcomes within this sample space is:
52 = 1 52
### Exercises
Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.
1. What is the sample space for choosing an odd number from 1 to 11 at random? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} {1, 3, 5, 7, 9 11} None of the above. RESULTS BOX:
2. What is the sample space for choosing a prime number less than 15 at random? {2, 3, 5, 7, 11, 13, 15} {2, 3, 5, 7, 11, 13} {2, 3, 5, 7, 9, 11, 13} All of the above. RESULTS BOX:
3. What is the sample space for choosing 1 jelly bean at random from a jar containing 5 red, 7 blue and 2 green jelly beans? {5, 7, 2} {5 red, 7 blue, 2 green} {red, blue, green} None of the above. RESULTS BOX:
4. What is the sample space for choosing 1 letter at random from 5 vowels? {a, e, i, o, u} {v, o, w, e, l} {1, 2, 3, 4, 5} None of the above. RESULTS BOX:
5. What is the sample space for choosing 1 letter at random from the word DIVIDE? {d, i, v, i, d, e} {1, 2, 3, 4, 5, 6} {d, i, v, e} None of the above. RESULTS BOX:
This lesson is by Gisele Glosser. You can find me on Google.
Lessons On Probability Introduction to Probability Certain and Impossible Events Sample Spaces The Complement of an Event Mutually Exclusive Events Addition Rules for Probability Independent Events Dependent Events Conditional Probability Practice Exercises Challenge Exercises Solutions
Related Activities Crossword Puzzles Pre-Made Worksheets Probability Forums Probability Goodies Game |
# Dividing Decimals
Site: Saylor Academy Course: RWM101: Foundations of Real World Math Book: Dividing Decimals
Printed by: Guest user Date: Monday, June 17, 2024, 3:20 AM
## Description
Read this text. Pay attention to the "How To" boxes which give brief step-by-step summaries of how to divide decimals. Complete the practice problems and check your answers.
## Divide Decimals
Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.
To understand decimal division, let's consider the multiplication problem
$(0.2)(4)=0.8$
Remember, a multiplication problem can be rephrased as a division problem. So we can write
$0.8 \div 4=0.2$
We can think of this as "If we divide 8 tenths into four groups, how many are in each group?" Figure $5.5$ shows that there are four groups of two-tenths in eight-tenths. So $0.8 \div 4=0.2$.
Figure 5.5
Using long division notation, we would write
Notice that the decimal point in the quotient is directly above the decimal point in the dividend.
To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.
#### HOW TO
##### Divide a decimal by a whole number.
Step 1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
Step 2. Divide as usual.
In everyday life, we divide whole numbers into decimals-money-to find the price of one item. For example, suppose a case of $24$ water bottles cost $\ 3.99$. To find the price per water bottle, we would divide $\ 3.99$ by $24$, and round the answer to the nearest cent (hundredth).
#### Divide a Decimal by Another Decimal
So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let's look at the same multiplication problem we looked at earlier, but in a different way.
$(0.2)(4)=0.8$
Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, "How many times does $0.2$ go into $0.8$ ?" Because $(0.2)(4)=0.8$, we can say that $0.2$ goes into $0.8$ four times. This means that $0.8$ divided by $0.2$ is 4.
$0.8 \div 0.2=4$
We would get the same answer, $4$, if we divide $8$ by $2$, both whole numbers. Why is this so? Let's think about the division problem as a fraction.
$\frac{0.8}{0.2}$
$\frac{(0.8) 10}{(0.2) 10}$
$\frac{8}{2}$
$4$
We multiplied the numerator and denominator by 10 and ended up just dividing 8 by 4. To divide decimals, we multiply both the numerator and denominator by the same power of 10 to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven't changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right.
We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
It may help to review the vocabulary for division:
#### HOW TO
##### Divide decimal numbers.
Step 1. Determine the sign of the quotient.
Step 2. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.
Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Step 4. Write the quotient with the appropriate sign.
In Example 5.23, we will divide a whole number by a decimal number.
Source: Rice University, https://openstax.org/books/prealgebra/pages/5-2-decimal-operations
## Exercises
#### EXAMPLE 5.19
Divide: $0.12 \div 3$
#### TRY IT 5.37
Divide: $0.28 \div 4$.
#### TRY IT 5.38
Divide: $0.56 \div 7$.
#### EXAMPLE 5.20
Divide: $\ 3.99 \div 24$.
#### TRY IT 5.39
Divide: $\ 6.99 \div 36$
#### TRY IT 5.40
Divide: $\ 4.99 \div 12$
#### EXAMPLE 5.21
Divide: $-2.89 \div(3.4)$
#### TRY IT 5.41
Divide: $-1.989 \div 5.1$
#### TRY IT 5.42
Divide: $-2.04 \div 5.1$
#### EXAMPLE 5.22
Divide: $-25.65 \div(-0.06)$.
#### TRY IT 5.43
Divide: $-23.492 \div(-0.04)$
#### TRY IT 5.44
Divide: $-4.11 \div(-0.12)$
#### EXAMPLE 5.23
Divide: $4 \div 0.05$.
#### TRY IT 5.45
Divide: $6 \div 0.03$
#### TRY IT 5.46
Divide: $7 \div 0.02$
#### EXAMPLE 5.19
##### Solution
$0.12 \div 3$ Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. Divide as usual. Since 3 does not go into 0 or 1 we use zeros as placeholders. $0.12 \div 3 = 0.04$
#### TRY IT 5.37
$0.07$
#### TRY IT 5.38
$0.08$
#### EXAMPLE 5.20
##### Solution
$\ 3.99 \div 24$ Place the decimal point in the quotient above the decimal point in the dividend. Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place. Round to the nearest cent. $\ 0.166 \approx \ 0.17$ $\ 3.99 \div 24 \approx \ 0.17$
This means the price per bottle is $17$ cents.
#### TRY IT 5.39
$\ 0.19$
#### TRY IT 5.40
$\ 0.42$
#### EXAMPLE 5.21
##### Solution
Determine the sign of the quotient. The quotient will be negative. Make the divisor the whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places to the right. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero. Write the quotient with the appropriate sign. $-2.89 \div(3.4)=-0.85$
#### TRY IT 5.41
$-0.39$
#### TRY IT 5.42
$-0.4$
#### EXAMPLE 5.22
##### Solution
$-25.65 \div(-0.06)$ The signs are the same. The quotient is positive. Make the divisor a whole number by 'moving' the decimal point all the way to the right.'Move' the decimal point in the dividend the same number of places. Divide.Place the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. $-25.65 \div(-0.06)=427.5$
#### TRY IT 5.43
$587.3$
#### TRY IT 5.44
$34.25$
#### Example 5.23
##### Solution
$4 \div 0.05$ The signs are the same. The quotient is positive. Make the divisor a whole number by 'moving' the decimal point all the way to the right. Move the decimal point in the dividend the same number of places, adding zeros as needed. Divide. Place the decimal point in the quotient above the decimal point in the dividend. Write the quotient with the appropriate sign. $4 \div 0.05=80$
We can relate this example to money. How many nickels are there in four dollars? Because $4 \div 0.05=80$, there are 80 nickels in $\ 4$.
#### TRY IT 5.45
$200$
#### TRY IT 5.46
$350$ |
Integral calculus
The indefinite integral
The indefinite integrals containing quadratic polynomial (trinomial)
Integration by parts rule
The rule for differentiating the product of two differentiable functions leads to the integration by parts formula.
Let f (x) and g (x) are differentiable functions, then the product rule gives
[ f (x) g (x)]' = f (x) g (x)' + g (x) f ' (x),
by integrating both sides
Since the integral of the derivative of a function is the function itself, then
and by rearranging obtained is
the integration by parts formula.
By substituting u = f (x) and v = g (x) then, du = f ' (x) dx and dv = g' (x) dx, so that
To apply the above formula, the integrand of a given integral should represent the product of one function and the differential of the other.
The selection of the function u and the differential dv should simplify the evaluation of the remaining integral.
In some cases it will be necessary to apply the integration by parts repeatedly to obtain a simpler integral.
57. Evaluate
Use similar methods to solve this integral as in the preceding example. Hence, the solutions depend on the sign of the leading coefficient a and the sign or the value of the vertical translation y0.
Example: 57 a ) Evaluate
Let solve given integral by separating the derivative of the quadratic polynomial in the numerator.
Solution:
58. Evaluate
Use the substitution mx + n = 1/ t .
Example: 58 a ) Evaluate
Solution:
59. Evaluate
Solution:
The integrand function to be real, the quadratic polynomial must be positive, therefore
a ) if a > 0 and y0 is not then
see the solutions of the integrals, example 31 and 36. By using above substitutions
Note that the sign of the vertical translation y0 affects the solution, i.e., changes the sign of its second term.
b ) if a < 0 and y0 > then
see the solution of the integral example 8 above. After applying the substitutions
Functions contents G |
Lecture 1: Systems of Linear Equations
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1 MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables ½ ax + by = A cx + dy = B. Recall that a pair of two numbers (x 0, y 0 ) is said to be a solution if x = x 0 and y = y 0 satisfy both equations simultaneously. Graphically, we know that the graph of any linear equation of two variables is a straight line in xy plane. Therefore, a solution (x 0, y 0 ) is the intersection of two lines represented by two linear equations. We know that there are three possibilities (1) two line intersect at exactly one point (the system admits a unique solution), () two lines are identical (there are in nite many solutions), and () two lines are parallel but di erent (there is no solution.) Example 1.1. Consider a system of two equations with two unknowns ½ x y = 1 x + y =. (1) There are two commonly used methods to solve linear systems elimination method and substitution method. We now use the elimination method by adding both equation to obtain y =. Substitute into the rst equation of system (1), we nd x = y 1 =. The system has exact one solution (, ). 1
2 Thick line x + y = Example 1.. Consider another system ½ x y = 1 x + y =. () Adding both equations leads to 0 =. This contradictory equation indicates that system () cannot possibly have a solution. Thick line x + y = Example 1.. Consider the third system ½ x y = 1 x + y = 1. ()
3 The second equation is a multiple of the rst equation (by -1). Therefore, this system has only one independent equation. Consequently, there are in nitely many solutions x = 1 + t y = t for any choice of t. These three examples demonstrated three possible situations for systems of linear equations in D one unique solution (good systems), or no solution (the system contradicts to itself), or in nite many solutions (two equation are equivalent, called degenerated systems). This observation and the elimination method (one of the most popular method to solve systems in D) extends to general situations. ² General Linear Systems. A system of m linear equations with n variables, denoted by x 1, x,..., x n, reads as > a 11 x 1 + a 1 x a 1n x n = b 1 a 1 x 1 + a x a n x n = b ()... > a m1 x 1 + a m x a mn x n = b m a ij is called a coe cient. It is in the ith equation and is associated with x j. A solution of system () consists of n ordered numbers (x 1, x,..., x n ) satisfying all m equations. The set of all possible solutions is called a solution set. General systems can be solved using the method of elimination following the same idea for D systems. Example 1.. Solve the following system of equations with unknowns x 1 x + x = 0 x x = x 1 + x + 9x = 9. () Solution We use the same method of elimination. The last equation is replaced by the sum of itself and times the rst equation ( x 1 + x + 9x = 9) + (x 1 x + x = 0) 0 x + 1x = 9 while maintain the other equations. The system reduces to x 1 x + x = 0 x x =. (6) x + 1x = 9
4 Next, We add Equation # to # in (6) (replace # by the sun of Equation # and #) to arrive at x 1 x + x = 0 x x = x = 1 We now solve x from the last equation to get x Equation #, we nd x µ 1 = =) x = =.. = 1. Substitute it into Finally, substituting x = 1, x = into the rst equation, we obtain 1 µ µ x = 0 =) x 1 = The solution is (x 1, x, x ) = µ 6 1,, 1. ² Argumented Matrix This whole process can be simpli ed using symbolic means. To this end, we introduce the coe cient matrix for system () as an array (or table) of m rows and n columns a 11 a 1... a 1n 6 a 1 a... a n () a m1 a m... a mn m n We call it a m n matrix. m n is referred as the dimension of the matrix. However, this matrix contains only the left-hand side of system (). The entire information of system () can be found in the following table of m rows by (n + 1) columns 6 a 11 a 1... a 1n j b 1 a 1 a... a n j b j... a m1 a m... a mn j b m. () It is called Augmented matrix. This is a m (n + 1) matrix. The vertical dash line is for the purpose of reminding us that the last column represents the righe-hand side of the system. A system of linear equations is represented by its Augmented matrix.
5 For instance, in Example above, the system x 1 x + x = 0 x x = x 1 + x + 9x = 9 has the augmented matrix 1 1 j 0 0 j 9 j 9 ² Elementary Row Operations The elimination method basically consists of the following row operations for the system 1. Replace one equation by the sum of itself and a multiple of another equation. For instance, in solving Example above, the rst stem is to replaced the third equation by the sum of itself and times the rst equation The resulting system is X ( x 1 + x + 9x = 9) + (x 1 x + x = 0) 0 x + 1x = 9. x 1 x + x = 0 x x = x + 1x = 9.. (9) However, if we perform the same row operation on the augmented matrix replace the third row in its augmented matrix by the sum of itself and times the rst row, i.e., the updated third row is 9 j 9 then we obtain a new matrix j 0 = 0 1 j 9, 1 1 j 0 0 j 0 1 j 9 which is exactly the augmented matrix for (9). Conclusion the row operation for the system is equivalent to the same type of row operation for the augmented matrix. Some times we also need two other row operations X,
6 . Interchange two equations (equivalent to exchanging two rows). One equation is replaced by a non-zero multiple of itself (equivalent to replacinging one rows by a non-zero multiple of itself). De nition 1.1. The following operations are called elementary row operations 1. Replace one row by the sum of itself and a multiple of another row.. Interchange two rows. One row is replaced by a non-zero multiple of itself. De nition 1.. Two matrices are called row equivalent if one is obtained from the other by a series of elementary row operations. Two linear systems are called equivalent if their augmented matrices are row equivalent. Theorem 1.1 If the augmented matrices of two linear systems are row equivalent, then the solution sets of these two linear systems are identical. In short, elementary row reductions do not alter the solution set. Example 1.. Solve system () using row operations. Solution Augmented matrix 0 j 1 1 j 0 9 j 9 We now perform a series of row operation in the way equivalent to what we did before 1 1 j 0 0 j 9 j 9 R 1 + R! R! 1 1 j 0 0 j 0 1 j 9 R + R! R! 1 1 j 0 0 j. 0 0 j 1 Notice that here, in the symbol the left-hand, R 1 + R! R! R 1 + R 6
7 stands for the sum of row # and times row #1. The right-hand side, R, means row # is updated by the result of row operation. The corresponding system is x 1 x + x = 0 x x = x = 1 which is exactly the same as before. We can continue row operations 1 1 j j 0 0 j 6 R /! R 0 j 0 0 j 1! j j j 0 R + R! R! j j 1 R /! R j! j j j 1 R + R 1! R 1! j j 1 ( 1) R R 1! R 1! j j j 1 = j 1 (Reduced Echelon Form) =) x 1 1 x = j 1 x 1 1. Example 1.6. Solve x x = x 1 x + x = 1 x 1 x + x = 1 (10) Solution Write down the Matrix form and perform row operations 0 1 j j 1 R! R 1 j j! j 1 j 1 R 1 /! R 1! 1 / 1 j 1/ 0 1 j j 1 ( 1/) R + R! R! ( ) R 1 + R! R! 1 / 1 j 1/ 0 1 j j /. 1 / 1 j 1/ 0 1 j 0 1/ j / 1/
8 We convert back to the system > x 1 x + x = 1 x x = > 0 = =) impossible, means no solution. In general, one can always solve a system of linear equations through a series of elementary row reductions till the augmented matrix reduces to Upper Triangle form, and then solve unknowns from bottom up. A general process will be summarized in the next section. ² Homework 1 1. Solve the following linear systems directly by the method of elimination (a) (b) x 1 x + x = x 1 x + 9x = x 1 + 6x x = x 1 x + x = x 1 x + 9x = x 1 9x + 1x =. For the following linear systems, nd their coe cient matrices and augmented matrices. Then solve the system by performing elementary row operations. (a) (b) x 1 x + x + x = x 1 x + 9x x = x 1 + 6x x + x = x 1 x + x = x 1 x + 9x = x 1 9x + 1x = 0. For each of the following statements, determine whether it is true or false. If your answer is true, state your rationale. If false, provide an counterexample (the example contradicting the statement). (a) Every elementary row operation is reversible. (b) Elementary row operations on an augmented matrix never change the solution set of the associated linear system. (c) Two matrices are row equivalent if they have the same number of rows and columns. (d) Two linear systems are equivalent if they have the same solution set.
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# Heidi Montag Under The Microscope (11/08/2019)
How will Heidi Montag get by on 11/08/2019 and the days ahead? Let’s use astrology to undertake a simple analysis. Note this is for entertainment purposes only – take it with a grain of salt. I will first work out the destiny number for Heidi Montag, and then something similar to the life path number, which we will calculate for today (11/08/2019). By comparing the difference of these two numbers, we may have an indication of how good their day will go, at least according to some astrology practitioners.
PATH NUMBER FOR 11/08/2019: We will consider the month (11), the day (08) and the year (2019), turn each of these 3 numbers into 1 number, and add them together. How? Let’s walk through it. First, for the month, we take the current month of 11 and add the digits together: 1 + 1 = 2 (super simple). Then do the day: from 08 we do 0 + 8 = 8. Now finally, the year of 2019: 2 + 0 + 1 + 9 = 12. Now we have our three numbers, which we can add together: 2 + 8 + 12 = 22. This still isn’t a single-digit number, so we will add its digits together again: 2 + 2 = 4. Now we have a single-digit number: 4 is the path number for 11/08/2019.
DESTINY NUMBER FOR Heidi Montag: The destiny number will consider the sum of all the letters in a name. Each letter is assigned a number per the below chart:
So for Heidi Montag we have the letters H (8), e (5), i (9), d (4), i (9), M (4), o (6), n (5), t (2), a (1) and g (7). Adding all of that up (yes, this can get tedious) gives 60. This still isn’t a single-digit number, so we will add its digits together again: 6 + 0 = 6. Now we have a single-digit number: 6 is the destiny number for Heidi Montag.
CONCLUSION: The difference between the path number for today (4) and destiny number for Heidi Montag (6) is 2. That is smaller than the average difference between path numbers and destiny numbers (2.667), indicating that THIS IS A GOOD RESULT. But don’t get too excited yet! As mentioned earlier, this is for entertainment purposes only. If you want a forecast that we do recommend taking seriously, check out your cosmic energy profile here. Go ahead and see what it says for you – you’ll be glad you did.
### Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene.
#### Latest posts by Abigale Lormen (see all)
Abigale Lormen
Abigale is a Masters in Business Administration by education. After completing her post-graduation, Abigale jumped the journalism bandwagon as a freelance journalist. Soon after that she landed a job of reporter and has been climbing the news industry ladder ever since to reach the post of editor at Tallahasseescene. |
# Does the Bodmas rule apply to fractions?
## Does the Bodmas rule apply to fractions?
The BODMAS rule states that division, multiplication, addition, and subtraction to be performed in the order brackets. We also learnt to use the BODMAS rule to simplify fractions.
How do you solve fractions with Bodmas?
Before applying the BODMAS rule to this question, you will have to convert all the mixed fractions to proper fractions. You can do these by multiplying the whole number with first the denominator and second add the result with the numerator and finally dividing your total result with the denominator.
### How do you do exponents with fractions?
To solve fractions with exponents, review the rules of exponents. You’ll distribute the exponent to the full fraction if indicated. Then, you’ll multiply the full fraction, the base, by itself the number of times directed by the exponent. Both exponents and fractions are important algebraic concepts.
Is Bodmas always correct?
Originally Answered: Should we follow BODMAS or PEMDAS in mathematical calculations? It doesn’t matter. PEMDAS and BODMAS both count division and multiplication as a single step. The ultimate rule is to go from left to right, so either way the answer is 9.
#### Is Bodmas and Bidmas the same?
The BODMAS acronym stands for brackets, orders, division, multiplication, addition, subtraction. It is sometimes known as BIDMAS (with ‘Indices’ used instead of ‘Orders’) or the PEMDAS rule in America (with ‘Parenthesis’ and ‘Exponents’).
How do you calculate Bodmas without brackets?
If there are no brackets, start solving from ‘order’ or ‘of’ followed by Division or multiplication (whatever comes first from left to right) then by addition or subtraction (whatever comes first from left to right).
## Can an exponent be a fraction?
If an exponent of a number is a fraction, it is called a fractional exponent. Exponents show the number of times a number is replicated in multiplication. For example, 42 = 4×4 = 16.
What is the Bedmas method?
BEDMAS is an acronym to help remember an order of operations in algebra basics. When you have math problems that require the use of different operations (multiplication, division, exponents, brackets, subtraction, addition) order is necessary and mathematicians have agreed on the BEDMAS/PEMDAS order. |
# 1.01 Identify odd and even numbers
Lesson
## Ideas
We can use what we know about patterns with numbers to help us determine if a number is even or odd.
### Examples
#### Example 1
Write the next number in the pattern:
2, \,4, \,6, \,8, \,10, \,12, \,⬚
Worked Solution
Create a strategy
Count how much is added each time and add it to the last given number.
Apply the idea
We can see that 2+2=4 and 4+2=6, and so on. This means we need to add 2 to 12, that is 12+2=14.
2, \,4, \,6, \,8, \,10, \,12, \,14
Idea summary
To find the next number with an increasing pattern, we need to find how much is added each time and add it to last given number.
## Odd and even numbers
What are even and odd numbers? How can we determine if a number is even or odd? Let's find out.
### Examples
#### Example 2
Carl has 13 blocks and wants to divide them into two equal groups that are as big as possible. When doing this, Carl doesn't break any of the blocks into pieces.
a
How many blocks will there be in each group?
Worked Solution
Create a strategy
Share the blocks into two groups, placing one block into the groups one at a time.
Apply the idea
We can start putting one block into each group as shown.
We can see that there are 6 blocks in each group with one block left over.
b
Is 13 an even number?
Worked Solution
Create a strategy
If a number is even, then we can equally share it into two equal groups without anything left over.
Apply the idea
We found from part (b) that when we made two equal groups, there was 1 block left over.
So, 13 is not an even number.
Idea summary
• All even numbers end in 2,4,6,8 or 0.
• All odd numbers end in 1,3,5,7 or 9.
Even numbers can be split into two equal groups with no remaining numbers left over.
### Outcomes
#### MA2-8NA
generalises properties of odd and even numbers, generates number patterns, and completes simple number sentences by calculating missing values |
# How to find the length of an arc
how to find the length of an arc
How to Find the Length of an Arc
To find the length of an arc, you can use the formula derived from the arc length formula in terms of radians. Here’s how you can do it:
1. Understand the Concepts:
Before diving into calculations, it’s crucial to understand the basic concepts. An arc is a portion of the circumference of a circle, while the circumference is the complete boundary of a circle. The length of an arc is a fraction of the circumference, determined by the central angle formed by the arc.
2. Measure the Angle:
Begin by measuring the central angle formed by the arc. The angle is measured in degrees or radians, depending on the context. Make sure you’re consistent with the units throughout the calculation.
3. Convert to Radians (if necessary):
Most arc length formulas use radians. If your angle is given in degrees, you may need to convert it to radians. Recall that 1 \text{ radian} = \frac{180}{\pi} \text{ degrees}.
4. Use the Arc Length Formula:
Once you have the angle in radians, you can use the arc length formula:
\text{Arc Length} = \theta \times r
Where:
• (\theta) is the central angle (in radians),
• (r) is the radius of the circle.
5. Perform the Calculation:
Simply multiply the central angle (in radians) by the radius of the circle to find the arc length.
Example:
Let’s say we have a circle with a radius of 10 units and a central angle of (\frac{\pi}{3}) radians. To find the length of the corresponding arc, we use the formula:
\text{Arc Length} = \frac{\pi}{3} \times 10 = \frac{10\pi}{3} \text{ units}
6. Round the Result (if necessary):
Depending on the context of the problem, you may need to round the arc length to a certain number of decimal places or significant figures. |
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Reasoning and Proof | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Geometry Concepts Go to the latest version.
# Chapter 2: Reasoning and Proof
Created by: CK-12
0 0 0
## Introduction
This chapter explains how to reason and how to use reasoning to prove theorems about angle pairs and segments. This chapter also introduces the properties of congruence, which will also be used in proofs. Subsequent chapters will combine what you have learned in Chapters 1 and 2 and build upon them.
## Summary
This chapter teaches students how to make conjectures and provide counterexamples. From there, it focuses on rewriting statements in if-then form and finding converses, inverses, and contrapositives. Two types of reasoning, inductive and deductive, are explored. Finally, the properties of equality and congruence are reviewed and practice for completing two-column proofs is provided.
### Symbol Toolbox for Chapter
$\rightarrow$ if-then
$\land$ and
$\therefore$ therefore
$\sim$ not
$\lor$ or
### Chapter Keywords
• Inductive Reasoning
• Conjecture
• Counterexample
• Conditional Statement (If-Then Statement)
• Hypothesis
• Conclusion
• Converse
• Inverse
• Contrapositive
• Biconditional Statement
• Logic
• Deductive Reasoning
• Law of Detachment
• Law of Contrapositive
• Law of Syllogism
• Right Angle Theorem
• Same Angle Supplements Theorem
• Same Angle Complements Theorem
• Reflexive Property of Equality
• Symmetric Property of Equality
• Transitive Property of Equality
• Substitution Property of Equality
• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality
• Distributive Property
• Reflexive Property
• of Congruence
• Symmetric Property
• of Congruence
• Transitive Property of Congruence
### Chapter Review
Match the definition or description with the correct word.
1. $5 = x$ and $y + 4 = x$, then $5 = y +4$ — A. Law of Contrapositive
2. An educated guess — B. Inductive Reasoning
3. $6(2a + 1) = 12a +12$ — C. Inverse
4. 2, 4, 8, 16, 32,... — D. Transitive Property of Equality
5. $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{AB}$ — E. Counterexample
6. $\sim p \rightarrow \sim q$ — F. Conjecture
7. Conclusions drawn from facts. — G. Deductive Reasoning
8. If I study, I will get an “$A$” on the test. I did not get an $A$. Therefore, I didn’t study. — H. Distributive Property
9. $\angle A$ and $\angle B$ are right angles, therefore $\angle A \cong \angle B$. — I. Symmetric Property of Congruence
10. 2 disproves the statement: “All prime numbers are odd.” — J. Right Angle Theorem — K. Definition of Right Angles
### Texas Instruments Resources
In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9687.
Jul 17, 2012 |
# What Is 69 percent of 35 + Solution with Free Steps?
The 69 percent of 35 is equal to 24.15. This solution is estimated by multiplication of 0.35 by 69.Â
The calculation of 69 percent of 35 may be used in many examples. For example lets say that you want to purchase apples at the rate of 35 dollars per dozen. Now you find out that Walmart is offering a discount of 69%. Since you know that 69% of 35 is equal to 24.15 dollars, you can easily know that you can purchase the same amount of apples for a discounted price of 24.15 dollars.
The main objective of this question is to find the value which is 69 percent of 35.
## What is 69 percent of 35?
The 69 percent of 35 is 24.15. This answer is calculated by multiplication of 0.35 by 69
The percentage is the part of 100 that can be written as either a number or ratio. It doesn’t have a unit and the only symbol for it is “%.”. We can rearrange the percentage formula in order to find the percent value which is 69 percent of 35. Thus:
a = ( Percentage x b ) x 100
Where a is the actual value and b is the total value.
## How to calculate 69% of 50?
Following are the steps used to represent that which value is 69 percent of 35.
## Step 1
We are given that:
The total value is 35.
The percentage is 69.
While we have to find the actual value.
## Step 2
Let represent the actual value by “a” which is an unknown.
## Step 3
We know that the percentage is calculated as:
Percentage = ( a / b ) x 100
Where a is the part of the whole number while b represents the whole number.
Rearranging the percentage formula according to our requirement results in:
a = ( b x Percentage ) / 100
By putting the values, we get”
a = ( 35 x 69 ) / 100
a = ( 35 x 69 ) / 100
a = 2415 / 100
Thus dividing by 100 results in:
a = 24.15.
Hence, 69 % of 35 is 24.15
## Step 4
To visualize what 69% of 35 looks like, we may look at the corresponding Pie Chart presented above. If we split the whole value into 100 equal parts, then 69% of 35 is 24.15, which is in the blue area, and 31% of 100 is in the yellow area.
A percentage is a number and maybe even a ratio written as just a fraction of 100. The word “percentage” comes from the Latin word “per centum,” which means “by a hundred.” A symbol represents the percentage as ” % “.
With the percentage formula, you can figure out how much of a whole something is in terms of 100. You can show a number as just a fraction of 100 by using the concept of percentage. Mathematically the percentage can be calculated as:
Percentage = ( a / b ) x 100
Where a is the part of the whole number while b represents the whole number.
We know that one percent, which is written as 1″%“, is a hundredth of the whole, so 100% is whole while 200% is twice the given amount.
All Images/Mathematical drawings are created with Geogebra. |
Work Calculus – Definition, Definite Integral, and Applications
Work, in physics, indicates the amount of force acting through a given distance. In this article, we’ll highlight the mathematical definition of work and learn how we can define work in terms of definite integrals. We’ll also get to see how essential integral calculus is in physics and engineering in our discussion.
Work in calculus is simply equal to the area under the curve of $\boldsymbol{F(x)}$, where $\boldsymbol{F(x)}$ is simply the function representing the force acted upon from $\boldsymbol{x =a}$ to $\boldsymbol{x = b}$.
We’ll show you how to calculate for an object’s work by evaluating definite integrals, so keep your notes on integral properties and antiderivative formulas handy. For now, let’s do a quick recap of what work is and learn how definite integrals come into place!
What is work in calculus?
Work in calculus reflects the area under the curve of $F(x)$. In our introductory physics class, work is simply the product of the force acting upon the object over a given distance (or displacement).
\begin{aligned}W &= Fd\end{aligned}
Keep in mind that in SI metric system, force is measured in Newtons ($N$), the distance is in meters ($m$), and work is in terms of $N \cdot m$ or in joules ($J$). For example, if a person exerts a force of $10 \phantom{x}N$ to lift an object $40 \phantom{x}m$ off the floor, the amount of work done is as shown below.
\begin{aligned}W &= F\cdot d\\&= (10 \phantom{x}N) \cdot (40 \phantom{x}m)\\&= 400 \phantom{x}N\cdot m\\&= 400 \phantom{x}J\end{aligned}
Now, let’s extend the definition of work by considering the fact that the force applied can be variable.
Work definition in calculus
In the earlier discussion, we’ve shown how work can be calculated when given a constant value for the force exerted. However, there are instances, when force is not constant. Suppose that we’re observing an object that is moving along the positive horizontal direction from $x=a$ to $x =b$.
Let’s say $F(x)$ is the function representing the amount of force exerted over the interval, $[a, b]$. If $F(x)$ is continuous throughout the interval, we can approximate the amount of work done by partitioning the interval into subintervals with the following endpoints: $\{x_0, x_1, x_2, …, x_n\}$ and a uniform width of $\Delta x$. Hence, we can estimate the total amount of work as shown below:
\begin{aligned}W &\approx \sum_{i =1}^{n} f(c_i)\Delta x_i \end{aligned}
To find a better approximation, as we have learned before, can be acquired by evaluating the quantity as $n\rightarrow \infty$.
\begin{aligned}W &=\lim_{n\rightarrow\infty} \sum_{i =1}^{n} f(c_i)\Delta x_i\\ &= \int_{a}^{b} F(x)\phantom{x}dx \end{aligned}
This means that work exerted on a moving object is simply the area under the curve of the function for force, $F(x)$ from $x=a$ to $x =b$.
How to solve work problems in calculus?
Now that we understand how we can define work in terms of the definite integral of the force function, $F(x)$, let’s break down the steps to solve problems involving work:
• Identify the function that represents the force, $F(x)$, exerted from $x=a$ to $x= b$.
• Set up the expression for work in terms of $F(x)$: $W = \int_{a}^{b} F(x)\phantom{x}dx$.
• Evaluate the definite integral by applying key integral properties and formulas.
For example, if we want to calculate the amount of work done given $F(x) = 2x + 1$ Newtons over the interval, $[2, 8]$. Hence, we can calculate the amount of work done as shown below.
\begin{aligned}W &= \int_{2}^{8} (2x +1)\phantom{x}dx\\&= \int_{2}^{8}2x \phantom{x}dx + \int_{2}^{8}1 \phantom{x}dx\\&= 2\left[\dfrac{x^2}{2}\right]_{2}^{8} – [x]_{2}^{8}\\&= [(8^2 -2^2) – (8 -2)]\\&= 54 \end{aligned}
This means that the work done to move the object is equal to $54$ Newton-meters.
Example 1
A particle is moving along the $x$-axis and located $x$ feet from the origin. If a force of $x^2 + x + 4$ pounds is being acted upon the article, calculate the total amount of work done to move the particle from $x = 2$ to $x = 6$.
Solution
We can see that the force acting upon the particle is equal to $F(x) = x^2 + x +4$. Hence, use this expression to calculate the amount of work done to move the particle over the interval, $[2, 6]$.
\begin{aligned} W&= \int_{2}^{6} (x^2 + x + 4) \phantom{x}dx\end{aligned}
Evaluate the definite integral to find $W$.
\begin{aligned} W&= \int_{2}^{6} x^2\phantom{x}dx +\int_{2}^{6} x\phantom{x}dx +\int_{2}^{6} 4 \phantom{x}dx\\ &= \left[\dfrac{x^3}{3} \right ]_{2}^{6} + \left[\dfrac{x^2}{2} \right ]_{2}^{6} + [4x]_{2}^{6}\\&= \left[\left(\dfrac{6^3}{3} – \dfrac{2^3}{3} \right ) +\left(\dfrac{6^2}{2} – \dfrac{2^2}{2} \right )+ (4\cdot 6 – 4\cdot 2) \right ]\\&= \dfrac{208}{3} + 16 + 16\\&= \dfrac{304}{3}\end{aligned}
This means that the total amount of work done upon the particle is $\dfrac{304}{3}\phantom{x}\text{ft-lb}$.
Example 2
We need a force of $30 \phantom{x}\text{N}$ to hold a spring that has been stretched from its natural length of $0.20 \phantom{x}\text{m}$ to a length of $0.45 \phantom{x}\text{m}$. Calculate the amount of work done if we stretch the spring from $0.25 \phantom{x}\text{m}$ to $0.48 \phantom{x}\text{m}$.
Hint: Recall that Hooke’s Law states that the force needed to maintain the stretch of $x$ units is $F(x) = kx$, where $k$ is the spring constant.
Solution
Calculate the sprint constant by using the given values: $F(x) = 30 \phantom{x}\text{N}$ and $x = 0.45 -0.20 = 0.25 \phantom{x}\text{m}$.
\begin{aligned} k&= \dfrac{F(x)}{x}\\&= \dfrac{30}{0.25}\\&= 120\end{aligned}
Using $k = 120$, we have $F(x) = 120x$. Let’s use this expression for the force needed to maintain a stretch from $0.25 \phantom{x}\text{m}$ to $0.48 \phantom{x}\text{m}$. Hence, we have the following expression for the amount of work needed:
\begin{aligned}W &= \int_{0.25}^{0.48} 120x \phantom{x}dx \end{aligned}
Evaluate the definite integral to find the exact value of work in $\text{N}\cdot \text{m}$.
\begin{aligned}W &= 120\int_{0.25}^{0.48} x \phantom{x}dx\\&= 120\left[\dfrac{x^2}{2} \right ]_{0.25}^{0.48}\\&= 60\left[x^2 \right ]_{0.25}^{0.48}\\&= 60[(0.48)^2 – (0.25)^2]\\&= 10.074 \phantom{x}\text{N}\cdot \text{m}\\&= 10.074 \phantom{x}\text{J} \end{aligned}
This means that the total amount of work done is $10.074 \phantom{x}\text{N}\cdot \text{m}$ or $10.074 \phantom{x}\text{J}$.
Practice Questions
1. A particle is moving along the $x$-axis and located $x$ feet from the origin. If a force of $2x^2 + 4x$ pounds is being acted upon the article, calculate the total amount of work done to move the particle from $x = 1$ to $x = 5$.
2. We need a force of $40 \phantom{x}\text{N}$ to hold a spring that has been stretched from its natural length of $0.10 \phantom{x}\text{m}$ to a length of $0.35 \phantom{x}\text{m}$. Calculate the amount of work done if we stretch the spring from $0.15 \phantom{x}\text{m}$ to $0.60 \phantom{x}\text{m}$.
1. $\dfrac{392}{3} \phantom{x}\text{ft-lb}$
2. $27 \phantom{x}\text{N}\cdot \text{m}$ or $27 \phantom{x}\text{J}$ |
# Negative Exponents
## Any value to the zero power equals 1, convert negative exponents
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Negative Exponents
All the students in a class were randomly given an expression, and they were asked to make pairs, with one boy and one girl per pair. The students were asked to divide the boy's expression by the girl's expression. Bill and Jenna paired up, with Bill having \begin{align*}10x^4\end{align*} and Jenna having \begin{align*}5x^5\end{align*}. Also, Tim and Meg paired up, with Tim having \begin{align*}7y^3\end{align*}and Meg having \begin{align*}14y^3\end{align*}. What is Bill and Jenna's quotient? How about Tim and Meg's quotient?
### Zero Power and Negative Exponents
Previously, we have dealt with powers that are positive whole numbers. Now, you will learn how to solve expressions when the exponent is zero or a negative number.
For all real numbers \begin{align*}\chi, \chi \neq 0, \chi^0=1\end{align*}. This means that any number raised to the 0th power is 1.
#### Let's simplify \begin{align*}\frac{\chi^4}{\chi^4}\end{align*}:
\begin{align*}\frac{\chi^4}{\chi^4} = \chi^{4-4} = \chi^0 = 1\end{align*}. This example is simplified using the Quotient of Powers Property.
#### Simplifying Expressions with Negative Exponents
The next objective is negative exponents. When we use the quotient rule and we subtract a greater number from a smaller number, the answer will become negative. The variable and the power will be moved to the denominator of a fraction. You will learn how to write this in an expression.
#### Let's simplify \begin{align*}\frac{x^4}{x^6}\end{align*}:
\begin{align*}\frac{x^4}{x^6} =x^{4-6}=x^{-2}=\frac{1}{x^2}\end{align*}. Another way to look at this is \begin{align*}\frac{\chi \cdot \chi \cdot \chi \cdot \chi}{\chi \cdot \chi \cdot \chi \cdot \chi \cdot \chi \cdot \chi}\end{align*}. The four \begin{align*}\chi\end{align*}s on top will cancel out with four \begin{align*}\chi\end{align*}s on the bottom. This will leave two \begin{align*}\chi\end{align*}s remaining on the bottom, which makes your answer look like \begin{align*}\frac{1}{\chi^2}\end{align*}.
If you want to simplify this expression any further, you need to use the Negative Power Rule for Exponents.
The Negative Power Rule for Exponents states that \begin{align*}\frac{1}{\chi^n} = \chi^{-n}\end{align*} where \begin{align*}\chi \neq 0\end{align*}.
#### Now, let's rewrite the expression \begin{align*}\chi^{-6} \gamma^{-2}\end{align*} using only positive exponents:
\begin{align*}\chi^{-6} \gamma^{-2}= \frac{1}{\chi^6} \cdot \frac{1}{\gamma^2} = \frac{1}{\chi^6 \gamma^2}\end{align*}. The negative power rule for exponents is applied to both variables separately in this example.
#### Finally, let's write the following expressions without fractions:
1. \begin{align*}\frac{2}{x^2}\end{align*}
\begin{align*}\frac{2}{x^2}=2x^{-2}\end{align*}
1. \begin{align*}\frac{x^2}{y^3}\end{align*}
\begin{align*}\frac{x^2}{y^3}=x^2y^{-3}\end{align*}
Notice that in question 1, the number 2 is in the numerator. This number is multiplied with \begin{align*}x^{-2}\end{align*}. It could also look like \begin{align*}2 \cdot \frac{1}{x^2}\end{align*} to be better understood.
### Examples
#### Example 1
Earlier, you were told about a class where students had to make pairs based on the expressions that they were randomly given. The pairs had to be one girl and one boy and they were required to divide the boy's expression by the girl's expression. In the pair of Bill and Jenna, Bill is given \begin{align*}10x^4\end{align*} and Jenna is given \begin{align*}5x^5\end{align*}. In the pair of Tim and Meg, Tim has \begin{align*}7y^3\end{align*} and Meg has \begin{align*}14y^3\end{align*}. What is Bill and Jenna's quotient? What is Tim and Meg's quotient?
Bill and Jenna's quotient is:
\begin{align*}\frac{10x^4}{5x^5}&= 2x^{4-5}\\ &=2x^{-1}\\ &=\frac{2}{x}\end{align*}
Tim and Meg's quotient is:
\begin{align*}\frac{7y^3}{14y^3}&=\frac{1}{2}\cdot y^{3-3}\\ &=\frac{1}{2}\cdot y^0\\ &= \frac{1}{2}\end{align*}
#### Example 2
Simplify \begin{align*}\left( \frac{x^2y^{-3}}{x^5y^2}\right)^{2}\end{align*}, giving the answer with only positive exponents.
\begin{align*}\left( \frac{x^2 y^{-3}}{x^5 y^2} \right)^2 = \left(x^2 x^{-5} y^{-3} y^{-2}\right)^2=\left(x^{2-5} y^{-3-2}\right)^2=(x^{-3} y^{-5})^2= (x^{-3})^2 (y^{-5})^2 &= x^{(-3)(2)} y^{(-5)(2)} = x^{-6} y^{-10}=\frac{1}{x^6y^{10}} \end{align*}
### Review
Simplify the following expressions. Be sure the final answer includes only positive exponents.
1. \begin{align*}x^{-1} \cdot y^2\end{align*}
2. \begin{align*}x^{-4}\end{align*}
3. \begin{align*}\frac{x^{-3}}{x^{-7}}\end{align*}
4. \begin{align*}\frac{1}{x}\end{align*}
5. \begin{align*}\frac{2}{x^2}\end{align*}
6. \begin{align*}\frac{x^2}{y^3}\end{align*}
7. \begin{align*}\frac{3}{xy}\end{align*}
8. \begin{align*}3x^{-3}\end{align*}
9. \begin{align*}a^2b^{-3}c^{-1}\end{align*}
10. \begin{align*}4x^{-1}y^3\end{align*}
11. \begin{align*}\frac{2x^{-2}}{y^{-3}}\end{align*}
12. \begin{align*}\left(\frac{a}{b}\right)^{-2}\end{align*}
13. \begin{align*}(3a^{-2}b^2c^3)^3\end{align*}
14. \begin{align*}x^{-3} \cdot x^3\end{align*}
Simplify the following expressions without any fractions in the answer.
1. \begin{align*}\frac{a^{-3}(a^5)}{a^{-6}}\end{align*}
2. \begin{align*}\frac{5x^6y^2}{x^8y}\end{align*}
3. \begin{align*}\frac{(4ab^6)^3}{(ab)^5}\end{align*}
Evaluate the following expressions to a single number.
1. \begin{align*}3^{-2}\end{align*}
2. \begin{align*}(6.2)^0\end{align*}
3. \begin{align*}8^{-4} \cdot 8^6\end{align*}
In 21 – 23, evaluate the expression for \begin{align*}x=2, y=-1, \text{and } z=3\end{align*}.
1. \begin{align*}2x^2-3y^3+4z\end{align*}
2. \begin{align*}(x^2-y^2)^2\end{align*}
3. \begin{align*}\left(\frac{3x^2y^5}{4z}\right)^{-2}\end{align*}
4. Evaluate \begin{align*}x^24x^3y^44y^2\end{align*} if \begin{align*}x=2\end{align*} and \begin{align*}y=-1\end{align*}.
5. Evaluate \begin{align*}a^4(b^2)^3+2ab\end{align*} if \begin{align*}a=-2\end{align*} and \begin{align*}b=1\end{align*}.
6. Evaluate \begin{align*}5x^2-2y^3+3z\end{align*} if \begin{align*}x=3, \ y=2,\end{align*} and \begin{align*}z=4\end{align*}.
7. Evaluate \begin{align*}\left(\frac{a^2}{b^3}\right)^{-2}\end{align*} if \begin{align*}a=5\end{align*} and \begin{align*}b=3\end{align*}.
8. Evaluate \begin{align*}3 \cdot 5^5 - 10 \cdot 5+1\end{align*}.
9. Evaluate \begin{align*}\frac{2 \cdot 4^2-3 \cdot 5^2}{3^2}\end{align*}.
10. Evaluate \begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}\end{align*}.
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English Spanish
TermDefinition
exponents of zero For all real numbers $\chi, \chi \neq 0, \chi^0=1$.
Negative Power Rule for Exponents $\frac{1}{\chi^n} = \chi^{-n}$ where $\chi \neq 0$.
Negative Exponent Property The negative exponent property states that $\frac{1}{a^m} = a^{-m}$ and $\frac{1}{a^{-m}} = a^m$ for $a \neq 0$.
quotient rule In calculus, the quotient rule states that if $f$ and $g$ are differentiable functions at $x$ and $g(x) \ne 0$, then $\frac {d}{dx}\left [ \frac{f(x)}{g(x)} \right ]= \frac {g(x) \frac {d}{dx}\left [{f(x)} \right ] - f(x) \frac{d}{dx} \left [{g(x)} \right ]}{\left [{g(x)} \right ]^2}$.
Zero Exponent Property The zero exponent property says that for all $a \neq 0$, $a^0 = 1$. |
# Problem Solving In Math For Grade 1
The site offers three different levels of difficulty for each strand.Factor Investigation challenges students to list all factors of the numbers 1-25 and identify the numbers as abundant, deficient, perfect, prime. Examples: If 8 3 = 11 is known, then 3 8 = 11 is also known.
The site offers three different levels of difficulty for each strand.Factor Investigation challenges students to list all factors of the numbers 1-25 and identify the numbers as abundant, deficient, perfect, prime. Examples: If 8 3 = 11 is known, then 3 8 = 11 is also known.
Here you will find a wide range of free printable addition Worksheets, which will help your child practice solving a range of addition problems using numbers with a sum of up to 20.
At First Grade, children love to play and explore Math with fun Math activities and games.
A.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
(Commutative property of addition.) To add 2 6 4, the second two numbers can be added to make a ten, so 2 6 4 = 2 10 = 12.
Children will enjoy completing these Math games and worksheets whilst learning at the same time.
During First Grade, your child's understanding of place value and numbers will grow, so that by the end of First Grade, most children should be able to count, read and write numbers up to 100. Students can refer to this list when playing Factor Blaster or Factor Game.The Million Dollar Mission asks students to decide which salary is the better offer for one month's work: one million dollars or one cent on the first day, two cents on the second day, four cents on the third day, etc.Spinner Game presents students with a frequency table of spins and requires students to create a bar graph of the results and draw a spinner that would yield those results. Students must justify their proposed spinner, explaining how it fits the data. Teachers should ask students to explain how they know what comes next to develop students' ability to explain their thinking.Shape Patterns: shape patterns that require students to draw what comes next.Unten findest du eine Liste mit allen Mathe-Kompetenzen für die 1. Die Kompetenzen sind nach Kategorien geordnet und du kannst dir für jede eine Beispielaufgabe ansehen.Halte dafür einfach den Mauszeiger über die jeweilige Kompetenz.Name That Number - 2 is designed to measure student understanding of place value as it is used in the Everyday Math Game of the same name.Open-ended Math Problems from the Franklin Institute Online offers monthly problems in Number Theory; Geometry; Measurement; Patterns, Algebra & Functions; Data, Statistics & Probability. |
## Solve Lesson 44 Page 61 Math 10 SBT – Kite>
Topic
One wants to design a rectangular flower bed inscribed in a circular plot of land with a diameter of 50 m (Figure 23). Determine the size of the rectangular flower garden so that the total distance traveled around the flower garden is 140 m.
Solution method – See details
Set the length of one side of the rectangle to $$x$$(m) ($$0 < x < 50$$).
Express the remaining side and perimeter of the rectangle in terms of x.
Detailed explanation
Set the length of one side of the rectangle to $$x$$(m) ($$0 < x < 50$$).
The length of the diagonal of the rectangle = Diameter of the circle = 50m.
The length of the remaining side of that rectangle is $$\sqrt {{{50}^2} – {x^2}} = \sqrt {2500 – {x^2}}$$ (m)
Then, the total distance traveled around the flower garden equal to the perimeter of the rectangle is: $$2\left( {\sqrt {2500 – {x^2}} + x} \right) = 140$$ (m)
We have the equation: $$2\left( {\sqrt {2500 – {x^2}} + x} \right) = 140 \Leftrightarrow \sqrt {2500 – {x^2}} + x = 70 \Rightarrow \sqrt {2500 – {x^2}} = 70 – x$$
$$\begin{array}{l} \Leftrightarrow \left\{ \begin{array}{l}x > 0\\70 – x \ge 0\\2500 – {x^2} = {\left( { 70 – x} \right)^2}\end{array} \right \Leftrightarrow \left\{ \begin{array}{l}0 < x \le 70\\2500 – {x^2} = {x ^2} – 140x + {70^2}\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}0 < x \le 70\\2{x^2} – 140x + 2400 = 0\end{array} \right \Leftrightarrow \left\{ \begin{array}{l}0 < x \le 70\\\left[\begin{array}{l}x=30\\x=40\;\end{array}\right\quad\end{array}\right\end{array}$$[\begin{array}{l}x=30\x=40\;\end{array}\right\quad\end{array}\right\end{array}\)
If $$x = 40$$ then the remaining side length is 30 (m) and vice versa.
So the size of the flower garden is 30 x 40 (m) |
# Multiplying mixed numbers!
Multiplying mixed numbers is what you will learn here with some easy to follow examples.
A mixed number is any number that has the following format:
9
3 / 4
The reason it is called mixed is because it is a mixture of a whole number and a fraction
In our example, the whole number is 9
The fraction is
3 / 4
It is extremely important to convert a mixed number into an improper fraction before doing the multiplication
We show you next how to convert a mixed number into an improper fraction using the mixed number above
Step 1. Multiply the whole number by the denominator of the fraction. (9 × 4 = 36)
Step 2. Add the result of step 1 to the numerator of the fraction (36 + 3 = 37)
The fraction is
37 / 4
Now, you are ready to multiply mixed numbers with a couple of good examples
Example #1:
2 1 / 6 × 4 3 / 2
Example #1:
2 1 / 6 × 4 3 / 2
Convert each mixed number by following the steps outlined above
Here is how for 2
1 / 6
Step 1. Multiply the whole number by the denominator of the fraction. (2 × 6 = 12)
Step 2. Add the result of step 1 to the numerator of the fraction (12 + 1 = 13)
The fraction is
13 / 6
Here is how for 4
3 / 2
Step 1. Multiply the whole number by the denominator of the fraction. (4 × 2 = 8)
Step 2. Add the result of step 1 to the numerator of the fraction (8 + 3 = 11)
The fraction is
11 / 2
Now just multiply the fractions. Do this by multipling the numerators together and the denominators together
13 / 6 × 11 / 2 = 13 × 11 / 6 × 2 = 143 / 12
13 / 6 × 11 / 2 = 13 × 11 / 6 × 2 = 143 / 12
Example #2:
7 5 / 5 × 4 3 / 3
Convert the mixed numbers
7 5 / 5 = 7 × 5 + 5 / 5 = 40 / 5
4 3 / 3 = 4 × 3 + 3 / 3 = 15 / 3
40 / 5 × 15 / 3 = 40 × 15 / 5 × 3 = 600 / 15
and
600 / 15
= 40
Example #2:
7 5 / 5 × 4 3 / 3
Convert the mixed numbers
7 5 / 5 = 7 × 5 + 5 / 5 = 40 / 5
4 3 / 3 = 4 × 3 + 3 / 3 = 15 / 3
40 / 5 × 15 / 3 = 40 × 15 / 5 × 3 = 600 / 15
and
600 / 15
= 40
Sometimes, multiplying mixed numbers is a piece of cake with some good observations. If you had noticed that 40 divided by 5 is 8 and 15 divided by 3 is 5.
You can just multiply 8 and 5 to get 40
Looking at the example #2 again.
Noticed that 5 divided by 5 is 1 and 3 divided by 3 is 1.
Remember that between a whole number and a fraction, there is always a + sign
The problem just becomes 7 + 1 × 4 + 1 = 8 × 5 = 40
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# Finding the equation of circle with a given center and a tangent line [closed]
Find the equation of circle that passes through the point $(2,2)$ and tangent to the line $x=1$ and $x=6$.
## closed as off-topic by Shailesh, Anurag A, Claude Leibovici, TastyRomeo, WatsonJan 8 '17 at 12:24
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shailesh, Anurag A, Claude Leibovici, TastyRomeo, Watson
If this question can be reworded to fit the rules in the help center, please edit the question.
Hint: Since the tangents are parallel, the centre of the circle lies at $O (\frac {7}{2}, k)$ and the radius equals $r= \frac {6-1}{2}=\frac {5}{2}$.
Can you take it from here?
Since both your tangents $x=1$ and $x=6$ are parallel, the diameter of the circle is the distance between them i.e. is $5$ and the radius is $5/2$. Therefore the x coordinate of center is $1+5/2 = 7/2$
We know that equation of a circle is $(x-a)^2 + (y-b)^2 = r^2$
Substituting known values gives $(x-7/2) + (y-b)^2 = (5/2)^2$
We also know that the circle passes through $(2, 2)$, we get,
$(2-7/2)^2 + (2-b)^2 = (5/2)^2$
$(2-b)^2 = (5/2)^2 - (-3/2)^2 = 4$
$(2-b) = \pm 2$
$b = 2 \pm 2$
$=>$ $b=0$ or $b=4$
So there are 2 possible circles
The 2 equations of the circles would be $(x-7/2)^2 + y = (5/2)^2$ and $(x-7/2)^2 + (y-4)^2 = (5/2)^2$
• Where did you get b=4? – Sheena Pascual Jan 8 '17 at 10:48
• @SheenaPascual I updated my answer. – Shreyash S Sarnayak Jan 8 '17 at 12:50 |
Surface Area of a Sphere Formula
# Surface Area of a Sphere Formula
## Surface Area of a Sphere
A sphere is a three-dimensional (3D) shape which has a curved surface. It has no edges and no corners. The area occupied by its curved surface is called the surface area of a sphere. The examples of sphere include football, basketball, soccer ball, globe, etc.
Let us study to find the formula to calculate the surface area of a sphere.
We can find the formula to calculate the surface area of a sphere using the formula to calculate the surface area of a cylinder.
Let us take a sphere and a cylinder with the same radius and the same height.
If the radius of the sphere is r, then the height of the cylinder, h = 2r as shown in the above figure.
Suppose we made this cylinder with a paper, then this cylinder is open from top and bottom. If we open up the cylinder and cover up the sphere with the paper obtained after opening it, then we see that the paper covers the whole surface of the sphere. It means that the curved surface area of the cylinder is equal to the surface area of sphere.
Hence, the surface area of the sphere = Curved surface area of the cylinder
= 2Ï€rh
Here, the height of the cylinder, h = 2r
Thus, the surface area of the sphere = 2Ï€r(2r) = 4Ï€r2
Surface area of a sphere = 4Ï€r2
### Surface Area of a Sphere Formula
Surface area of a sphere = 4Ï€r2
### Surface Area of a Sphere Example
Example 1: Find the surface area of a sphere whose radius is 7 cm.
Solution: Given: r = 7 cm
Surface area of a sphere = 4Ï€r2
= 4 × 22/7 × 7 × 7
= 616 sq. cm
Example 2: How much material is required to make a football of radius 14 cm.
Solution: Given: r = 14 cm
Surface area of a sphere = 4Ï€r2
= 4 × 22/7 × 14 × 14
= 2464 sq. cm
Example 3: If the surface area of a globe is 9856 sq. cm, find the diameter of the globe.
Solution: Given: Surface area = 9856 sq. cm
Surface area of a sphere = 4Ï€r2
9856 = 4 × 22/7 × r2
r2 = 68992/88 = 784
r = 28 cm
Diameter = 2r = 2 × 28 = 56 cm |
# Module :MA0001NP Foundation Mathematics Lecture Week 3.
## Presentation on theme: "Module :MA0001NP Foundation Mathematics Lecture Week 3."— Presentation transcript:
Module :MA0001NP Foundation Mathematics Lecture Week 3
Significant Figures and Rounding Figures Algebraic Expression, Simplifying Algebraic Expressions & Factorizing
Algebraic Expressions An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols. Here are some examples of algebraic expressions. 5x²+2x-3, x+2, 1/3 xy-5y, 7(x-2)
Consider the example:5x²+x-7 The terms of the expression are separated by addition. There are 3 terms in this example and they are 5x², x and -7. The coefficient of a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1. The last term, -7, is called a constant since there is no variable in the term.
Distributive Property a ( b + c ) = ba + ca To simplify some expressions we may need to use the Distributive Property Do you remember it? Distributive Property
Examples Example 1: 6(x + 2) Distribute the 6. 6 (x + 2) = x(6) + 2(6) = 6x + 12 Example 2: -4(x – 3) Distribute the –4. -4 (x – 3) = x(-4) –3(-4) = -4x + 12
Practice Problem Try the Distributive Property on -7 ( x – 2 ). Be sure to multiply each term by –7. -7 ( x – 2 ) = x(-7) – 2(-7) = -7x + 14 Notice when a negative is distributed all the signs of the terms in the ( )’s change.
Examples with 1 and –1. Example 3: (x – 2) = 1( x – 2 ) = x(1) – 2(1) = x - 2 Notice multiplying by a 1 does nothing to the expression in the ( )’s. Example 4: -(4x – 3) = -1(4x – 3) = 4x(-1) – 3(-1) = -4x + 3 Notice that multiplying by a –1 changes the signs of each term in the ( )’s.
Like Terms Like terms are terms with the same variables raised to the same power. Hint: The idea is that the variable part of the terms must be identical for them to be like terms.
Examples Like Terms 5x, -14x -6.7xy, 02xy The variable factors are identical. Unlike Terms 5x, 8y The variable factors are not identical.
Combining Like Terms Recall the Distributive Property a (b + c) = b(a) +c(a) To see how like terms are combined use the Distributive Property in reverse. 5x + 7x = x (5 + 7) = x (12) = 12x
Example All that work is not necessary every time. Simply identify the like terms and add their coefficients. 4x + 7y – x + 5y = 4x – x + 7y +5y = 3x + 12y
Collecting Like Terms Example
Both Skills This example requires both the Distributive Property and combining like terms. 5(x – 2) –3(2x – 7) Distribute the 5 and the –3. x(5) - 2(5) + 2x(-3) - 7(-3) 5x – 10 – 6x + 21 Combine like terms. - x+11
Simplifying Example
Evaluating Expressions Remember to use correct order of operations. Evaluate the expression 2x – 3xy +4y when x = 3 and y = -5. To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number.
Example Evaluate 2x–3xy +4y when x = 3 and y = -5. Substitute in the numbers. =2(3) – 3(3)(-5) + 4(-5) Use correct order of operations. =6 + 45 – 20 =51 – 20 =31
Evaluating Example Remember correct order of operations. Substitute in the numbers.
Common Mistakes IncorrectCorrect
Factoring Algebraic Expression To factor a number means to rewrite it as the product of smaller numbers. To factor an algebraic expression means to rewrite it as the product of simpler algebraic expressions. We factor algebraic expressions to simplify the expressions and to help solve equations. The number 36 can be factored several different ways
Factoring Algebraic Expressions To factor an algebraic expression we start by looking for common factors in each of its terms. If there are factors common to each term we can factor them out of each term. Here each term has 2 as a common factor and also x as a common factor. When we factor 2x out of each term we get This expression cannot be factored any further. To be sure that you have factored correctly do the multiplication 2x(3x-1) and see that you get the original expression back again.
Factoring Algebraic Expressions Even when there is no factor common to each term of an algebraic expression we can often still factor it into two or more simpler algebraic expressions. For example: can be factored into This occurs frequently when we want to factor quadratic expressions. A quadratic expression is one of the form
Factoring Algebraic Expressions To factor the expression start by looking at the factor pairs of 12. We are looking for a pair of factors which add up to equal 8. Because 12 is positive we are looking for factor pairs which are either both positive or both negative; because we want them to add up to a positive 8 we only need to look at the positive factor pairs of 12. The positive factor pairs of 12 are: 12 and 1 6 and 2 4 and 3 Since 6 + 2 = 8 this is the pair we are looking for and we can factor the original expression into:
Factoring by Grouping Now we look at how to factor algebraic expressions that have more than three terms. To do this we use a technique called factoring by grouping. We will group the terms into two (or more) groups and factor each group separately. We hope that this results in a factor common to each group which can then be factored out of each group. Factor Group the first two terms together and the second two terms together and factor each group. Group and factor Group and factor
Practice Factor
Factoring Quadratics when a≠1 Start by multiplying the leading coefficient (6) to the constant (-20) (6)(-20) = -120 Now look at factor pairs of -120 which add up to -7. One factor must be positive and the other must be negative. Also, the larger factor must be the negative one and the smaller factor must be the positive one
Factoring Quadratics when a≠1 The factor pairs of -120 which meet the requirements are -120 and 1-60 and 2 -40 and 3-30 and 4 -24 and 5-20 and 6 -15 and 8-12 and 10 Because -15 + 8 = -7 this is the pair that we are looking for and we can rewrite Now use factoring by grouping
Practice Factor 35 + (-3) = 32
Practice Factorize Do any of those pairs add up to 22 ? Rewrite and factor! Look for f
Practice Factorize Do any of those pairs add up to 22 ? Rewrite and factor! Look for f |
Home>Mathematics>The Correct Terms For Multiples: 6, 7, And 8 Revealed!
Mathematics
# The Correct Terms For Multiples: 6, 7, And 8 Revealed!
Written by: Clarey Rosenberg
Discover the accurate terms for multiples of 6, 7, and 8 in this comprehensive guide on mathematics. Gain valuable insights and expand your knowledge.
## Introduction
Multiples are an essential concept in mathematics, often encountered in various calculations and real-life scenarios. Understanding the correct terminology for multiples is crucial for effective communication and problem-solving. In this article, we will delve into the precise terms for multiples of 6, 7, and 8, shedding light on these often overlooked but significant details.
When discussing multiples, it's common to refer to numbers that are products of a given number and another integer. For instance, the multiples of 6 include 6, 12, 18, 24, and so on, while the multiples of 7 comprise 7, 14, 21, 28, and beyond. Similarly, the multiples of 8 encompass 8, 16, 24, 32, and beyond. While these patterns are familiar to many, the specific terms used to describe these multiples may not be as widely known.
By uncovering the precise terms for these multiples, we aim to provide a comprehensive understanding of mathematical terminology, enriching the knowledge of students, educators, and enthusiasts alike. Join us as we unravel the correct terms for multiples of 6, 7, and 8, unveiling the intricacies of mathematical language and enhancing our mathematical fluency.
## The Correct Term for 6
In mathematics, the multiples of a number play a significant role in various calculations and applications. When it comes to the multiples of 6, it's essential to understand the specific terminology associated with these numerical relationships. The term used to describe the multiples of 6 is "six times table." This terminology is fundamental in conveying precise mathematical concepts and facilitating clear communication among mathematicians, educators, and students.
The six times table encompasses a sequence of numbers that are obtained by multiplying 6 by integers, resulting in a series of multiples. This sequence includes 6, 12, 18, 24, 30, and continues indefinitely. Understanding the term "six times table" enables individuals to articulate the concept of multiples of 6 concisely and accurately, fostering a deeper comprehension of mathematical principles.
Moreover, the six times table is a foundational element in arithmetic and multiplication, serving as a building block for more advanced mathematical operations. By mastering the multiples of 6 within the context of the six times table, students develop a solid foundation in mathematics, paving the way for enhanced problem-solving skills and mathematical fluency.
Furthermore, the terminology "six times table" is integral in educational settings, where educators utilize it to introduce and reinforce the concept of multiples to students. By incorporating this precise terminology, educators create a cohesive learning environment, enabling students to grasp the intricacies of multiplication and numerical patterns with clarity and precision.
In real-world applications, the understanding of the six times table facilitates various calculations, such as determining quantities, measurements, and proportions. Whether it's calculating the number of items in multiple sets of 6 or analyzing patterns in data, the ability to recognize and apply the terminology "six times table" is invaluable in practical scenarios.
In summary, the term "six times table" encapsulates the multiples of 6, providing a succinct and accurate description of this fundamental mathematical concept. By embracing this terminology, individuals gain a deeper appreciation for the significance of multiples in mathematics and equip themselves with the language necessary to articulate and comprehend numerical relationships with clarity and precision.
## The Correct Term for 7
In the realm of mathematics, the concept of multiples holds substantial importance, influencing various calculations and mathematical operations. When exploring the multiples of 7, it is essential to grasp the precise terminology associated with these numerical relationships. The specific term used to describe the multiples of 7 is the "seven times table." This terminology serves as a foundational element in articulating and comprehending the multiples of 7, playing a pivotal role in mathematical communication and education.
The seven times table encompasses a sequence of numbers derived by multiplying 7 by integers, resulting in a series of multiples. This sequence includes 7, 14, 21, 28, 35, and extends indefinitely. Understanding the term "seven times table" is instrumental in conveying the concept of multiples of 7 with clarity and precision, fostering a deeper understanding of mathematical principles and numerical patterns.
Moreover, the seven times table serves as a fundamental building block in arithmetic and multiplication, laying the groundwork for more advanced mathematical concepts and operations. Mastery of the multiples of 7 within the framework of the seven times table equips students with a solid foundation in mathematics, nurturing their problem-solving abilities and mathematical fluency.
In educational settings, the terminology "seven times table" is employed by educators to introduce and reinforce the concept of multiples to students. By incorporating this specific terminology, educators create a cohesive learning environment, enabling students to comprehend the intricacies of multiplication and numerical relationships effectively.
Furthermore, the understanding of the seven times table extends beyond the confines of the classroom, finding relevance in real-world applications. Whether it involves calculating quantities, analyzing data, or solving practical problems, the ability to recognize and apply the terminology "seven times table" is invaluable in various scenarios, emphasizing the practical significance of mathematical concepts in everyday life.
In summary, the term "seven times table" encapsulates the multiples of 7, providing a concise and accurate description of this fundamental mathematical concept. Embracing this terminology enables individuals to articulate and comprehend numerical relationships with clarity and precision, fostering a deeper appreciation for the significance of multiples in mathematics.
## The Correct Term for 8
In the realm of mathematics, the concept of multiples serves as a fundamental building block, underpinning various calculations and numerical relationships. When delving into the multiples of 8, it is paramount to grasp the precise terminology associated with these numerical iterations. The specific term used to describe the multiples of 8 is the "eight times table." This terminology holds significant importance in articulating and comprehending the multiples of 8, playing a pivotal role in mathematical communication, education, and real-world applications.
The eight times table encompasses a sequence of numbers obtained by multiplying 8 by integers, resulting in a series of multiples. This sequence includes 8, 16, 24, 32, 40, and extends indefinitely. Understanding the term "eight times table" is instrumental in conveying the concept of multiples of 8 with clarity and precision, fostering a deeper understanding of mathematical principles and numerical patterns.
Moreover, the eight times table serves as a foundational element in arithmetic and multiplication, laying the groundwork for more advanced mathematical concepts and operations. Mastery of the multiples of 8 within the framework of the eight times table equips students with a solid foundation in mathematics, nurturing their problem-solving abilities and mathematical fluency.
In educational settings, the terminology "eight times table" is employed by educators to introduce and reinforce the concept of multiples to students. By incorporating this specific terminology, educators create a cohesive learning environment, enabling students to comprehend the intricacies of multiplication and numerical relationships effectively.
Furthermore, the understanding of the eight times table extends beyond the confines of the classroom, finding relevance in real-world applications. Whether it involves calculating quantities, analyzing data, or solving practical problems, the ability to recognize and apply the terminology "eight times table" is invaluable in various scenarios, emphasizing the practical significance of mathematical concepts in everyday life.
In summary, the term "eight times table" encapsulates the multiples of 8, providing a concise and accurate description of this fundamental mathematical concept. Embracing this terminology enables individuals to articulate and comprehend numerical relationships with clarity and precision, fostering a deeper appreciation for the significance of multiples in mathematics.
## Conclusion
In conclusion, the exploration of the correct terms for multiples of 6, 7, and 8 has unveiled the precise terminology that encapsulates these fundamental mathematical concepts. By delving into the specific terms – "six times table," "seven times table," and "eight times table," we have gained a deeper understanding of the language that underpins the communication and comprehension of multiples in mathematics.
Understanding the terminology for multiples is not merely an exercise in semantics; it is a gateway to enhanced mathematical fluency and problem-solving capabilities. By embracing the precise terms, individuals, particularly students and educators, can articulate and comprehend numerical relationships with clarity and precision. The utilization of these terms fosters effective communication, enabling seamless exchanges of mathematical ideas and concepts.
Moreover, the significance of the correct terms for multiples extends beyond the realm of theoretical mathematics. The practical applications of the six, seven, and eight times tables are ubiquitous, permeating various aspects of everyday life. Whether it involves calculating quantities, analyzing data, or solving real-world problems, the ability to recognize and apply these specific terminologies is invaluable.
In educational settings, the incorporation of the accurate terms for multiples enriches the learning experience, providing students with a solid foundation in arithmetic and multiplication. Educators play a pivotal role in introducing and reinforcing these terms, cultivating a cohesive learning environment that empowers students to navigate the intricacies of numerical relationships with confidence.
Furthermore, the exploration of the correct terms for multiples serves as a testament to the interconnectedness of mathematics with the world around us. The language of mathematics, encapsulated in these precise terms, transcends theoretical boundaries and finds resonance in practical scenarios, emphasizing the relevance and applicability of mathematical concepts in diverse contexts.
Ultimately, the journey through the correct terms for multiples of 6, 7, and 8 has not only enriched our mathematical vocabulary but has also deepened our appreciation for the role of precise terminology in fostering mathematical fluency and real-world problem-solving. By embracing these terms, we pave the way for a more nuanced and comprehensive understanding of multiples, contributing to a broader fluency in the language of mathematics. |
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# ZingPath: Volume
Searching for
## Volume
Learn in a way your textbook can't show you.
Explore the full path to learning Volume
### Lesson Focus
#### Problem Solving Involving Volumes of Prisms
Geometry
How to apply the mathematical problem solving process to solve problems involving the volume of a prism is explained.
### Now You Know
After completing this tutorial, you will be able to complete the following:
• Solve problems involving the volume of rectangular prisms.
• Calculate the volume of rectangular prisms.
### Everything You'll Have Covered
The volume of a three-dimensional figure is a measure of the space it occupies. Volume is measured in cubic units. A rectangular prism is a three-dimensional figure that has rectangular bases.
Volume of a prism V is the product of the area of its base B (l x w) and the height. The formula for the volume of a rectangular prism is
V = B x h or V = l x w x h
In this Activity Object, students will use V = l x w x h. Provide students with a several example problems using whole numbers less than ten for each of the dimensions to ensure that students can solve volume of rectangular prism problems.
A cube is used in this Activity Object. A cube is a rectangular prism in the same way that a square is a rectangle. Be sure students realize that a cube is also a rectangular prism whose dimensions are the same length. Therefore, its volume is found by multiplying the length of any dimension by itself twice.
• base (b) - any side or flat surface of the object
• cube - a rectangular prism whose length, width, and height are the same number of units
• dimension - a measure of length, width or height
• height (h) - the perpendicular distance to the base
• length (l)- the distance along a line or shape from one point to another
• prism - a figure made of two parallel faces that are polygons of the same shape and sides that are parallelograms
• rectangular prism - a prism whose base is a rectangle
• volume - the measure of the amount of space inside of a solid figure
• width (w)- the horizontal measurement to the length
### Tutorial Details
Approximate Time 30 Minutes Pre-requisite Concepts Students know the properties of rectangular prisms and cubes, and the volume of rectangular prisms and cubes. Course Geometry Type of Tutorial Problem Solving & Reasoning Key Vocabulary problem solving, volumes of prisms, volume |
# Path Counting
In the city of Trafficville there is a region where all the streets are one-way. If a car drives into this region, it can only drive south or east. (It’s really not very practical. The city council is debating what should be done about it.)
Restaurant owner Rosa Gustaroni, who is going to work this morning, wants to drive from the bakery (at the green star) to her restaurant (at the orange star). How many ways can she get there? Which route is the shortest?
In this problem, it’s just as important to remember what we can’t do as to remember what we can. The big limitation here is the bit about the one-way streets. It means Ms. Gustaroni can’t do this:
In fact, we can do even better. We know that she can’t go on streets north or west of her house without driving the wrong way along a one-way street. And we also know that if she drives too far south or east, she won’t be able to get back to the restaurant without taking a big detour. So we can simplify our diagram like so:
Well now, that looks a lot more manageable.
Now, if we look hard enough we discover a key point: whatever route Ms. Gustaroni takes, she has to go exactly four blocks east and exactly three blocks south. No more, no less. That makes a total of seven steps, always – and that answers one of the questions we are asked. There is no “shortest” route.
What about the other question? How many ways are there to get from the bakery to the restaurant? We know that there will be seven total steps, and that three of them will be southward. Hey, this looks familiar – we’re asked to pick three of the seven steps to be southward! We’ve bumped into our old friend, choosing.
$$\dbinom{7}{3} = \dfrac{7!}{3!(7-3)!} = \dfrac{5040}{6\cdot 24} = 35$$
That means that there are $35$ ways for Ms. Gustaroni to drive from the bakery to the restaurant. (We’d get the same answer if we’d chosen the $4$ eastward steps – read more on choosing if you don’t understand why.)
Let’s put a twist on this problem.
There has been a traffic accident at the intersection where the red X is, and the police have blocked off the area so that cars must go around the accident. Rosa Gustaroni still needs to get to her restaurant from the bakery. Now how many ways are there for her to make the trip?
Well, we could either do an awful lot of casework, or we could use a trick called complementary counting. In complementary counting, you count the total number of things, and then subtract the things you don’t want. This problem has a pretty clear definition of what we don’t want, and we already know the total number of things, so it seems like complementary counting is a good thing to try.
Okay, so what exactly is a bad route? Clearly, it’s one that goes through the blocked-off intersection. So we want to count how many paths go from the bakery to the intersection to the restaurant. That means we’re really counting two things: 1) the number of paths from the bakery to the intersection, and 2) the number of paths from the intersection to the restaurant.
Let’s tackle these one at a time.
1. To get from the bakery to the intersection, you have to take a total of three steps, two southward and one eastward. Choosing one eastward step out of three total steps:
$$\dbinom{3}{1} = \dfrac{3!}{1!(3-1)!} = \dfrac{6}{2} = 3.$$
1. To get from the intersection to the restaurant, you have to take a total of four steps, one southward and three eastward. Choosing three eastward steps out of four total steps:
$$\dbinom{4}{3} = \dfrac{4!}{3!(4-3)!} = \dfrac{24}{6} = 4.$$
So there are $3$ ways to get from the bakery to the intersection, and $4$ ways to get from the intersection to the restaurant. Do these choices depend on each other in any way? No, so to get the total we multiply (because for each option of the first, all options of the second are possible). Then there are $3 \cdot 4 = 12$ bad routes.
The total number of routes that Ms. Gustaroni could take, as we already calculated, is $35$. The number of bad routes, we just figured out, is $12$. So the number of good routes is
$$35 - 12 = 23.$$
So Rosa Gustaroni still has plenty of possible ways to get to work. |
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 3.3: The Complement of an Event
Difficulty Level: At Grade Created by: CK-12
## Learning Objectives
• Know the definition of the complement of an event.
• Using the complement of an event to calculate the probability of an event.
• Understanding the complementary rule.
Definition
The complement A\begin{align*}A'\end{align*} of an event A\begin{align*}A\end{align*} consists of all the simple events (outcomes) that are not in the event A\begin{align*}A\end{align*}.
Let us refer back to the experiment of throwing one die. As you know, the sample space of a fair die is S={1,2,3,4,5,6}\begin{align*}S = \left \{1, 2, 3, 4, 5, 6 \right \}\end{align*}. If we define the event A\begin{align*}A\end{align*} as
A:{observe an odd number}\begin{align*}\text{A:} \left \{\text{observe an odd number}\right \}\end{align*}
Then, A={1,3,5}\begin{align*}A = \left \{1, 3, 5 \right \}\end{align*}, which includes all the simple events of the set S\begin{align*}S\end{align*} that are odd. Thus, the complement of A\begin{align*}A\end{align*} is the set of simple events that will not occur in A\begin{align*}A\end{align*}. So A\begin{align*}A'\end{align*} will include all the elements that are not odd in the sample space of the set S\begin{align*}S\end{align*}:
A={2,4,6}.\begin{align*}A' = \left \{2, 4, 6 \right \}.\end{align*}
The Venn diagram is shown below.
This leads us to say that the event A\begin{align*}A\end{align*} and its complement A\begin{align*}A'\end{align*} are the sum of all the possible outcomes of the sample space of the experiment. Therefore, the probabilities of an event and its complement must sum to 1\begin{align*}1\end{align*}.
## The Complementary Rule
The sum of the probabilities of an event and its complement must equal 1\begin{align*}1\end{align*}.
P(A)+P(A)=1\begin{align*}P(A) + P(A') = 1\end{align*}
As you will see in the following examples below, it is sometimes easier to calculate the probability of the complement of an event rather than the event itself. Then the probability of the event, P(A)\begin{align*}P(A)\end{align*}, is calculated using the relationship:
P(A)=1P(A)\begin{align*}P(A) = 1 - P(A')\end{align*}
Example:
If you know that the probability of getting the flu this winter is 0.43\begin{align*}0.43\end{align*}, what is the probability that you will not get the flu?
Solution:
First, ask the question, what is the probability of the simple event? It is
P(A)={you will get the flu}=0.43\begin{align*}P(A) = \left \{ \text{you will get the flu} \right \} = 0.43\end{align*}
The complement is
P(A)={you will not get the flu}=1P(A)=10.43=0.57\begin{align*}P(A') = \left \{ \text{you will not get the flu} \right \} = 1 - P(A) = 1 - 0.43 = 0.57\end{align*}
Example:
Two coins are tossed simultaneously. Here is an event:
A:{observing at least one head }\begin{align*}\text{A:} \left \{\text{observing at least one head }\right \}\end{align*}
What is the complement of A\begin{align*}A\end{align*} and how would you calculate the probability of A\begin{align*}A\end{align*} by using the complementary relationship?
Solution:
Since the event A\begin{align*}A\end{align*} is observing all simple events A={HH,HT,TH}\begin{align*} A = \left \{HH, HT, TH \right \}\end{align*}, then the complement of A\begin{align*}A\end{align*} is defined as the event that occurs when A\begin{align*}A\end{align*} does not occur, namely, all the events that do not have heads, namely,
A={observe no heads}={TT}\begin{align*}A' = \left \{\text{observe no heads} \right \} = \left \{ TT \right \}\end{align*}
We can draw a simple Venn diagram that shows A\begin{align*}A\end{align*} and A\begin{align*}A'\end{align*} in the toss of two coins.
The second part of the problem is to calculate the probability of A\begin{align*}A\end{align*} using the complementary relationship. Recall that P(A)=1P(A)\begin{align*}P(A) = 1 - P(A')\end{align*}. So by calculating P(A)\begin{align*}P(A')\end{align*}, we can easily calculate P(A)\begin{align*}P(A)\end{align*} by subtracting it from 1\begin{align*}1\end{align*}.
P(A)=P(TT)=1/4\begin{align*}P(A') = P(TT) = 1/4\end{align*}
and
P(A)=1P(A)=11/4=3/4.\begin{align*}P(A) = 1 - P(A') = 1 - 1/4 = 3/4.\end{align*}
Obviously, we could have gotten the same result if we had calculated the probability of the event of A\begin{align*}A\end{align*} occurring directly. The next example, however, will show you that sometimes it is easier to calculate the complementary relationship to find the answer that we are seeking.
Example:
Here is a new kind of problem. Consider the experiment of tossing a coin ten times. What is the probability that we will observe at least one head?
Solution:
Before we begin, we can write the event as
A={observe at least one head in ten tosses}\begin{align*}\text{A} = \left \{\text{observe at least one head in ten tosses}\right \}\end{align*}
What are the simple events of this experiment? As you can imagine, there are many simple events and it would take a very long time to list them. One simple event may look like this: HTTHTHHTTH,\begin{align*}HTTHTHHTTH,\end{align*} another THTHHHTHTH,\begin{align*}THTHHHTHTH,\end{align*} etc. Is there a way to calculate the number of simple events for this experiment? The answer is yes but we will learn how to do this later in the chapter. For the time being, let us just accept that there are 210=1024\begin{align*}2^{10} = 1024\end{align*} simple events in this experiment.
To calculate the probability, each time we toss the coin, the chance is the same for heads and tails to occur. We can therefore say that each simple event, among \begin{align*}1024\end{align*} events, is equally likely to occur. So
\begin{align*}P(\text{any simple event among}\ 1024) = \frac{1} {1024}\end{align*}
We are being asked to calculate the probability that we will observe at least one head. You may find it difficult to calculate since the heads will most likely occur very frequently during \begin{align*}10\end{align*} consecutive tosses. However, if we calculate the complement of \begin{align*}A\end{align*}, i.e., the probability that no heads will be observed, our answer may become a little easier. The complement \begin{align*}A'\end{align*} is easy, it contains only one simple event:
\begin{align*}A' = \left \{ TTTTTTTTTT \right \}\end{align*}
Since this is the only event that no heads appear and since all simple events are equally likely, then
\begin{align*}P(A') = \frac{1} {1024}\end{align*}
Now, because \begin{align*}P(A) = 1 - P(A'),\end{align*} then
\begin{align*}P(A) = 1 - P(A') = 1 - \frac{1} {1024} \approx 0.999 = 99.9 \%\end{align*}
That is a very high percentage chance of observing at least one head in ten tosses of a coin.
## Lesson Summary
1. The complement \begin{align*}A'\end{align*} of an event \begin{align*}A\end{align*} consists of all the simple events (outcomes) that are not in the event \begin{align*}A\end{align*}.
2. The Complementary Rule states that the sum of the probabilities of an event and its complement must equal \begin{align*}1\end{align*}, or for an event \begin{align*}A,\end{align*} \begin{align*}P(A) + P(A') = 1.\end{align*}
## Review Questions
1. A fair coin is tossed three times. Two events are defined as follows: \begin{align*}& \text{A:} \left \{ \text{At least one head is observed} \right \} \\ & \text{B:} \left \{ \text{The number of heads observed is odd} \right \}\end{align*}
1. List the sample space for tossing a coin three times
2. List the outcomes of \begin{align*}A.\end{align*}
3. List the outcomes of \begin{align*}B.\end{align*}
4. List the outcomes of the events \begin{align*}A \cup B, A', A \cap B.\end{align*}
5. Find \begin{align*}P(A), P(B), P(A \cup B), P(A'), P(A \cap B).\end{align*}
2. The Venn diagram below shows an experiment with five simple events. The two events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are shown. The probabilities of the simple events are: \begin{align*}P(1) = 1/10, P(2) = 2/10, P(3) = 3/10, P(4) = 1/10, P(5) = 3/10.\end{align*} Find \begin{align*}P(A'), P(B'), P(A' \cap B), P(A \cap B), P(A \cup B'), P(A \cup B), P [(A \cap B)']\end{align*} and \begin{align*}P[(A \cup B)'].\end{align*}
1. all: \begin{align*}\left \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \right \} \end{align*}
2. \begin{align*}\mathrm{A:} \left \{ HHH, HHT, HTH, THH, HTT, THT, TTH \right \}\end{align*}
3. \begin{align*}\mathrm{B:} \left \{ HHH, HTT, THT, TTH \right \}\end{align*}
4. \begin{align*}A \cup B\end{align*} same as \begin{align*}A,\end{align*} \begin{align*}A': \left \{ TTT \right \},\end{align*} \begin{align*}A \cap B\end{align*} same as \begin{align*}B\end{align*}
5. \begin{align*}P(A) = P(A \cup B) = 7/8, P(B) = P(A \cap B) = 1/2, P(A') = 1/8\end{align*}
1. \begin{align*}4/10, 2/10, 3/10, 5/10, 9/10, 7/10, 5/10, 1/10.\end{align*}
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# What Are the Metric Units for Area?
This entry uses the metric system. For the imperial system click here.
When you want to describe the size of a surface, you are talking about an area. Examples of this are the size of a floor, a piece of paper, or a wall. How many doormats can you fit on the floor of your living room? How many pieces of paper can you fit in the hallway? How many towels can you fit on the floor of your bedroom?
Because towels can come in many different sizes, it can be difficult to agree on how big a surface is if you measure it in the number of towels you can put on it. Which towel should you use? A bath towel or a kitchen towel? Because there isn’t any standard towel size, we agreed to use a square with one meter on each side. We call this square a square meter. Because we all agree on the length of a meter, we also agree on the size of the square meter.
Area is measured in square meters (m2). A square meter is as large as a square where every side has a length of one meter. If you want to measure the area of smaller things, you can use square centimeters (cm2). Below you can see an example of an area with a size of $1$ m2.
You can also have surfaces with an area of $1$ cm2. The figure below is an example of this. It has a length and width of $1$ cm2 each.
You might have already learned that $1$ m is the same as $100$ cm. But $1$ m2 is not the same as $100$ cm2. There’s actually space for $10\phantom{\rule{0.17em}{0ex}}000$ squares of $1$ cm2 in the first figure with size $1$ m2. Take a look at the figure below. You can see that the blue square ($1$ cm2) can fit $100$ times into the white square. The white square can fit $100$ times into the pink square, which has the size $1$ m2. That means that there is space for $100\cdot 100=10\phantom{\rule{0.17em}{0ex}}000$ of the blue squares ($1$ cm2) in the pink square of $1$ m2.
Example 1
Eric is going to paint the pattern below on one of the walls at his school, and he has to calculate how much paint he needs.
If every square is $\text{}1\text{}\phantom{\rule{0.17em}{0ex}}{\text{m}}^{2}$, how many square meters of paint does he need for every color?
Eric needs $8$ m2 of purple paint, $8$ m2 of pink paint, $8$ m2 of orange paint, $8$ m2 of red paint, $16$ m2 of blue paint and $16$ m2 of green paint.
Example 2
Cathy needs to cut out the pattern below from colored pieces of paper.
How many square centimeters of paper does she need of each color if each square is $\text{}1\text{}\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}$?
Cathy needs $7$ cm2 of yellow paper, $7$ cm2 of purple paper and $67$ cm2 of blue paper.
So, if you want to measure smaller surfaces, you can use square decimeters, square centimeters or square millimeters. If you want to measure larger surfaces, you can use square kilometers. As one meter is ten decimeters, one square decimeter (dm2) is a square with sides equal to one tenth of a meter (a decimeter). You might think that one square meter has space for ten square decimeters. But that’s not the case! You can fit $100$ square decimeters in one square meter. You can see the reason for this in the figure below.
In the same way, one square centimeter (cm2) is a square where each side has a length of one hundredth of a meter (one centimeter). As there are $100$ cm in a meter, you can fit a whole $100\cdot 100=10\phantom{\rule{0.17em}{0ex}}000$ square centimeters in a square meter.
Similarly, one square millimeter (mm2) is a square where each side has a length of one thousandth of a meter (one millimeter). As there are $1000$ mm in a meter, you can fit a whole $1000\cdot 1000=1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$ square millimeters in a square meter!
When you need to measure the surface of really big surfaces, for example cities and countries, it can be smart to use larger units. In that case, the unit square kilometers (km2) is often used. A square kilometer is a square where each side has a length of a thousand meters (one kilometer).
Notice how I write and do calculations with units. I begin by writing down the expression. Then I check the bubble figure to see what I have to divide or multiply by to get to the unit the exercise asks for. Then, I solve the exercise step by step. The trick is to switch the unit in the expression with this: The multiplication or division sign (depending on which way I’m moving in the bubble figure), the number the bubble figure shows me, and the corresponding unit. In that way, I can work my way through the bubble figure until I get where I need to go.
Example 3
Square meters to square decimeters:
$4\phantom{\rule{0.17em}{0ex}}{\text{m}}^{2}=\left(4\cdot 100\right)\phantom{\rule{0.33em}{0ex}}{\text{dm}}^{2}=400\phantom{\rule{0.17em}{0ex}}{\text{dm}}^{2}$
Square millimeters to square centimeters:
$4000\phantom{\rule{0.17em}{0ex}}{\text{mm}}^{2}=\left(4000÷100\right)\phantom{\rule{0.33em}{0ex}}{\text{cm}}^{2}=40\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}$
Square decimeters to square centimeters:
$40\phantom{\rule{0.17em}{0ex}}{\text{dm}}^{2}=\left(40\cdot 100\right)\phantom{\rule{0.33em}{0ex}}{\text{cm}}^{2}=4000\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}$
Square centimeters to square decimeters:
$400\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}=\left(400÷100\right)\phantom{\rule{0.33em}{0ex}}{\text{dm}}^{2}=4\phantom{\rule{0.17em}{0ex}}{\text{dm}}^{2}$
Square centimeters to square millimeters:
$400\phantom{\rule{0.17em}{0ex}}{\text{cm}}^{2}=\left(400\cdot 100\right)\phantom{\rule{0.33em}{0ex}}{\text{mm}}^{2}=40\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}{\text{mm}}^{2}$
Square decimeters to square meters:
$400\phantom{\rule{0.17em}{0ex}}{\text{dm}}^{2}=\left(400÷100\right)\phantom{\rule{0.33em}{0ex}}{\text{m}}^{2}=4\phantom{\rule{0.17em}{0ex}}{\text{m}}^{2}$
Math Vault
Would you like to solve exercises about area? Try Math Vault! |
# What Are the 7 Indeterminate Forms? (L’hopital’s Rule)
Video Series: Calculus 1 (Explore, Discover, Learn) Series
(D4M) — Here is the video transcript for this video.
00:00
[Music] who is lahapital and why does she have a rule what are
indeterminate forms i go over all these questions with many many examples
in fact i give at least one instance of each of the seven types of indeterminate
forms hi everyone i’m dave welcome back to the
calculus explore discover learn series and in this video i’m going to cover
indeterminate forms and i’m going to give lots of examples and uh
towards the end we’re going to see how we can make a change of variable
00:01
when finding limits and we’re going to see when the hopital’s rule isn’t useful
and then at the very end we’ll talk about some exercises
so stick around and let’s get started okay so up first are the indeterminate
forms what are they and so we’re going to see
that there are seven different types i’m going to show them to you right now now
expressions here and the idea is that we’re going to be looking at trying to
find limits and sometimes when you’re finding limits
you get these expressions like infinity over infinity
zero zero infinity minus infinity zero times infinity infinity to
zero power zero to the zero power and then one to the infinity power so
there you go there’s seven types of indeterminate forms there
00:02
you’ll see that we’re going to go through each of these types and
then we’ll do do some even more examples at the end there
so let’s get started um you know what do i mean by
um we say that we have a limit that has an
indeterminate form so for example this limit right
here as x goes to infinity we’re looking at three x
minus one over x to the third now if we look at the numerator and we look
at three x minus one as x is going to infinity we can see
that we have indeterminate form because when we look at 3x minus 1 that’s going
to infinity and then the denominator is also going to infinity so
as the x goes to infinity the numerator goes to infinity
and the denominator goes to infinity so this is our first type of
indeterminate form here this is our first example
00:03
now one thing i’m going to try to emphasize throughout is to not use an
equal sign with an indeterminate form and you’ll see why
but i just want to bring that to your attention as soon as i can
so let’s look at uh our l’hopital’s rule it’s going to be our main tool
of this episode and it’s going to say that
when you were trying to take the limit of a quotient of function so f over g
and we can say that that limit of that quotient
is equal to the limit of the quotient of the derivatives
and and and in some cases this is very useful
but there are some conditions in order to make it work and so
what are those conditions so it’s it’s important to actually verify those
conditions um especially because when you’re
working out lots of limit problems and lots of exercises
you tend to like just try to solve the problem solve the problem
00:04
but it’s important to not forget what the underlying conditions are
to use the habitat’s rule so what is the habitat’s rule well we have
some conditions right so what are our first conditions so
first of all we have two functions and we’re going to assume that they’re both
differentiable now that is not a difficult hypothesis to remember because
the whole point of habitat’s rule is finding limits
taking derivatives and so if your derivative doesn’t exist
then it’s going to come to your attention but you also need to remember
the derivative is not zero there on that on some open interval
and some open interval containing c right so we’re approaching c
so we’re looking at the limit as x approaches c
of f of x over g of x right so we’re taking the limit of a quotient
now when you’re looking at the limit of that quotient you might realize you have
an indeterminate form and it might be zero zero or it
00:05
might be infinity over infinity those were the first two types of
indeterminate forms that i mentioned now if you also know that the limit of
the quotient of the derivatives exists or if that limit is in infinite and
it is infinite sorry then and here’s the big conclusion of habitat’s rule
big conclusion very impactful so we can find the limit of the quotient f over g
by finding the limit of the quotient of the derivatives
and like i said we’ll we’ll use this right here
um this equal sign with this limit right here when we start working out lots of
examples when we start doing calculations we’ll um you know use this conclusion
here a great deal but you have to remember to always check
does it produce an intermediate form zero of zero or infinity over infinity
00:06
and does this limit exist or is it infinite
if all those conditions hold then we can say the limit of the quotient of the
functions is equal to the quotient the limit of the quotients of the derivatives
now an interesting thing about l’hopital’s rule
is that it’s not just as x approaches c so we could go back through this um
the habitat’s rule here theorem for x approaches c
and we have x approaches c we have x approaches c
we have x approaches c so you can replace that
by limit from the right or limit from the left or limit at infinity
or limit at minus infinity so for example if we wanted to use the habitat’s rule
as x goes to infinity then we would have x goes to infinity here
and we would have x goes to infinity here and x goes to infinity here and here
and that statement that would be another version of l’hopital’s rule
so l’hopital’s rule think of it as x approaches c
00:07
and those other four cases over there as well okay so l who was he well here’s a
picture of him and his name is firmly established for
calculating limits involving indeterminate forms
so people know his name throughout it’s not like a
regional local thing people people associate his name with this theorem and
the interesting thing is he didn’t actually come up with the rule
but he did publish it in one of his books one of his first books and he had a
famous book that was one of the first cases of differential calculus in print
and his book went a long way towards uh bringing calculus to the masses
and so he had this book out in which he put lahapatel’s world in there
00:08
and so well let’s look at some examples now let’s get started [Music]
okay so up first um let’s try to put all this together that that i was just
mentioning um you know what are indeterminate forms
how do we use the hapathos rule and let’s start putting it into practice
so first of all why do we say that zero over zero is an indeterminate form
so if i look at something like this this limit right here is 1. you should be
comfortable with that and the first episode we did in the series was over limits
and we talked about that limit right there um but also the limit as 2x
over x as x goes to zero that limit is two
and so you know that one is not equal to two
00:09
now as i mentioned at the beginning we do not use an equal sign
when we’re using indeterminate forms and let me illustrate why
so we could go here and say the limit as x approaches zero
of x over x which we know is one um but on the other hand maybe you’ll say
oh this is zero of zero right so uh let me start off by writing it like this
i i like to write like this it’s like a joke almost one is equal to
two prove it all right one is equal to the limit as x approaches zero of x over
x and this is zero over zero and that is also the same thing as this
limit right here of 2x over x but everyone knows that this limit is two
00:10
so there we go one is equal to two of course we know one is not equal to two
and this right here is the problem right here whenever you’re going to use
an equal sign with an indeterminate form you’re going to run into problems so my
advice to you is to do not use indeterminate forms with an equal sign
equal sign you want to have numbers you’re using equal sign with numbers for
example all of this right here is a number it’s just one
and this this right here is not equal to in zero zero
so this would be you know something that you would not want to write down
anywhere and you don’t even have to have both sides
just don’t write indeterminate form down with an equal sign
you see that a lot of students making that mistake
so i want to bring that to your attention never use any of the seven
indeterminate forms with an equal sign it’s just it just doesn’t make sense all
right so but why do we call it an indeterminate form because if you have
00:11
the limit of this expression right here we see it like this
this limit right here has now i’m going to abbreviate
indeterminate form with if has indeterminate form zero of zero because
we approach the numerator whatever it is it’s just x is zero and x approaches
zero of the denominator whatever it is and then that limit is zero so the limit
of the numerator is zero and the limit of the denominator is zero
so we have an indeterminate form so instead of writing equals zero zero you
can write this now some people will say okay just write this in fact some people
might even abbreviate it further they might just say type zero
zero right so i’ve seen this in print in some
00:12
calculus textbooks this limit has type zero to zero
but whatever you do don’t put an equal sign between these two
it’s just not defined doesn’t make any sense
so you know some people will say this some people want you to write it all out
some people will just want you to write this out
so it really just depends on who you’re communicating with
all right so let’s look at our first example here
evaluate the limit as x approaches zero of sine x over x
and so it says to use the habitats rule now you might remember we kind of talked
where i just said that limit is one and we’ll explain why later
or i think actually we gave a geometric argument as to why that limit is one
or perhaps a graphing argument in any case let’s use the habitat’s rule
00:13
so when i go find this limit right here what i’m going to notice is that the
limit of the numerator is going to zero the limit of the denominator is going to
zero so it looks like we have indeterminate form
zero over zero so i’m going to put that out here
indeterminate form zero over zero now i’m thinking to myself because the
limit of the numerator goes to zero and the limit of the denominator goes to
1. but i’m not going to write all that out i’m just going to say indeterminate
form 0 of 0. now equals so what we’re going to do is
we’re going to take the derivative of the numerator which is cosine
over the derivative of the denominator okay so that’s the habitat’s rule right
there and now we can find this limit the limit of the numerator cosine zero
so it’s going to one over one which is just one
so that limit right there is one and we’re using by the hopital’s rule
00:14
so maybe you want to use the lower l rule however you want to write it
any case there’s the first example there let’s look at another one and now we
have the limit as we’re approaching zero of e to the two x
minus one over x so how do we even know that we should use l’hopital’s rule well
the limit of the numerator right so the limit of the numerator is going to
zero because we have e to the 2x 2x is going to zero so e to the zero is one
so this is going to be one minus one in the numerator
so the limit of the numerator is going to zero the limit of the
denominator is going to zero so i’m going to say this right here has
indeterminate form zero over zero so i’m going to use the hobitos rule and
00:15
so what’s the limit of the numerator is going to be 2 e to the 2x
over 1 and now as the limit approaches 0 as x approaches 0 the limit is
two over one so it’s two so there we go there
so in the last example we saw the limit as we approach zero of sine x over x
this has indeterminate form zero of zero we found the limit to be one
but on the other hand this limit right here is two
so if you have indeterminate form zero zero sometimes you might get one
sometimes you might get two in fact you can get any
number that you want this is why we call it an indeterminate form
we found this limit it’s two and we found this limit it’s one
so even though they both had indeterminate forms zero over zero
you can still find their limits using l’hopital’s rule
00:16
all right so let’s look at this right here
so we’re justified in calling infinity minus infinity in indeterminate form
why well what is this limit right here this limit is 2x over x this limit is 2
right so that’s just one now if you have indeterminate form infinity
over infinity the value of the limit can be anything now
here let’s make the same argument again this limit is one and this is the limit
as x goes to infinity x over x this has indeterminate form and if you write
indeterminate form with an equal sign bad things happen because
00:17
this is the same indeterminate form and now i can put 2x over x this has
indeterminate form also infinity over infinity
if i write an equal sign here that’s wrong and you can see why
this limit is two this limit is one one is not equal to two if you write
indeterminate form with an equal sign you’ll get contradictions
all right so let’s not do that all right let’s look at an example
evaluate this limit using the habitat’s rule and so we have here
indeterminate form yes or no what is the form
so a to the x minus b to the x that’s going to
what so a to the x as x goes to zero that’s going to a to the zero which is one
so it’s going to one minus one so it’s going to zero over zero
so we have an indeterminant form zero over zero [Music]
00:18
okay so we had um we can use the hopital’s rule here
so the derivative of a to the x will be natural log of a times a to the x minus
the derivative of b to the x is natural log of b
times b to the x and then derivative of x is just one
and now we have this is going to one and this is going to one and that’s
already one so this will be natural log of a
minus natural log of b and if you wish you could write it as natural log of a
over b [Music] okay so there’s another limit where we
have indeterminate form zero over zero let’s see if we can find one where we
have indeterminate form infinity over infinity here’s a good one
so let’s look at here the limit as x goes to infinity now when
00:19
you’re looking at natural log of x i always think it’s important to have a
understanding of what natural log looks like
so it’s just going up it’s more curvier than that but it’s just going up
and so as x as x goes to infinity this is just going to infinity
that’s just going up up up up up up right so
the numerator is going to infinity the denominator is going to infinity
so we’re going to say we have indeterminate form infinity over infinity
so we’re going to try to use the hapatel’s rule here
so derivative of natural log of x is one over x derivative of x is one
and so this limit right here is just zero so this limit is just zero
00:20
let’s look at another example and let’s let’s see if we can do this limit here
we’re going to approach minus infinity this time so here we go we have the limit
as we approach minus infinity and we have the x square so the square is
taking care of the negative signs here think of that as like negative 100
negative thousand negative a million but
whenever you’re squaring those negatives
it’s going to become positive so they so the numerator is going to infinity
and then we have e to the minus x now x’s are negative so when i do a negative
an x that’ll be a positive so this is going to positive infinity also so we have
indeterminate form uh infinity over infinity indeterminate form infinity over
infinity and so we’re going to try to use the habitats rule here
so we’re going to take the derivative of the x squared so 2x
00:21
and the derivative here will be minus e to the minus x
and so what do we get here x is going to infinity and so when we
look here we’re going to get here minus infinity over minus infinity and so
you know we’re going to have indeterminate form again infinity over infinity
now if you are not sure about that move the minus up here if you wish and
now the numerator is going to positive infinity and the denominator is
going to positive infinity so let’s use the habitat’s rule again
and derivative here is minus 2 and the derivative here is
negative e to the minus x now the numerator went from an x square to an x to
00:22
a two but this is substantially different because
the numerator is going to minus two and the denominator is going to
um blow up so this whole thing right here is going to be zero
so we did it we found this limit right here is just
zero and this is an illustration of you can use the habitats rule more than once
i used it twice here okay next example um let’s see here okay the next part
[Music] okay so now we’re going to look at infinity minus infinity and zero
00:23
times infinity and these are two additional indeterminate forms but with these
two types of indeterminate forms we’ll be able to
take an indeterminate form that looks like one of these two types here
infinity minus infinity and convert them to one of these two types over here
and so here’s an example of one right here so let’s look at this up close
we have limit as we go to one from the right
one over natural log of x so what is that part right there doing
when we are um looking at this part right here right natural log
as we’re approaching one from the right so this is looking like it’s going to
zero so this whole thing looks like it’s going to go to infinity here
and then we have minus one over x minus one and then this right here is the same
thing we’re going to be going to infinity here and so we’re going to have
00:24
infinity minus infinity and so we’re going to have indeterminate form
infinity minus infinity now we’re going to try to use the hoppitas
rule but in order to use lopita’s rule we have to have an
f over g we have to have a quotient right now we don’t have a quotient we
have a difference so i’m going to try to combine these together
into one fraction one quotient so i’m gonna multiply by x minus one
x minus one over x minus one and natural log of x
over natural log of x and this is what we get let’s put x minus 1 first
all right so getting common denominator now what’s happening as x approaches one
from the right this is approaching zero that’s approaching
zero so this is approaching zero and if i look at the denominator that’s
00:25
approaching zero so this is indeterminate form zero zero
so we converted this in determined form into this indeterminate form
and now we can apply the hopital’s rule so derivative of the x is one and then
minus one over x and in the denominator here we have the product rule
the derivative of this part one times natural log of x
plus now this part right here x minus one
times the derivative of natural log of x okay now you know we can do l’hopital’s
rule multiple times um should we or are we done let’s see
x approaches one well we’re going to get zero
x approaches one we’re getting zero plus zero so it looks like we’re still
getting zero over zero so what i want to do now is to try to
simplify this algebraically before i start taking derivatives so i’m
00:26
going to multiply through top and bottom by an x just to make it
a little bit simpler so multiply the top by x i’m going to get x minus 1
and multiply the denominator by x i’m going to get
x times natural log of x and then multiply this by x
so now i’m going to get plus and then x minus 1.
and so now what happens we still get zero over
zero zero so we’re still getting zero over zero
so we still have indeterminate form zero of zero
so i’m gonna apply the habitats rule again in determinant form all right so
let’s apply the habitats rule again so i’m looking at the derivative here so
i get a one [Music] and i’m looking here i need to take the product rule so
00:27
derivative of the x times the second plus the first times one over x plus one
and so there’s the limit of the quotient of the derivatives
and now what’s happening now we’re getting a one here
this is zero and that’s a one plus a one so we’re getting one half there we go
so we used the hoppital’s rule twice and we had this indeterminate form at first
but we were able to convert it to this indeterminate type
here using some algebra so that’s the basic idea behind these
two indeterminate forms infinity minus infinity and infinity times zero
so you want to do some algebra to get it to a familiar indeterminate form
so we can use the hopital’s rule all right so let’s look at another example
and on this example here what are we getting what’s our indeterminate form
00:28
[Music] and so it looks like the first part is
positive infinity right it’s just an x and then minus square root of x squared
minus 3x so this is going to infinity as x goes to infinity
that square is going to dominate right for example when x is
you know 1 million that’ll be 1 million squared minus
3 million right so this will this will dominate so this is going to be infinity
minus infinity so i’m going to put indeterminate form
infinity minus infinity here and let’s see what we can do now
when we’re trying this problem right here one approach
might be to try to use a conjugate um when you try that i think that you’re
going to get it won’t be as easy as trying to factor out
00:29
an x squared here and then trying to factor out an x from that
that’ll be a shorter process i think so i’m going to do that
so i’m going to try to factor out an x squared here
so i’ll do this in steps here in case you’re
not seeing how to do that so this would be 1 minus 3 over x 3 over x good
so when you factor out an x squared here you just get a 1
when you factor out an x squared here well there aren’t enough x’s so i have
to divide all right so the point is is that if i
multiply through x squared times 1 is x squared and the x squared times the
minus 3 over x the x’s cancel and i get a minus 3x left
all right and so now i can take the square root of the x squared
and so i can get an x here so i guess i’ll write that out
00:30
so this will be x minus x square root and then i’ll have 1 minus 3 over x left
here all right so both of these i can factor out an x
so now we can write it right here x times and then we have 1 minus
square root of 1 minus 3 over x all right so we went from this one right here
to this one right here and this is infinity
and that so x is going to infinity so that’s going to
zero so this will be square root of one one minus one so that’s going to
infinity times 0 or in other words 0 times infinity so that’s actually the
next indeterminate form we’re going to talk about
is 0 times infinity indeterminant form 0 times infinity and what happens if you
get something like this well we’re going to move that x down here
00:31
so let’s write it like this [Music] 1 minus square root 1 minus 3 over x
over now that’s x in the numerator so it’s 1 over x in the denominator
now what we see is we’re getting 0 down here
and this is going to 0 so that’s going to 1 minus one
so we have indeterminate form zero over zero
so finally we get it into a form where we can use l’hopital’s rule
so now we’re going to start taking some derivatives
so here we go take the derivative here [Music]
and so the derivative of that is zero and then we can have a minus sign right
here now we have a one-half power here so the one-half is going to come down
and i’m going to have 1 minus 3 over x to the minus 1 half power
00:32
times the derivative of the inside part here derivative of the one is zero
and i already have a minus here so and then i’m going to have a
three here and so let’s just write that as minus three
and then x to the minus two right so that the minus 3 gives us like
a constant and then we have minus 1 and we reduce the power by 1. and then
in the denominator we have minus 1 over x squared
so i wrote x to the minus 2 here but we could just write it as 1 over x squared
the point is that these are going to cancel right here
and we’re going to get a negative and a negative so that’ll be a positive 3 over
1. now x is going to infinity so this part right here is going to 0
so that’s 1 to some power so all this is 1 and these cancel with each other
00:33
and so it looks like we’re getting looks like we’re getting here
um minus three over two but um oh yeah i see what happened so
this minus three here right so it’s minus three
x the minus one third in here and so when i look at that right there i’m
getting three x to the minus two so that should be a positive there that
should just be a three so the minuses uh cancel the x squares cancel
so it looks like we’re just getting three over two
okay good so this limit right here is just equal to three over two
and the first thing we did is we noticed that this was one of the seven
indeterminate forms and we’re gonna see why in a second but this is infinity
00:34
minus infinity don’t think of that as zero it’s not in
fact we got three over two so this is infinity minus infinity so
what was the algebra that i did well i factored out an x squared here
and the reason why is because i have a square root
so i can take square root of the x squared and i can take square root of
the second factor here then i can pull an x out of these two
and then i can pull that x to the numerator to the denominator
so finally i get something which i can use the hopital’s rule for
i get zero over zero so using l’hopital’s rule
taking the derivative of the numerator taking the derivative of the denominator
and then now checking the form again x is going to infinity what’s happening
we’re gonna get zero so this is one to some power and these right here cancel
cancel and so this right here ends up with negative negative
00:35
also cancel so end up with 3 over 2. all right very good let’s look at
another one um actually um you know when we
look at these two right here we can see why infinity minus infinity is an
indeterminate form so we had um the example we just found was the minus
three over two and then the previous one we worked on was one half
and so now you can try to prove to somebody that one half is equal to
three over two you and and that might you know
a lot of people heard oh how one equals two or zero equals one
will try to surprise them one half is equal to
three halves watch watch my proof and i’ll prove to them
by showing them this limit this limit is one half as we showed before
00:36
and that is infinity minus infinity and that’s also equal to this limit
right here the limit as we go to infinity of x minus square root of x squared
minus 3x so that’s also infinity minus infinity
that’s the same indeterminate form and we found this limit right here to be
three halves so once you show them that this limit is one half
and once you show them that this limit is three halves then you can
try to convince them that one half is equal to three halves
they may not know that you cannot use equal sign with indeterminate form
this is the step right here that is incorrect as well as this equal sign
right here so this is the limit equals to an indeterminate form
don’t do that if you use indeterminate form with an equal sign
00:37
you can prove anything you want a contradiction all right so next example
evaluate using the habitats rule right so here we go
what indeterminate form do we have if any so we have e to the minus x this is
going to positive infinity so this is going to zero right here
and this is going to infinity square root of a large number is still a large
number this is growing larger and larger you
know what square root looks like it’s just going up
it’s just increasing right so this is zero
times infinity that’s our indeterminate form zero times infinity now what are we
gonna do if we have zero times infinity we’re going to try to use the hopital’s
rule but first we have to have indeterminate form 0 over 0
or infinity over infinity i’m going to have a negative x here so
00:38
i’m going to say this just e to the x now i have infinity over infinity is my
indeterminant form so now we can try to use the hopital’s rule [Music]
so let’s see here derivative of the square root of x
that’s minus one over two square roots of x derivative of e to the x
is just e to the x and let’s see does that help us or not
as x’s go to infinity this denominator is growing really large so the numerator
is going to zero and the denominator is going to um infinity
so let’s see if we can rewrite this actually make it a little bit more clear
this will be 1 over 2 square roots of x times e to the x
that’s a much easier way to think about it and the reason why
is because as x goes to infinity what is the denominator doing
it’s just getting larger and larger and larger there’s no subtraction or nothing
00:39
like that so one over something that’s growing
larger and larger and larger this is just zero so we found this limit right here
is zero so here’s one example where we have an
indeterminate form infinity zero times infinity
and we get as our limit the value of zero let’s look at another example
probably tell where i’m going with this let’s find this limit now
using the hopital’s rule okay so what indeterminate form do we have
so the x square is going to infinity as x is
growing larger and larger and larger x squared is growing larger and larger and
larger and one over four x squared [Music]
that’s going to zero sine of zero zero so we have
indeterminate form of zero times infinity now you might say infinity times 0
00:40
it doesn’t matter i could easily write this like this
and do the same limit it’s just that it’s not very clear here if i write it
like this it’s that we’re taking the sign of only this
it just looks prettier if you write it like this in either case you have the
same indeterminate form what is zero times infinity have you thought about that
zero times anything is zero now well zero times any number is zero
but zero times infinity well that’s an indeterminate form
why well our last example we got a limit with indeterminate form zero times
infinity what is this limit going to be all right so
one approach to find this limit right here is to
you know do the same as we did before take this right here the x squared
and divide by it so here we’re going to have sine of 1 over 4x squared [Music]
and if we have x squared in the numerator then we have
00:41
1 over x squared in the denominator so let’s see what we get when we try to
apply l’hopital’s rule actually let’s write our indeterminate form here
right so this is going to zero still this is going to infinity that’s going
to zero sine of zero that’s going the numerator is going to zero
in the limit of the denominator x is growing larger and larger so that
expression is going to zero so we have indeterminate form of zero
zero so now i’m going to try to apply the hopital’s rule here
and let’s let’s see what we get x is going to infinity
and let’s take the derivative of the numerator so derivative of sine is cosine
leave the inside alone times the derivative of that
now let’s look at that over here what’s 4x squared to the minus 1
what’s the derivative of that right that’s that’s the same thing as 1 over right
what’s the derivative so it’ll be minus 1 times 4x squared
00:42
minus 2 times what here 8x right so and we bring the minus 1 down we
leave this inside alone and then times the derivative and then
we reduce the power and then times the derivative here which will be
eight x okay so let’s write that as minus eight x over um
sixteen x to the fourth perhaps that’s one way to write
it and then here we have a um [Music] we have a minus two over
x squared uh x to the third okay so very good
so what is this limit here now as x just goes
to infinity this is still going to zero but cosine of zero is
00:43
one so that that part right there’s going to one
with a one-half here so perhaps we’ll write this out in one more step here
let’s write this out here just to make it clear
let’s write this out as cosine of one over four x squared
and then what do we have here a the minuses are going to cancel and so
here we have a 1 over 2x to the third and here we have a minus 2x to the third
a minus is cancelled [Music] okay so now maybe the limit might be a
little bit easier that’s still going to zero
its cosine of zero that’s going to one now the x
to the thirds cancel so i’m getting one over two
00:44
over another two so we get a one fourth out of all this
so there’s the limit right there’s one fourth
so this limit right here is one fourth and now let’s put together the last two
examples all right so hopefully everybody’s good that’s one-fourth
all right so now um we can say here [Music]
if we look at the example that we just did that limit was one-fourth of course
one-fourth is not zero but we found that limit right there to be zero
so in both of these cases we had indeterminate form zero times infinity
but as you can see if your limit has the indeterminate form zero times infinity
the limit might be any value at all and we worked out two cases there
okay so just because you have indeterminate form zero times
00:45
infinity don’t think you’re always going to get zero don’t think you’re going to
get you have to check and see each limit is different all right next part
[Music] okay so now we’re going to look at the indeterminate forms
infinity to the zero power zero to the zero power
and one to the infinity power and so that’s the
uh topic now that’s the indeterminate forms that we have now
are those three right there so these three are going to be handled um
all similarly so that’s why i broke them into three
buckets the first type right here is zero over zero infinity minus
infinity over infinity um you can use the habitat’s rule or you
know you can check to see these two types over here you want to
00:46
use some type of algebra to reduce it to these two types over here
now when i’m when we’re looking at these three types over here
we won’t be able to use some algebra always but what we’ll be able to do is
to use logarithms logarithms uh help out when you’re
solving when you’re working with problems with exponents
so this limit right here is said to have indeterminate form
zero infinity to the zero if the limit of the base function f of x is infinity
and the limit of the exponent function is zero
and similarly we have zero to the zero power
and we have one to the infinity power so those are the three types of
indeterminate forms and the um technique that we’re about to use
is the same for all three types is that we’re going to use this identity here
of f of x to the g of x is always equal to e that’s the number e [Music]
00:47
to the g of x times natural log of x natural log of f of x now where does
that come from so let’s you know see how easy it is to understand that
um here if i’m looking at um let’s say we have
two functions f of x to the g of x power and let’s call that function l of x
right so i’m going to take natural log of both sides
natural log of f of x to the g of x and here’s why natural logs are so useful
is because i can bring this g of x down this is g of x
times natural log of f of x so the natural log of l of x
00:48
is g of x times the natural log of f of x and now we can use e we can use f of x
to the g of x equals that’s our l and the l i can write it as e to the
natural log of l of x and we just saw that natural log of l to the x is this
so this will be e to the g of x natural log of f of x so here’s how we
handle the limits if we’re taking the limit say as x approaches a
of f of x here to the g of x if i want to take the limit of this
i can take the limit of this instead so i’ll take the limit as we approach
00:49
a of all this uh g of x times natural log of f of x
now this may look more complicated because now we have e to
all of this but actually remember e is a continuous function
so we can do e and then bring the limit up to the exponent
so that’s x approaches a of g of x natural log of f of x
and just to make sure it’s an exponent i’ll emphasize that by writing it like
that right so e to this power right here so if i want to take this limit of this
f of x to a g of x it’s the same thing as e to the limit of this guy g of x
times natural log of x so you might ask well this limit
doesn’t look as hard but actually it is this limit is going to be much easier
because it’s multiplication here and as we’ve seen in some examples
we can work with multiplication by changing it to
00:50
to division and doing some algebraic manipulation
we’re going to be able to get a l’hopital’s rule
working here by looking at this whereas we won’t be able to get lobsters rule
working directly if we use this right here so that’s the kind of the idea there
so with these indeterminate forms infinity to the zero
zero to the infinity and one to the infinity
we can apply this identity here and we know the exponential function is
continuous so if we want to take this limit and that’s the
given limit that we’re interested in then we can take the limit of g e to the
g of x times natural log of x f of x and because e is a continuous function
we can bring that limit up to the numerator so we need to find the limit
of that expression there all right then possibly apply the habitat’s rule
so let’s see three examples one of each of the
00:51
indeterminate forms and see how this all works okay so up first is
sine x to the x here we go so we have the limit as we approach as zero
to the right of sine x over x now is this indeterminate form
so sine x as x goes to zero that’s going to zero
and the exponents going to zero so this is going to zero to the zero power
so that’s one of these seven indeterminate forms and so i’m going to write
indeterminate form zero to the zero so that’s my first step now to
to find this limit i’m first going to go find this limit
00:52
here we’re going to approach 0 from the right
and it’s going to be the exponent times natural log of the base function
i’m going to first try to find this limit so this is just the exponent here and
the natural log of the base function here
so i’m going to see if this limit exists
first now when i’m looking at this limit i’ve got
zero here and what do we get over here for this part right here think of that
part right there so x is going to zero so sine is sine of zero
so that’s going to zero and the natural log of zero
that’s going to be infinity right remember what the natural log looks like
as you’re approaching zero here this is going down to zero
uh to to infinity there so we’re going to have indeterminate form here
i’ll put it down here indeterminate form of 0 times infinity
and but we’re going to break this up and do some algebra here
00:53
we’re just going to move that x down to the denominator so natural log of sine x
over 1 over x right it’s an x in the numerator
so it’s 1 over x in the denominator now what’s happening so now we’re
getting infinity over infinity so i’m going to write indeterminate form
infinity over infinity here and now finally we have one of the forms we can
apply the hopital’s rule to we have one numerator we have one
denominator it’s indeterminate form infinity over infinity so now i’m going
to try to apply the hopital’s rule so we have x approaches zero and what’s
the derivative of the side ln the sine x it’s one over sine
times the derivative of sine which is cosine
and then derivative of one over x is minus one over x squared
all right good now are we getting anything that we can work with
00:54
maybe let’s manipulate it a little bit let’s say we have tangent right we have
cosine over sine which is cotangent and i’m going to
move that to the denominator and i’m going to move the minus x squared to the
numerator because right now we have complex
fractions so i can write this as minus x squared over tangent
just by simplifying algebraically now what’s happening as x is going to zero
we got zero over tangent of zero that’s zero so we’ve got zero over zero again
so now we have another indeterminant form i’ll put that here
indeterminate form zero over zero so let’s try the hopital’s rule
limit of the numerator is minus 2x limit of uh sorry derivative of minus x
squared is minus 2x derivative of tangent is uh secant squared
and now let’s see if we can find the limit we got
00:55
0 over secant squared of 0 that’s 1. so we’re getting 0 over 1 so we get zero
so we got zero out for this limit let’s go back and make sure that
we understand everything that’s happening here
to find this limit though so here we go we have the limit
as we approach zero of sine x to the x that’ll be equal to e to the limit
as x approaches zero of this guy right here x natural log of sine x
and we found this limit we weren’t sure if this limit existed
00:56
we found this limit if we trace it all the way through here we get
zero so this limit is zero up here so we’re getting e
to the zero and e to the zero is just one
so all that work and we got the limit as x approaches zero
sine x to the x is one so this was a computation here or calculation here
where we were given this limit and we first found this limit here it’s
going to be the exponent x times the natural log of the base
and we go and find that limit using l’hopital’s rule hopefully
we found it now it may happen that this limit doesn’t exist right here
so if you work it and you realize oh that limit doesn’t exist
well then this approach doesn’t work here but
we found that limit to exist it’s in fact zero
so the limit of this right here is going to be e to that limit right there
00:57
which we found to be zero so we got one here
all right so let’s do another example now let’s look at x to the one over x and
let’s see here what we got going on here now all right our our base here is x
that’s going to infinity and our exponent one over x that’s going to zero so it
looks like we have infinity to the zero power
and that’s one of our three types here that we’re working on
so i’m going to say this has indeterminate form of infinity to the zero
that’s our indeterminate form so in order when you get one of these three
types here we’re going to um first compute the limit
00:58
we take the exponent 1 over x times natural log of the base which is just x
i’m going to go find this limit first so what is this limit this is
natural log of infinity i’m sorry that’s going to infinity so this is
natural log is blowing up so that’s infinity over infinity
so we have i’ll put it down here we have indeterminate form
infinity over infinity so right away we can
try to use the hopper tells rule to find this limit here
so limit of the numerator just natural log of x so we get 1 over x
and limit of the denominator which is just x so this is just 1.
so this limit is 0 here all right so when we try to find this
limit we notice it has indeterminate form so i’m going to take this limit first
the exponent times natural log of the base function
00:59
and i find that limit if i can we did and so now we can come back and find
this limit here the limit as we approach positive infinity
x to the one over x is e to the limit of this one right here
so x is approaching positive infinity of one over x
natural log of x and we found this limit of this one right here
to be zero so we have e to zero which is one so this limit here
is also one this limit right here is also one okay so next example
this one looks really fun let’s see what we can do here we have um sine
01:00
4x and to the cotangent x power this one looks most exciting so far
but we’ll see one that looks even better here in a minute
but anyways what is the indeterminate form that we think we have here
we’re approaching zero from the right so we have 1 plus sine 4x
and so this is going to zero so that’s going to one plus zero so that’s going
to one and what about cotangent x so if you look at the graph of cotangent
x and you realize that this is some infinity right here so
we’re going to have indeterminant form 1 to the infinity and that’s one of the
three types there that’s our last type and so what we’re going to do is we’re
going to first find the limit as we approach zero from the right we have our
exponent here oops sorry we have our exponent here cotangent x times
01:01
natural log of the base function 1 plus sine 4x
so in order to find this limit here i first want to find this limit here how
do i know that because of the indeterminate form
if you get this indeterminate form here then try to do the limit
of the exponent times natural log the base so let’s see if we can find this
limit here now this is going to be infinity and that’s going to be 0
so we’re going to have another indeterminate form here
of the type 0 to 0 times infinity but i’m just going to write it out here
this is natural log of 1 plus sine 4x over let’s just say tangent x
so as x goes to zero tangent x is going to zero
01:02
and this is going to be sine four x x is going to zero so that’s going to zero
that’s going to one and natural log of that’s going to zero also
so now i’ll put here indeterminate form zero over zero all right so perfect
so now we can try to use the hopital’s rule so here we go
limit as x approaches zero from the right
and now let’s take the derivative of the numerator
which will be one over one plus sine four x times
the derivative of the inside part here derivative of one is zero
so here we have cosine four x and then times the derivative of the
angle four x so four i’ll just put the four over here
and now all over the derivative of tangent is secant squared
01:03
all right good so again we have one to the infinity
so i take the limit of this and just you know changing cotangent is one over
tangent and then indeterminate form so take the derivative derivative of the
numerator over the derivative of the denominator
now let’s see what indeterminate form we have
so in the denominator we’re getting zero going to zero
secant squared of zero right that’s one so we don’t have indeterminate form here
on the on this part here so we have what um
one over one so for all this here we’re getting four
no more indeterminate forms all right so the limit of this expression right here
this function right here is four so now we’re ready to find the
limit of the original limit here that we found here so here we go the limit
as we approach zero from the right of one plus sine four x
01:04
to the cotangent x power is e to the limit as we approach zero from the right
of the exponent function times the natural log of the base function
one plus sign ln of one plus sine four x we found this limit right here to be
four so this is e to the fourth and there we are
so there’s one of each of those types now um i
only found one limit of each of those three types
infinity to zero zero to the zero one to the infinity
in the homework you’ll see more of those
types and you’ll see different values so here’s one example where
the limit of an indeterminate form of indeterminate type indeterminate form type
one to the infinity i got four so you should it should be easy to find a
01:05
limit of another function that has indeterminate form
and you get a different value than four you don’t always get four
all right let’s look at some more examples now okay so on this example here
i want to show you how to make a change variables doing a limit now you don’t
always need to make a change of variables
in fact when you’re working out textbook problems
it’s rare but you know sometimes you’ll see a function
and it just looks better if you make a substitution
and so i want to show you how to do that in case you haven’t seen that yet
and we’re also going to use the hopital’s rule we do this
so when i’m looking at this function right here
01:06
i see a 1 over x i see another 1 over x and i see an expression that has 1 over
x to the third so i’m thinking to myself can i simplify that
by making a change of variables and so we’re going to make a change of
variables here i’m going to say that u is equal to one over x
and when you’re making a change of variables it’s
it’s a sometimes c you know try to make a change and see if it helps you
if it doesn’t maybe you may need to make a different change of variables
and see if that helps you of course you don’t always need to make a change of
variables but for this problem i’m going to try
now as soon as you come up with a new variable i’m coming up with variable u
here we need to change the limit so what does u go to as
x goes to what does x go to positive infinity
so as x goes to positive infinity what does u go to
right so as x is getting larger and larger and larger one over that that’s
01:07
going to zero so when i change my limit instead of writing
x goes to positive infinity now i need to say u goes to u goes to zero
okay so we’re going to write this out as
the limit as x goes to positive infinity and we have x to the third
times all of this right here sine of 1 over x minus 1 over x and then
plus 1 6 x to the third now to be equal to now this is everything with x’s in it
when i make my change of variable it should only have u’s in it
so this will be the limit as u goes to zero
instead of x goes to infinity we’ll have u goes to zero now
now what is x to the third i need to rewrite that with
01:08
u’s now obviously that’s a u and that’s a u but we need to look at
that and we need to look at that and we need to write them in terms of u
so this is what u is and so i’ll come down here and say
uh x is one over u then right it’s multiplying right so x is one over u so
x to the third is one to the third over u to the third
right so and then if i want to put a a six in here right so
we can do that thing too all right so here we go x to the third is just one
over u to the third here so that’s one over u to the third here
and so what are we gonna get here we’re gonna get sine of u minus u plus
one sixth and then we have one over x to the third which is just u
01:09
right this x to the third here is just one over u to the third
uh so this will be to the third here yeah and then this x this x to the third
right that’s one over u to the third so i’ll be using the third down here
this x to the third is u to the third down here
all right so there we went from all x’s to all u’s
now right so this x to the third here is one over u to the third so this x to
the third is just one over u to the third
right so i just want to put it down here all right so what’s happening now we’ve
made a change of variable it looks a little bit easier now
what and we have indeterminate form don’t we this is going to zero so that’s
going to zero a zero a zero so the whole numerator is
01:10
going to zero and the denominator is going to zero
so this is indeterminate form zero over zero
so i need to take some limits uh sorry i need to take some derivatives right the
hopital’s rule now we can see why i was motivated to make a change of variable
when i start taking derivatives they’re a lot easier
if we start taking derivatives of all this
it’s not going to get easier it’s going to be pretty easy to take derivatives
here derivatives of sine is cosine minus one and then that’s a six so
that’ll be what uh one half u squared [Music] all over 3u squared
all right so now let’s try to that’s a cosine so now let’s try to find our limit
so u is going to zero so that’s going to one
01:11
one minus one so that’s all going to zero and he’s going to zero so that’s going
to zero zero so we’re getting indeterminate form of zero again
zero of zero so let’s try the hopital’s rule again
remember you can plot the hopital’s rule twice
can we work can we apply it three times well you can apply it as many times as
you want cos as long as the hypothesis is satisfied each time you use it
all right cosine of of my of u is minus sine u and then derivative here’s zero
and then here we have a u and then here we have a six u
here we go now we went from sine cosine sine so it seems like we’re kind of
repeating ourselves there let’s see what happens here we got 0 and we got 0
we’re going to zero so it looks like we have indeterminate form again
01:12
zero over zero so the question is now after doing it three times
you may ask yourself am i making any progress
or am i stuck in some kind of loop and the habitat’s rule is never going to work
well we’re getting simpler over here aren’t we
i mean we went from a cubic squared first power right
so let’s try it one more time [Music] use going to zero derivative of minus sign
is minus cosine plus one and then over six and let’s see what we get now we’re
going to get um so let’s see here um [Music]
01:13
i just realized that this problem has an x to the fifth in it
and that wrote i wrote here x to the third so we need to go back and fix the
problem sometimes people make mistakes like that i’m not immune
there’s a fifth and so what is x to the fifth here it’s just going to be
u to the fifth so this is the fifth here so luckily i’m using
erase marker here so we’ll be able to fix this very quickly
so this will be u to the fifth now we still have indeterminate form but
this is a little different here this will be
five u to the fourth so five u to the fourth here
we still have indeterminant form here so this will be 20 u to the third
and after we use it a couple times we realize the power is still
01:14
decreasing here so this will be 20 u to the third we still have indeterminant
form so now this will be 60 u squared so 60 u squared and
used going to zero here so now i’m getting minus one
plus one and now we’re still getting zero so now we’re still getting
indeterminate form here so now after we do this so many times
you really start to wonder you know are we making any progress but
i think so because we went from sixth or fifth sorry six five four three two
all right so let’s see what we get next use going to zero since we have a
minus cosine so the derivative will be positive sine and the derivative of 1 is
0 and then here we’re getting 120 u now because we already know the limit of
01:15
sine u over u as u goes to zero we already know that limit is one
or if you want you could apply l’hopital’s rule one more time
so in either case you could get 101 over
and there’s the value of the limit there so even though i had to take um
l’hopital’s rule several times we kept getting indeterminate form
we eventually got it to work now if i had stayed with my original
variable x i wouldn’t have wanted to take all those derivatives all those
rational expressions um you know like 1 over x with all those negative exponents
so this was you know we had to do it several times but each of the
derivatives were very easy so those derivatives are very easy and then we just
substitute in and then find the limit there you know
01:16
even if you want to do l’hopital’s rule one more time
derivative of sine is cosine and then we have 120.
cosine of 0 is 1 we get 1 over 120. so all the derivatives are easy because
we made a change of variable so there’s a real good media example where
you apply the habitat’s rule several times and you
make a change of variable first all right so now let’s look at
our last example here so here we go we’re looking at the limit
as x goes to positive infinity and we’re looking at x plus sine x
plus x minus cosine x okay so when i’m first looking at this
i’m thinking to myself that this x is dominating right here so sine x
01:17
what is that going to right so remember sine x it just keeps repeating over and
over again right so you’re going to hit one you’re going to hit minus one
over and over again right so as you go out to infinity right here
right you’re you know this limit right here doesn’t exist
we’re not approaching a single height as we’re going to infinity
because it just keeps oscillating back and forth back and forth
however the x here is dominating as x goes to infinity like 10
20 30 40. while the sign is always at most
one right so that’d be like 10 million plus one
a hundred million plus one a hundred trillion plus one
so the x is dominating here so this looks like infinity
and the same thing with this guy right here now you might be wondering about
the minus sign but remember cosine hits -1 doesn’t it
right cosine goes like that it hits -1 down here so for some x values
01:18
we’re going to get you know like 100 million and then minus something between
0 and 1 is going to be insignificant so this x is going to dominate and this x
is going to dominate so it looks like we have indeterminate form infinity
over infinity so we’re going to try to apply the habitat’s rule so here we go
[Music] equals derivative of the x is one and then the derivative of sine x is
cosine x we have one minus and then we have a negative sign all right
and now what is this limit here well this limit right here actually does
not exist so l’hopital’s rule fails us in this example
this limit doesn’t exist now just because this limit doesn’t exist
01:19
doesn’t mean this limit doesn’t exist this is not an equal sign here
this equal sign that we we were using whenever we use the hopital’s rule
this was always a try and see i’m going to try to see if this limit exists
if this limit exists then they were equal to each other
but this limit doesn’t exist so the habitat’s rule fails l’hopital’s rule fails
to apply [Music] lava tells will doesn’t fail it’s just saying i can’t use it
on this example now how can we find this limit
well we can try an old trick that we were using before
and we can divide by the highest power so the x’s are dominating here so i’m
going to divide everything by an x so i’m going to say x over x
01:20
plus sine x over x and then x over x and then minus cosine over x
so this is going to one and this is going to zero now why is this going to zero
well x is going to infinity it’s getting larger and larger and larger
and this is at most one so this is this is going to zero here
and this is one minus zero this limit is one in fact
so if you were to go graph this you would see
a horizontal isotope of y equals one you would see that long term behavior okay
[Music] and that’s it part six is for some exercises let’s look at them
01:21
okay here’s some exercises for us to look at um
as usual in this series i give you some exercises for you to try
and when you try these exercises here see what you get and make some comments
and to the video below and tell me if you want me to work on
any of them or any of them extra hard any of them easy
but there’s some problems right there to work on for the habitats rule
several of them um and then here is exercise two where it tries to help you
understand the hobital’s rule and when you can use it
um and then on exercise three it’s a little bit of a puzzle you have to find
the value of a where that limit is finite non-zero
um and then we have some more puzzles here find find some parameters a and b
01:22
and um exercise seven there looks interesting um
maybe that limit was in l’hopital’s rule book
uh in the habitat’s book um and then eight there and then um now i want to say
uh thank you for watching um the next video is um indefinite integrals
what is an antiderivative and i look forward to seeing you in that video
button and subscribe to my channel now i want to turn it over to you
math can be difficult because it requires time and energy to become
skills i want you to tell everyone what you do
to succeed in your studies either way let us know what you think in the comments |
Solve the equation \dfrac{1}{x^2}+\dfrac{1}{(x+2)^2}=\dfrac{10}{9}
#### Question
Solve the equation
\dfrac{1}{x^2}+\dfrac{1}{(x+2)^2}=\dfrac{10}{9}
Collected in the board: Algebraic equation
Steven Zheng posted 1 year ago
Let
a=\dfrac{1}{x}
(1)
b=\dfrac{1}{x+2}
(2)
Substitute to original equation to get rid of fraction. We get a new equation.
a^2+b^2=\dfrac{10}{9}
(3)
In the meantime, take reciprocal of (1) and (2)
\dfrac{1}{a}=x
\dfrac{1}{b}=x+2
Subtraction of them gives
\dfrac{1}{a}-\dfrac{1}{b} = -2
Simplifying the equation gives
a-b=2ab
(4)
From (3), the following equation is valid after subtraction of 2ab from both sides
a^2+b^2-2ab=\dfrac{10}{9} -2ab
(a-b)^2 = \dfrac{10}{9} -2ab
Replacing ab with (4), we get a quadratic equation in terms of a-b
(a-b)^2 = \dfrac{10}{9} -(a-b)
Let p = a-b. The equation is simplified as
p^2+p- \dfrac{10}{9}=0
Solving the quadratic equation, we get two solutions
p_1=-\dfrac{5}{3}, p_2=\dfrac{2}{3}
When p = -\dfrac{5}{3}, let's add 2ab in both sides of (3)
a^2+b^2+2ab=\dfrac{10}{9} +2ab
(a+b)^2=\dfrac{10}{9} +2ab
Replace ab with the result of (4)
(a+b)^2=\dfrac{10}{9} +(a-b)
(5)
Since a-b=p= -\dfrac{5}{3}
(a+b)^2=\dfrac{10}{9} -\dfrac{5}{3}=-\dfrac{5}{9}<0
which is impossible. Therefore, p = -\dfrac{5}{3} is cancelled
When p = \dfrac{2}{3}, that is a-b = \dfrac{2}{3} substitute to (5)
(a+b)^2=\dfrac{10}{9} + \dfrac{2}{3} =\dfrac{16}{9}
Taking square root of both sides gives
a+b = \pm\dfrac{4}{3}
Plus a-b = \dfrac{2}{3}, we get two equation systems.
\begin{cases} a + b =\dfrac{4}{3} \\ a - b = \dfrac{2}{3} \end{cases}
and
\begin{cases} a + b =-\dfrac{4}{3} \\ a - b = \dfrac{2}{3} \end{cases}
Solving the first equation system, we get a = 1, and then x = 1
Solving the second equation system, we get a = -\dfrac{1}{3} , and then x =-3
The graph of y=\dfrac{1}{x^2}+\dfrac{1}{(x+2)^2}-\dfrac{10}{9} illustrates the points when y =0
Steven Zheng posted 1 year ago
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# Finding a cubic equation with transformed roots using substitution method
So the question goes like this: The roots of the cubic equation $$2z^3+5z^2-3z-2$$ are $$\alpha, \beta, \gamma$$
Find the cubic equation with roots $$2\alpha + 1, 2\beta + 1, 2\gamma + 1$$
The original way I solved this is by first finding the coefficients - e.g $$\alpha + \beta + \gamma = \frac{b}{a}$$ and so on, but it turns out this way takes quite a long time. My textbook states that there is another method: the substitution method. The method involves a new variable $$w = 2z + 1$$. We write $$z$$ in terms of $$w$$ and substitute into the original equations, so since $$z = \frac{w-1}{2}$$, we do the following substitutions $$2(\frac{w-1}{2})^3 + 5(\frac{w-1}{2})^2 + 3(\frac{w-1}{2}) -2 = 0$$ The explanation in the book is as follows: This is a transformation of $$z$$ in the same way as the new roots are a transformation of the original $$z$$ roots. I don't get this part. How are these new roots (e.g $$2\alpha + 1$$) related in any way to the $$z$$ variable?
## 2 Answers
Define a new polynomial, given by $$Q(w) = P\left(\frac{z-1}{2}\right)$$. Then $$Q(w)$$ has roots $$2 \alpha+1, 2 \beta + 1, 2 \gamma + 1$$, because subbing these into $$Q$$ outputs $$P(\alpha), P(\beta), P(\gamma)$$, all of which are $$0$$.
Just write the polynomial explicitly
$$p(z) = 2(z-\alpha)(z-\beta)(z-\gamma)$$
Now, you see
$$p\left(\frac{w-1}{2}\right) = 2(\frac{w-1}{2}-\alpha)(\frac{w-1}{2}-\beta)(\frac{w-1}{2}-\gamma)$$ $$= \frac 14(w-(2\alpha + 1))(w-(2\beta + 1))(w-(2\gamma + 1))$$ |
Metric Conversion Worksheet | Problems & Solutions
# Metric Conversion Worksheet
Metric Conversion Worksheet
• Page 1
1.
How many millimeters are there in 4 centimeters?
a. 40 mm b. 0.4 mm c. 4 mm d. 400 mm
#### Solution:
1 cm = 10 mm
4 cm = 4 × 10 mm
[Convert centimeters into millimeters.]
= 40 mm
[Simplify.]
40 millimeters are there in 4 centimeters.
2.
A family consumed 15 kg 500 g of flour in the month of December, 40 kg 450 g in the month of January and 36 kg 750 g in the month of February. What was their total consumption?
a. 94.4 kg b. 98.7 kg c. 93.8 kg d. 92.7 kg
#### Solution:
Total quantity of flour consumed for three months = 15 kg 500 g + 40 kg 450 g + 36 kg 750 g
= (15 kg + 500 g) + (40 kg + 450 g) + (36 kg + 750 g)
= 15 kg + 40 kg + 36 kg + 500 g + 450 g + 750 g
[Write the like terms together.]
= 91 kg + 1700 g
= 91 kg + (1700 × 1) g = 91 kg + (1700 × 1 / 1000) kg
[Substitute 1 g = 1 / 1000 kg.]
= 91 kg + 1.7 kg
[Work inside the grouping symbols.]
= 92.7 kg
3.
The distance to be covered in a race was 55 km 75 m. When Henry was at 23 km 44 m, he had to stop due to sprained foot. How far was he from the finishing line then?
a. 32 km 32 m b. 32 km 31 m c. 33 km 31 m d. None of the above
#### Solution:
The distance to be run in the race is 55 km 75 m.
The distance covered by the Henry is 23 km 44 m.
Distance left to reach the finishing line = Total distance - distance covered
= 55 km 75 m - 23 km 44 m
[Substitute the values.]
km m 55 075 23 044 32 31
Henry is 32 km 31 m far from the finishing line.
4.
Convert 39.5 km 59 m into meters.
a. 98.5 m b. 395590 m c. 39559 m d. 395900 m
#### Solution:
1km = 1000 m
[Convert km to m.]
39.5 km 59 m
[Original data]
= 39.5 km + 59 m = (39.5 × 1) km + 59 m
[Split the data.]
= (39.5 × 1000) m + 59 m
[Substitute 1 km = 1000 m.]
= 39500 m + 59 m
[Multiply]
= 39559 m
5.
Convert 20 liters into kiloliters.
a. 0.2 kiloliter b. 20 kiloliter c. 0.02 kiloliter d. 2 kiloliter
#### Solution:
1 liter = 1 / 1000 kiloliter
[Convert liter to kiloliter.]
20 liters = (1 × 20) liters = 1×20 / 1000 kiloliter
[Substitute 1 kiloliter = 1 / 1000 liter.]
20 liters = 2 / 100 = 0.02 kiloliter
[Simplify]
20 liters is 0.02 kiloliter.
6.
Convert 23 kilometers into meters.
a. 23 m b. 23000 m c. 2300 m d. 230 m
#### Solution:
1 km = 1000 meters
[Convert km to meters.]
23 kilometers = 23 × 1 km = 23 × 1000 m
[Substitute 1 km = 1000 meters.]
= 23000 m
[Multiply.]
7.
Convert 900 meters into kilometers.
a. 9 kilometers b. 90 kilometers c. 0.9 kilometers d. 0.09 kilometers
#### Solution:
1 meter = 1 / 1000 kilometers
[Convert meter to kilometer.]
900 meters = (900 × 1) m = 900×1 / 1000 kilometers
[Substitute 1 meter = 1 / 1000 kilometers.]
= 0.9 kilometers
[Simplify.]
8.
Convert 13 km 40 m into cm.
a. 130400 cm b. 13040000 cm c. 1304000 cm d. 13040 cm
#### Solution:
1 km = 1000 m and 1 m = 100 cm
[Convert km to m and m to cm.]
13 km 40 m
[Original data.]
= 13 km + 40 m = (13 × 1) km + 40 m
[Split the data.]
= (13 × 1000) m + 40 m
[Substitute 1 km = 1000 m.]
= (13000 + 40) m
[Multiply.]
= 13040 m
= (13040 × 1) m = (13040 × 100) cm
[Substitute 1 m = 100 cm.]
= 1304000 cm
[Multiply.]
9.
Convert 31 liters 500 milliliters into milliliters.
a. 31,005 mL b. 31,050 mL c. 315,000 mL d. 31,500 mL
#### Solution:
31 liters 500 milliliters = 31 L + 500 mL
31 liters = 31,000 milliliters
[1 L = 1,000 mL.]
31 L + 500 mL = 31,000 mL + 500 mL = 31, 500 mL
10.
Find the sum of 64 km 25 m and 43 km 3 m.
a. 106 km 28 m b. 107 km 28 m c. 106 km 29 m d. 107 km 29 m
#### Solution:
64 km 25 m + 43 km 3 m
[Original expression.]
The sum of given values is 107 km 28 m. |
An Introduction to Prime Numbers
The study of Prime numbers forms the basis of Number Theory.
A natural number is said to be Prime if it can be evenly divided by one and itself, for example: 2, 3, 5, 7, 11.The number 9 is not prime, as it can be divided by 3 other than 1 and 9. Prime numbers have only two distinct divisors, 1 and itself. The number which is not Prime is called Composite number.
There is no set formula to check the primality of the number, other than the definition itself. The most common algorithm to identify a prime number, is to start dividing the number by all odd numbers from 2(include it as an exception) up to its square root; if any divisor with zero remainder is found in the process then the number is Composite else Prime.
E.g. To check whether 53 is prime or not.
We’ll start dividing 53 with all odd numbers including two right up to the square root of 53 which is approximately 7; that is we’ll divide 53 by 2,3,5,7 and conclude 53 is prime.
The Greek Mathematician Eratosthenes has given an algorithm for finding Primes in a given range, called the Sieve of Eratosthenes. He used the elimination process, remove all the multiples of each number starting from 2.
To find all primes between 1 and 100.List all the numbers.
First eliminate all multiples of 2 i.e. 4,6,8,10,12,14 so on. Basically it eliminates all even numbers except 2.
Next eliminate all multiples of 3 i.e. 9,15,21 27 so on from the remaining numbers. Similarly multiples of 5 and 7 are eliminated.
The list of primes between 1 and 100 is
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
It is to be noted that, the natural number 1 is not Prime and 2 is the only even number which is Prime. Every even integer greater than 2 can be written as the sum of two Primes, states The Goldbach’s Conjecture, but has not yet been proved.
The fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number greater than 1 can be written as a unique product of prime numbers. It is used to find the Greatest Common divisor (GCD) and the Least Common Multiple (LCM) in Elementary Arithmetic.
Prime numbers play an important role in data security in today’s information age. When we multiply two Prime numbers we get a composite number which is divisible by only those two Primes other than one and itself, which is quite obvious.
E.g. 3*5 = 15; where 3 &5 are Primes and the product 15 is composite
with factors 1,3,5 and 15.
The concept of product of two Prime numbers is a composite number with only two factors other than one and itself forms the basis of modern cryptography. The encryption and decryption of data, transmitted over the internet is done using the above concept. To find the product of two Primes is easy but the reverse that is, given a number and finding its two Prime factors is reasonably tedious even by fast computers, especially if the number is few hundred digits long, it may take couple of computing years.
Euclid has proved that the set of prime numbers is infinite. The pursuit for the largest prime number has always been a craze and fascination among mathematicians across the globe. The Largest Prime number till date is the 44th Mersenne Prime, 232,582,657-1. It is 9,808,358 digits long, discovered by Dr. Curtis Cooper and Dr. Steven Boone’s of Central Missouri State University, in Sept 2006. The Electronic Frontier Foundation has kept US\$100,000 prize money for the discovery of a Prime with at least 10 million digits for which the quest is on. |
# 8.3 Probability Applications of Counting Principles
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1 8. Probability Applications of Counting Principles In this section, we will see how we can apply the counting principles from the previous two sections in solving probability problems. Many of the probability problems involving dependent events that were solved earlier by using tree diagrams can also be solved by using permutations and combinations. Permutations and combinations are especially helpful when the numbers involved are large. To compare the method of using permutations and combinations with the method of tree diagrams used in Section 7.5, the first example repeats Example 8 from that section. (It is a good idea to review that example from section 7.5 before/after going through the following example.) Example. The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: in Chicago, in Los Angeles, and in New York. Due to a lack of inspectors, they decide to inspect two plants selected at random, one this month and one next month, with each plant equally likely to be selected, but no plant selected twice. What is the probability that Chicago plant and Los Angeles plant are selected? Solution. We note that, although the plants are selected one at a time, with one labeled as the first plant and the other as the second, the probability that Chicago plant and Los Angeles plant are selected should not depend upon the order in which the plants are selected (the outcomes, Chicago followed by Los Angeles, and Los Angeles followed by Chicago are both contained in the required event). So, we may use combinations. The number of ways to select Chicago plant out of Chicago plants and Los Angeles plant out of Los Angeles plants is ( ) ( ) = = 6. The number of ways to select any plants out of 6 is ( ) 6 = 6!!! = 6 5!! = 6 5 = 5. Thus, the probability that Chicago plant and Los Angeles plant are selected is This agrees with the answer found earlier. 6 5 = 5. Example. From a group of nurses, are to be selected to present a list of grievances to management. (a) In how many ways can this be done? Solution. nurses can be selected from a group of in combinations, since the group of is an unordered set.) ( ) =! 0 9 8! = = 8!! 8! There are 75 ways to choose nurses from. ( ) 0 9 ways. = 75. (We use Fall 00 Page Penn State University
3 Solution. The container will not be shipped if the sample of engines contains at least defective engine, that is, or defective engines (note that the container contains only defective engines). If P ( defective) represents that probability of exactly defective engine in the sample, then There are P (not shipping) = P ( defective) + P ( defective). ( ) ways to choose the engines for testing: ( ) ( ) =! 9!! = 0. There are ways of choosing defective engine from the in the container, and for each ( ) 0 of these ways there are ways of choosing good engines from among the 0 good engines in the container. By the multiplication principle, there are ( ) ( ) 0 =!!! 0! = 5 = 90 8!! ways to choose a sample of engines containing defective engine. Thus, ( ) There are ( ) 0 container, and P ( defective) = 90 0 = 9. ways of choosing defective engines from the defective engines in the there are ( ) ways of choosing good engine from among the 0 good engines. Thus ( ) 0 = 0 = 0 ways of choosing a sample of engines containing defective engines. So, Finally, P ( defective) = 0 0 =. P (not shipping) = P ( defective) + P ( defective) = ( ) ( ) 0 ( ) + ( ) ( ) 0 ) ( = 9 + = Fall 00 Page Penn State University
4 Remark. Observe that in Example, the complement of finding or defective engines is finding 0 defective engines. Then instead of finding the sum P ( defective) + P ( defective), the result in Example could be found as P (0 defective). P (not shipping) = P (0 defective) = = 0 0 ( ) ( ) 0 0 ( ) = 0 0 = Example. In a common form of the card game poker, a hand of 5 cards is dealt to each player from a deck of 5 cards. There are a total of ( ) 5 = 5! =, 598, ! 5! such hands possible. Find the probability of getting each of the following hands. (a) A hand containing only hearts, called a heart flush. Solution. There are hearts, and 5 = 9 other cards in a deck. Thus there are ( ) ( ) 9 =! 0 9 = = ! 5! 5 different hands containing only hearts. Hence, the probability of a heart flush is ( ) ( ) P (heart flush) = ( ) = 5, 598, (b) A flush of any suit (5 cards of the same suit). Solution. There are suits in a deck, so P (flush) = P (heart flush) = (c) A full house of aces and eights ( aces and eights). Fall 00 Page Penn State University
5 ( ) Solution. There are ways to choose aces from among the in the deck, and ( ) ways to choose eights out of the in the deck. Thus, ( ) ( ) ( ) 0 P ( aces, eights) = ( ) = 6 5, 598, (d) Any full house ( cards of one value, of another). Solution. There are values in a deck King, Queen, Jack, 0, 9, 8, 7, 6, 5,, (,, ) Ace, and cards of each value. Thus, there are choices for the first value, and ways to choose cards from among the cards that have that value. This leaves choices for the second value (order is important here, since a full house of aces, and eights is not the same as a full house of eights, ( and ) aces). From the cards that have the second value, cards can be chosen in ways. The probability of any full house is then ( ) ( ) P (full house) = 0.00., 598, 960 Example 5. A music teacher has violin pupils, Fred, Carl, and Helen. For a recital, the teacher selects a first violinist and a second violinist. The third pupil will play with the others, but not solo. If the teacher selects randomly, what is the probability that Helen is first violinist, Carl is second violinist, and Fred does not solo? Solution. We use permutations to find the number of arrangements in the sample space. P (, ) =! = 6 (We can think of this as filling the positions of the first violin, second violin, and no solo.) The 6 arrangements are equally likely, since the teacher will select randomly. Now there is only one arrangement where Helen is first violinist, Card is second violinist, and Fred does not solo. Thus, the required probability is 6. Example 6. Suppose a group of n people is in a room. Find the probability that at least of the people have the same birthday. Solution. Same birthday refers to the month and the day, not necessarily the same year. Also, we ignore leap years, and assume that each day in the year is equally likely as a birthday. To see how to proceed, we look at the case in which n = 5 and find the probability Fall 00 Page 5 Penn State University
6 that no people from among the 5 people have the same birthday. There are 65 different birthdays possible for the first of the 5 people, 6 for the second (so that the people have different birthdays), 6 for the third, and so on. The number of ways that 5 people can have different birthdays is thus the number of permutations of 65 days taken 5 at a time or P (65, 5) = The number of ways that 5 people can have the same or different birthdays is = (65) 5. Finally, the probability that none of the 5 people have the same birthday is P (65, 5) 65 5 = Thus, the probability that at least of the 5 people do have the same birthday is = Now this result can be extended to more than 5 people. Generalizing, the probability that no people among n people have the same birthday is P (65, n) 65 n. Therefore, the probability that at least of the n people do have the same birthday is P (65, n) 65 n. The following table shows this probability for various values of n. Number of People, Probability That Two n Have the Same Birthday > 65 The probability that people among have the same birthday is 0.507, a little over half. This is quite surprising! Fall 00 Page 6 Penn State University
7 Example 7. Ray and Nate are arranging a row of fruit at random on a table. They have 5 apples, 6 oranges, and 7 lemons. What is the probability that all fruit of the same kind are together? Solution. Method : Ray can t tell individual pieces of fruit of the same kind apart. All apples look the same to him, as do all oranges and all lemons. So, in the denominator of the probability, he calculates the number of distinguishable ways to arrange the = 8 pieces of fruit, given that all apples are indistinguishable, as are all oranges and all lemons. 8! 5! 6! 7! =, 70, 688 As for the numerator, the only choice is how to arrange the kinds of fruit, for which there are! = 6 ways. Thus P (all fruit of the same kind are together) = 6, 70, 688 = Method : Nate has better eyesight than Ray and can tell the individual pieces of fruit apart. So in the denominator of the probability, he calculates the number of ways to arrange the 8 pieces of fruit, which is 8! = For the numerator he must choose how to arrange the kinds of fruit, for which there are! ways. Then there are 5! ways to arrange the apples, 6! ways to arrange the oranges, and 7! ways to arrange the lemons, for a total number of possibilities of Therefore,! 5! 6! 7! =, 6, 76, 000. P (all fruit of the same kind are together) =, 6, 76, = The results for Method and Method are the same. The probability does not depend on whether a person can distinguish individual pieces of the same kind of fruit. Fall 00 Page 7 Penn State University
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