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ribute 12x + 8 = 6x Move variables to one side 12x Subtract 12x from both sides 12x 6x Divide both sides by βˆ’ Our Solution 269 Example 362. 2x 3 βˆ’ 7x + 4 2 5 3) = 2(7x + 4) Distribute = Calculate cross product 15 = 14x + 8 Move variables to one side Subtract 10x from both sides Subtract 8 from both sides 5(2x βˆ’ 10x 10x...
feet and a height of 6 feet The flagpole has a shadow of 8 feet, but we donβ€²t know the height 3.5 6 8 x 3.5x = (8)(6) Multiply = This gives us our proportion, calculate cross product 3.5x = 48 Divide both sides by 3.5 3.5 3.5 x = 13.7ft Our Solution Example 365. In a basketball game, the home team was down by 9 points ...
exchange rate is approximately S0.70 to 1 Tala. At this rate, how many dollars would you get if you exchanged 13.3 Tala? 32) If you can buy one plantain for S0.49 then how many can you buy with S7.84? 272 33) Kali reduced the size of a painting to a height of 1.3 in. What is the new width if it was originally 5.2 in. ...
both sides 8x = 19 Divide both sides by 8 8 x = Our Solution 8 19 8 We will use the same process to solve rational equations, the only difference is our 274 LCD will be more involved. We will also have to be aware of domain issues. If our LCD equals zero, the solution is undefined. We will always check our solutions in ...
2 + 2x (x + 3)(x x + 3 = 0 or x 3 βˆ’ x = 3 + 2)( ( βˆ’ 3 or x = 1 Check solutions, LCD canβ€²t be zero 3 in (x + 1)(x + 2), it works 2)( 1) = 2 Check βˆ’ (1 + 1)(1 + 2) = (2)(3) = 6 Check 1 in (x + 1)(x + 2), it works 3 or 1 Our Solution x = βˆ’ βˆ’ In the previous example the denominators were factored for us. More often we will...
8(x βˆ’ 3)(x + 2), multiply each term (x βˆ’ 2)8(x x 3)(x + 2)8(x βˆ’ x + 2 3)(x + 2) 8(x 5 Β· = 3)(x + 2) βˆ’ 8 Reduce 8(x βˆ’ 2)(x + 2) + 8( βˆ’ x βˆ’ 2)(x βˆ’ 3) = 5(x βˆ’ 3)(x + 2) 8(x2 βˆ’ 8x2 x2 + x + 6) = 5(x2 4) + 8( 32 βˆ’ βˆ’ βˆ’ 8x2 + 8x + 48 = 5x2 8x + 16 = 5x2 8x 16 0 = 5x2 βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ FOIL x 6) Distribute βˆ’ 5x 30 Combine like te...
x2 1 βˆ’ 19) 21) x 23) 25) 27 = 5x + 20 6x + 24 βˆ’ x x 1 βˆ’ βˆ’ 2x x + 1 βˆ’ 2 x + 1 = 4x2 x2 βˆ’ 1 3 x + 5 = βˆ’ 8x2 x2 + 6x + 5 29) x x 31 = βˆ’ x2 βˆ’ βˆ’ 4x2 12x + 27 3 6 + x + 5 x + 3 = βˆ’ βˆ’ x2 2x2 3x βˆ’ 33) 4x + 1 x + 3 + 5x x 3 1 = 8x2 x2 + 2x βˆ’ βˆ’ 2) x2 + 6 1 + x x x βˆ’ 2 1 = 2x βˆ’ βˆ’ 6) x x βˆ’ βˆ’ 4 1 = 12 3 x + 1 βˆ’ 8) 6x + 5 2x2 2x βˆ’ 2...
not identical in appearance, but rather identical in value. Below are several fractions, each equal to one where numerator and denominator are identical in value = 100cm 1m = 1lb 16oz = 1hr 60 min = 60 min 1hr The last few fractions that include units are called conversion factors. We can make a conversion factor out ...
in them. Consider the following example. Example 373. A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second? 45mi hr 45mi hr 5280ft 1mi 45mi hr 5280ft 1mi 1hr 3600 sec 45 1 5280ft 1 1 3600 sec β€²β€²perβ€²β€² is the fraction bar, put hr in denominator To clear mi they must go in denom...
sheet has been included at the end of this lesson which includes several conversion factors for length, volume, mass and time in both English and Metric units. The process of dimensional analysis can be used to convert other types of units as well. If we can identify relationships that represent the same value we can ...
560 ft2 640 acres = 1 mi2 Metric 1 a = 100 m2 1 ha = 100 a English/Metric 1 ha = 2.47 acres English 1 lb = 16 oz 1 T = 2000 lb Weight (Mass) Metric 1 g = 1000 mg 1 g = 100 cg 1000 g = 1 kg 1000 kg = 1 t English/Metric 28.3 g = 1 oz 2.2 lb = 1 kg Time 60 sec = 1 min 60 min = 1 hr 3600 sec = 1 hr 24 hr = 1 day 284 7.8 Pr...
iliter of blood volume. If a person’s blood sugar level measured 128 mg/dL, how much is this in grams per liter? 24) You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost S18 per square yard. How much will the carpet cost? 25) A car travels 14 miles in 15 minutes. How fast is it going in m...
roots. Square roots are the most common type of radical used. A square root β€œunsquares” a number. For example, because 52 = 25 we say the square root of 25 is 5. The square root of 25 is written as 25√. World View Note: The radical sign, when first used was an R with a line through the tail, similar to our perscription...
fastest method, is to find perfect squares that divide evenly into the radicand, or number under the radical. This is shown in the next example. into · 5√ 36 180√ Example 378. 75√ 3√ 25 · 3√ · 5 3√ 25√ 75 is divisible by 25, a perfect square Split into factors Product rule, take the square root of 25 Our Solution If th...
x2y3z5 2√ 5 p Β· 15x2y3z5 2√ βˆ’ βˆ’ 18 is divisible by 9, a perfect square Split into factors Product rule, simplify roots, divide exponents by 2 Multiply coefficients Our Solution We can’t always evenly divide the exponent on a variable by 2. Sometimes we have a remainder. If there is a remainder, this means the remainder i...
) 42) βˆ’ βˆ’ p 8 32xy2z3 32m2p4q p 291 8.2 Radicals - Higher Roots Objective: Simplify radicals with an index greater than two. While square roots are the most common type of radical we work with, we can take higher roots of numbers as well: cube roots, fourth roots, fifth roots, etc. Following is a definition of radicals. ...
27, 54 is divisible by 27! 3√ 273√ Β· 3 Write as factors Product rule, take cubed root of 27 Our Solution 2 Β· 23√ 23√ 27 Just as with square roots, if we have a coefficient, we multiply the new coefficients together. Example 385. We are working with a fourth root, want fourth powers 484√ 3 24 = 16 Test 2, 24 = 16, 48 is di...
224n3 5 17) 224p5 p βˆ’ βˆ’ βˆ’ 3 p 3 p 3 19) 21) 23) 25) 27) 29) 7√ 3 896r 3√ 2 48v7 βˆ’ 3√ 7 320n6 135x5y3 32x4y4 βˆ’ βˆ’ 256x4y6 p 3 31) 7 81x3y7 βˆ’ p 3√ 33) 2 375u2v8 8 484√ 8) βˆ’ 10) 3 484√ 4√ 12) 5 243 14) 4√ 64n3 16) 5√ 18) 6√ 96x3 βˆ’ 256x6 20) 7√ 8 384b8 βˆ’ 3√ 22) 4 250a6 24) 26) βˆ’ 3√ 3√ 512n6 64u5v3 28) 3√ 1000a4b5 3 30) 189...
4 35√ 8 35√ βˆ’ We cannot simplify this expression any more as the radicals do not match. Often problems we solve have no like radicals, however, if we simplify the radicals first we may find we do in fact have like radicals. Example 390. 5 45√ + 6 18√ 5√ 2√ 3 5√ + 6 5 βˆ’ Β· Β· 15 5√ + 18 2√ βˆ’ 2√ 2 49 2 2 98√ + 20√ 5√ + 4 Β· ...
5√ 12) 5√ βˆ’ βˆ’ βˆ’ 14) 2 20√ + 2 20√ 2 54√ 3√ βˆ’ 3 27√ + 2 3√ 12√ βˆ’ 16) 18) 20) 22) βˆ’ βˆ’ βˆ’ βˆ’ 2 2√ βˆ’ 3 18√ 3 8√ βˆ’ 5√ βˆ’ 2√ + 3 8√ + 3 6√ 8√ + 2 8√ + 2 8√ 3 6√ + 2 18√ βˆ’ 24) 2 6√ 54√ 3 27√ 3√ βˆ’ βˆ’ βˆ’ 3√ 26) 3 135 3√ 81 βˆ’ βˆ’ 3 44√ + 3 324 4√ 3√ 135 + 2 644√ 30) 2 64√ + 2 44√ + 3 64√ 4√ 2 243 964√ + 2 964√ 32) βˆ’ 34) 2 484√ βˆ’ 4√ 3 ...
√ 5√ 2 64 βˆ’ 5√ 2 192 5√ 160 βˆ’ 39) 6√ 256 2 46√ βˆ’ βˆ’ βˆ’ 6√ 3 320 160 5√ βˆ’ βˆ’ 6√ 2 128 βˆ’ 36) 38) βˆ’ βˆ’ 297 8.4 Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals match. The prodcu...
4 + 6 30√ 5 2√ + 6 30√ βˆ’ βˆ’ 20 2√ + 6 30√ Β· βˆ’ 8 30√ 8 30√ βˆ’ βˆ’ 8 30√ 16 2√ βˆ’ βˆ’ βˆ’ 12 9 12 12 18√ 2√ Β· 3 2√ Β· 36 2√ 2 30√ βˆ’ βˆ’ Simplify radicals, find perfect square factors Take square root where possible Multiply coefficients Combine like terms Our Solution World View Note: Clay tablets have been discovered revealing much a...
the denominator we will rationalize it, or clear out any radicals in the denominator. in the denominator of our final answer. If there is a radical 299 We do this by multiplying the numerator and denominator by the same thing. The problems we will consider here will all have a monomial in the denominator. The way we cl...
iply numerator and denominator by 11√ 77√ 11 2 Β· 77√ 22 Multiply denominator Our Solution The same process can be used to rationalize fractions with variables. Example 402. Reduce coefficients and inside radical 6x3y4z 10xy6z3 4 18 4 p8 p 9 4 4 4√ 3x2 5y2z3 Index is 4. We need four of everything to rationalize, three mor...
( 5√ + 2√ ) 8) 10) 5 15√ (3 3√ + 2) 12) 15√ ( 5√ 3 3v√ ) βˆ’ 14) ( βˆ’ 2 + 3√ )( βˆ’ 5 + 2 3√ ) 16) (2 3√ + 5√ )(5 3√ + 2 4√ ) 18) ( βˆ’ 2 2p√ + 5 5√ )( 5p√ + 5p√ ) 20) (5 2√ 1)( βˆ’ βˆ’ 2m√ + 5) 22) 24) 26) 15√ 2 4√ 12√ 3√ 2√ 3 5√ 28) 4 3√ 15√ 30) 5 4 3xy4 p 8n2√ 10n√ 32) 34) 153√ 643√ 36) 24√ 2 644√ 38) 4 64m4n2 4√ 302 8.5 Radi...
denominator we must divide all terms by the same number. The problem can often be made easier if we first simplify any radicals in the problem. 2 20x5√ 12x2√ βˆ’ 18x√ Simplify radicals by finding perfect squares √ 2 4 3x2 4 Β· Β· √ 5x3 βˆ’ 2x√ 9 Β· 2 Β· 2x2 5x√ βˆ’ 3 2x√ 2x 3√ Simplify roots, divide exponents by 2. Multiply coeffic...
the these we get a difference of squares. Squaring middle gives the product 3 22. Our answer when multiplying conjugates 25 = will no longer have a square root. This is exactly what we want. 5 in the denominator, the conjugate would be 3√ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ Example 404. 2 3√ βˆ’ 5 2 3√ βˆ’ 3√ + 5 3√ + 5! 5 Multiply numerator and de...
fference of squares 6 + 3 3√ 2 5√ 3 βˆ’ 4 βˆ’ 15√ βˆ’ Simplify denominator 6 + 3 3√ 2 5√ 1 βˆ’ βˆ’ 15√ Divide each term by 1 6 + 3 3√ 2 5√ 15√ βˆ’ βˆ’ Our Solution 306 Example 408. 2 5√ 3 7√ βˆ’ 5 6√ + 4 2√ Multiply by the conjugate, 5 6√ 4 2√ βˆ’ 3 7√ 2 5√ βˆ’ 5 6√ + 4 2√ 5 6√ 5 6√ 4 2√ 4 2√! βˆ’ βˆ’ FOIL numerator, denominator is difference o...
9√ 3) 4 + 2 3√ 5 4√ 5) 2 5 5√ βˆ’ 4 13√ 7) 2√ 3 3√ βˆ’ 3√ 9) 5 3 5√ + 2√ 11) 2 5 + 2√ 13) 3 3 3√ 4 βˆ’ 15) 4 3 + 5√ 17) 19) 21) 23) 4 4 2√ βˆ’ 4 βˆ’ 1 1 + 2√ 14√ 7√ 2 βˆ’ 2√ βˆ’ ab√ b√ a βˆ’ a√ βˆ’ 25) a + ab√ a√ + b√ 27) 2 + 6√ 2 + 3√ 29) a b√ βˆ’ a + b√ 31) 33) 35) 3 2√ a a b√ 6 βˆ’ βˆ’ βˆ’ 2 3√ b b a√ 5√ 2 βˆ’ 3 + 5√ βˆ’ 2) βˆ’ 4 + 3√ 4 9√ 4) 2 3...
index. Another way to write division is with a fraction bar. This idea is how we will define rational exponents. Definition of Rational Exponents: a n m = ( am√ )n The denominator of a rational exponent becomes the index on our radical, likewise the index on the radical becomes the denominator of the exponent. We can us...
different problems, using different properties to simplify the rational exponents. Example 413 Need common denominator on aβ€²s(6) and bβ€²s(10) 4 5 1 2 a 6 b 10 a 6 b 10 Add exponents on aβ€²s and bβ€²s 5 7 a 6 b 10 Our Solution Example 414. Example 415 Multiply 3 4 by each exponent x 4 y 10 Our Solution x2y 2 3 2x 7 Β· 2 y0 x ...
exponential form. 5) 7) 1 ( 6x√ )3 1 ( n4√ )7 Evaluate. 2 3 9) 8 3 2 11) 4 6) v√ 8) 5a√ 1 4 10) 16 12) 100βˆ’ 3 2 Simplify. Your answer should contain only positive exponents. 3 2 xy 1 2)βˆ’ Β· 1 13) yx 1 3 1 2b 15) (a 17) a2b0 3a4 19) uv u Β· Β· 21) (x0y 1 3) 3 2)3 (v 3 2x0 23) a 7 4 b 3 4bβˆ’1 Β· 3bβˆ’1 25) βˆ’ 5 4 3y βˆ’ 1 3 yβˆ’1 2...
3 1 4 All exponents have denominator of 4, this is our new index x 4 y x3y 4 Our Solution p What we have done is reduced our index by dividing the index and all the exponents by the same number (2 in the previous example). If we notice a common factor in the index and all the exponnets on every factor we can reduce by...
y2 p xy 15 Common index: 15. Multiply first index and exponents by 3, second by 5 Add exponents Simplify by dividing exponents by index, remainder is left inside Our Solution p Just as with reducing the index, we will rewrite coefficients as exponential expressions. This will also allow us to use exponent properties to si...
moves to denominator p y6z5 x5 24 r 24 y6z5 x5 r 24 x19 x19 r! Cannot have denominator in radical, need 12xβ€²s, or 7 more Multiply numerator and denominator by 12√ x7 x19y6z5 x 24 p Our Solution 316 8.7 Practice - Radicals of Mixed Index Reduce the following radicals. 8 1) 16x4y6 12 p3) 64x4y6z8 6 p5) q 12 7) 16x2 9y4 ...
2 6√ x5 p 28) 4√ a3 3√ a2 30) 5√ a3b ab√ 32) 2x3y3 3 4xy2 34) √ p a4b3c4 3√ p ab2c 8x (y + z)5 3 4x2(y + z)2 p 3√ x2 p x5√ 5√ 3√ a4b2 ab2 5 x3y4z9 p xyβˆ’2z p 3 (2 + 5x)2 4 p (2 + 5x) 3x)3 3x)2 p 4 (5 3 p(5 p βˆ’ βˆ’ 36) 38) 40) 42) 44) 46) 317 8.8 Radicals - Complex Numbers Objective: Add, subtract, multiply, rationalize, a...
of a negative number is no longer undefined. We now are allowed to do basic operations with the square root of negatives. First we will consider exponents on imaginary numbers. We will do this by manipulating our definition of i2 = 1. If we multiply both sides of the definition βˆ’ by i, the equation becomes i3 = i. Then i...
subtraction can be combined into one problem. Example 430. (5i) βˆ’ (3 + 8i) + ( 8i 3 5i βˆ’ βˆ’ 4 + 7i) Distribute the negative 4 + 7i Combine like terms 5i 7 + 4i Our Solution βˆ’ βˆ’ βˆ’ βˆ’ 8i + 7i and 3 4 βˆ’ βˆ’ Multiplying with complex numbers is the same as multiplying with variables with one exception, we will want to simplify...
ator of a fraction, then we have a radical in the denominator! This means we will want to rationalize our denominator so there are no i’s. This is done the same way we rationalized denominators with square roots. Example 436. 7 + 3i 5i βˆ’ 7 + 3i 5i βˆ’ i i 7i + 3i2 5i2 βˆ’ 7i + 3( 5( βˆ’ 1) βˆ’ 1) βˆ’ Just a monomial in denominat...
οΏ½ βˆ’ 6√ βˆ’ (i 6√ )(i 3√ ) Multiply i by i is i2 = Simplify the radical Take square root of 9 Our Solution 18√ βˆ’ 2√ 9 Β· 3 2√ βˆ’ βˆ’ βˆ’ If there are fractions, we need to make sure to reduce each term by the same number. This is shown in the following example. Example 441. 15 βˆ’ 200 Simplify the radical first √ βˆ’ βˆ’ 20 √ √ 1 βˆ’ Β· ...
) 4i 10 + i βˆ’ 8 6i 7 βˆ’ 7 βˆ’ 7i 10 5i 6 βˆ’ i βˆ’ 16) ( βˆ’ i)(7i)(4 3i) βˆ’ 18) (8i)( βˆ’ 20) (3i)( 2i)( βˆ’ 3i)(4 2 8i) βˆ’ 4i) βˆ’ βˆ’ 8(4 22) βˆ’ 24) ( βˆ’ 26) ( βˆ’ 6i)(3 βˆ’ 2 + i)(3 8i) 2( 2 6i) βˆ’ 2i) βˆ’ βˆ’ (7i)(4i) βˆ’ 5i) βˆ’ βˆ’ 3 + 2i 3i βˆ’ 28) βˆ’ 30) βˆ’ 32) βˆ’ 4 + 2i 3i 5 + 9i 9i 34) 10 5i 36) 38) 40) 42) 9i 5i 1 βˆ’ 4 4 + 6i 9 8 βˆ’ βˆ’ 6i 8i 7i 6 βˆ’ 3...
sides to a third power. There is one catch to solving a problem with roots in it, sometimes we end up with solutions that do not actually work in the equation. This will only happen if the index on the root is even, and it will not happen all the time. So for these problems it will be required that we check our answer...
x)2 Evaluate exponents, recal (a order terms βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ 12)(x 10x + x2 Re 10x + 25 Make equation equal zero 4x 1 14x + 24 2 Subtract 4x and 1 from both sides Factor Set each factor equal to zero Solve each equation √ ( 4x + 1 4x + 1 = 25 4x + 1 = x2 4x 1 0 = x2 βˆ’ βˆ’ 0 = (x 12 = 0 or x βˆ’ x βˆ’ + 12 + 12 b)2 = a2 βˆ’ βˆ’ 2ab ...
βˆ’ 8 βˆ’ 4√ βˆ’ βˆ’ βˆ’ 4√ = 0 Multiply 4√ = 0 Subtract 4√ = 0 Take roots 328 2 βˆ’ Subtract 2 = 0 0 = 0 True! It works x = 4 Our Solution When there is more than one square root in the problem, after isolating one root and squaring both sides we may still have a root remaining in the problem. In this case we will again isolate ...
x + 4√ 1 1 Even index! We will have to check answers Isolate the first root by adding x + 4√ Square both sides = x + 4√ √ ( 3x + 9 3x + 9 = x + 4 )2 = ( x + 4√ βˆ’ 2 x + 4√ βˆ’ 3x + 9 = x + 5 x x 5 5 βˆ’ 2x + 4 = βˆ’ βˆ’ βˆ’ 2 x + 4√ βˆ’ 2 x + 4√ βˆ’ 1)2 Evaluate exponents + 1 Combine like terms Isolate the term with radical Subtract ...
x√ βˆ’ βˆ’ 8) √ 2x + 2 = 3 + 2x √ 1 βˆ’ 10) √ 3x + 4 12) √ 7x + 2 14) √ 4x 3 βˆ’ 16) √ 2 3x βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ x + 2√ = 2 √ 3x + 6 = 6 √ 3x + 1 = 1 √ 3x + 7 = 3 331 9.2 Quadratics - Solving with Exponents Objective: Solve equations with exponents using the odd root property and the even root property. Another type of equation we can s...
, one equation for βˆ’ 4 βˆ’ 10 Divide both sides by 2 βˆ’ 2 5 Our Solutions βˆ’ In the previous example we needed two equations to simplify because when we 6. If the roots did took the root, our solutions were two rational numbers, 6 and not simplify to rational numbers we can keep the in the equation. βˆ’ Β± Example 452. (6x βˆ’ ...
9 Clear exponent first with even root property ( 5√ ( 4x + 1 )2 = 9√ Β± q Simplify roots 334 ) Β± = 3 Clear radical by raising both sides to 5th power 5√ 5√ ( 4x + 1 4x + 1 )5 = ( 4x + 1 = 4x + 1 = 243 or 4x + 1 = 1 1 1 Β± 3)5 243 243 1 Β± Β± βˆ’ βˆ’ βˆ’ βˆ’ 4x = 242 or 4x = 4 4 βˆ’ 4 121 2 x = Simplify exponents Solve, need 2 equati...
) (x + 1 2 )βˆ’ 2 3 = 4 19) (x βˆ’ 5 1)βˆ’ 2 = 32 21) (3x βˆ’ 4 2) 5 = 16 23) (4x + 2) 5 = 3 8 βˆ’ 2) x3 = 8 βˆ’ 4) 4x3 6) (x βˆ’ 2 = 106 βˆ’ 4)2 = 49 8) (5x + 1)4 = 16 10) (2x + 1)2 + 3 = 21 12) (x βˆ’ 3 1) 2 = 8 14) (2x + 3) 3 = 16 4 16) (x + 3)βˆ’ 3 = 4 1 18) (x βˆ’ 5 1)βˆ’ 3 = 32 3 20) (x + 3) 2 = 8 βˆ’ 22) (2x + 3) 2 = 27 3 24) (3 βˆ’ 4 2x) ...
, and a perfect square, the third term, c, can be easily found by the formula 2. This is shown in the following examples, where we b find the number that completes the square and then factor the perfect square. 1 2 · Example 459. x2 + 8x + c c = 2 1 2 · b 2 and our b = 8 8 = 42 = 16 The third term to complete the square...
2 2 48 = βˆ’ βˆ’ 48 Subtract 24 48 Divide by a or 2 2 x2 + 10x 24 Find number to complete the square: = βˆ’ 2 1 2 Β· b 2 10 = 52 = 25 Add 25 to both sides of the equation 1 2 Β· x2 + 10x βˆ’ = 24 + 25 + 25 x2 + 10x + 25 = 1 (x + 5)2 = 1 1x + 5)2 Β± 5 p Factor Solve with even root property Simplify roots Subtract 5 from both side...
completes the square and then rewrite as a perfect square. 1) x2 3) m2 5) x2 7) y2 βˆ’ βˆ’ βˆ’ βˆ’ 30x + __ 36m + __ 15x + __ y + __ 2) a2 4) x2 6) r2 8) p2 βˆ’ βˆ’ βˆ’ βˆ’ 24a + __ 34x + __ 1 9 17p + __ r + __ βˆ’ 14 = 4 6 βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ 29 16p 12n 24 = 3 = 6 1 = 0 52 = 0 57 = 4 βˆ’ 10x 6x + 47 = 0 8v + 45 = 0 βˆ’ 37 = 5 10...
7 = 7n + 6n2 βˆ’ 7k + 50 = 3 βˆ’ 2x2 + 3x βˆ’ 15 + 9m 3n + 6 + 4n2 5 + 7b2 + 3b 6x + 40 = 0 5a + 25 = 3 βˆ’ 11x = 8n + 60 = βˆ’ n = 10n + 15 26 + 10x p + 56 = βˆ’ 5 = βˆ’ 5x 33 = 0 40 = 8 56 = 10a 4x2 41 55 18 βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ 55) βˆ’ βˆ’ βˆ’ 8 3r2 + 12r + 49 = 6 6r2 βˆ’ βˆ’ 15b + 56 = 3b2 15v = 27 + 4v2 6v βˆ’ βˆ’ 342 9.4 Quadratics - Qu...
Β± b b2 p2a 4ac and we will get our two solutions. This formula is Quadratic Formula: if ax2 + b x + c = 0 then x = βˆ’ b Β± b2 p2a 4ac βˆ’ World View Note: Indian mathematician Brahmagupta gave the first explicit formula for solving quadratics in 628. However, at that time mathematics was not done with variables and symbols...
= 0 2)2 βˆ’ βˆ’ p 2(1(1)(13) βˆ’ 52 √ 4 2 48√ Β± βˆ’ 2 4i 3√ 2 2i 3 First set equation equal to zero Subtract 2x2 and 6x and add 5 a = 1, b = 2, c = 13, use quadratic formula βˆ’ Evaluate exponent and multiplication Evaluate subtraction inside root Simplify root Reduce fraction by dividing each term by 2 Our Solution When we use...
)(x x = 2 or x = 3 x2 + 2x = 4 βˆ’ 3) = 0 2 2 = 12 = 1 1 2 Β· 5√ 5√ x2 + 2x + 1 = 5 (x + 1) x2 βˆ’ x = 3 x = 3 Β± βˆ’ Β± 3x + 4 = 0 ( 3)2 βˆ’ p 2(1) i 7√ 2 Β± Β± βˆ’ 4(1)(4) The above table is mearly a suggestion for deciding how to solve a quadtratic. Remember completing the square and quadratic formula will always work to solve any...
3 βˆ’ βˆ’ 31) 4a2 64 = 0 βˆ’ 33) 4p2 + 5p 36 = 3p2 βˆ’ 3n 35) βˆ’ 37) 7r2 39) 2n2 5n2 βˆ’ 52 = 2 7n2 βˆ’ βˆ’ 36) 7m2 6m + 6 = m βˆ’ βˆ’ 12 = 3r βˆ’ 9 = 4 βˆ’ βˆ’ 38) 3x2 3 = x2 βˆ’ 40) 6b2 = b2 + 7 b βˆ’ 347 9.5 Quadratics - Build Quadratics From Roots Objective: Find a quadratic equation that has given roots using reverse factoring and reverse co...
In these cases we will use reverse completing the square. When there are radicals the solutions will always come in pairs, one with a plus, one with a. We will then set this minus, that can be combined into β€œone” solution using solution equal to x and square both sides. This will clear the radical from our problem. Β± ...
76. The solutions are 4 5i and 4 + 5i Write as β€²β€²oneβ€²β€² expression equal to x βˆ’ Β± 5i βˆ’ x x = 4 4 4 βˆ’ 5i 4 = 8x + 16 = 25i2 8x + 16 = βˆ’ Β± βˆ’ x2 x2 βˆ’ βˆ’ Isolate the i term Subtract 4 from both sides Square both sides i2 = 1 25 Make equal to zero βˆ’ + 25 + 25 Add 25 to both sides 8x + 41 = 0 Our Solution x2 βˆ’ Example 477. The...
βˆ’ 2 i 15√ Β± 2 351 9.6 Quadratics - Quadratic in Form Objective: Solve equations that are quadratic in form by substitution to create a quadratic equation. We have seen three different ways to solve quadratics: factoring, completing the square, and the quadratic formula. A quadratic is any equation of the form 0 = a x2 ...
3 = 0 or Raise both sides to 2 = aβˆ’ 1 = a 2)βˆ’ 1 1 3 2 Square both sides 2 and b for aβˆ’ Substitute b2 for aβˆ’ Solve. We will solve by factoring Set each factor equal to zero Solve each equation Solutions for b, still need a, substitute into b = aβˆ’ 1 power Simplify negative exponents 2 b = 3 or b = 1 or 3 = aβˆ’ 1 = a or (...
y = βˆ’ 1 We have y, we still need x. 1 = x βˆ’ + 7 7 Substitute into y = x 7 βˆ’ + 7 Add 7. Use common denominator as needed βˆ’ x = 26 3, 6 Our Solution Example 482. (x2 βˆ’ 6x)2 = 7(x2 7(x2 6x) + 12 6x)2 βˆ’ 7(x2 βˆ’ βˆ’ βˆ’ βˆ’ 7(x2 (x2 βˆ’ βˆ’ 12 Make equation equal zero 6x) 6x) + 12 Move all terms to left βˆ’ βˆ’ 6x) + 12 = 0 Quadratic for...
need x. Substitute into y = x3 Set each equation equal to zero 1 Factor each equation, difference of cubes First equation factored. Set each factor equal to zero First equation is easy to solve x x3 (x βˆ’ βˆ’ βˆ’ βˆ’ 1 = 0 or x3 βˆ’ 1)(x2 + x + 1) = 0 βˆ’ 1 = 0 or x2 + 12 βˆ’ p2(1) 2)(x2 + 2x + 4) = 0 (x βˆ’ 2 = 0 or x2 + 2x + 4 = 0 ...
= 0 βˆ’ 17x4 + 16 = 0 22) 2x4 x2 3 = 0 βˆ’ βˆ’ 10x3 + 16 = 0 24) x6 βˆ’ 26) 8x6 + 7x3 1 = 0 βˆ’ 4(x 28) (x 1)2 βˆ’ βˆ’ 1) = 5 βˆ’ 30) (x + 1)2 + 6(x + 1) + 9 = 0 32) (m 1)2 βˆ’ 34) (a + 1)2 + 2(a βˆ’ 5(m 1) = 14 βˆ’ 1) = 15 βˆ’ 1) = 3 36) 2(x 38) (x2 1)2 3)2 βˆ’ βˆ’ βˆ’ βˆ’ (x βˆ’ 2(x2 βˆ’ 3) = 3 40) (x2 + x + 3)2 + 15 = 8(x2 + x + 3) 42) (x2 + x)2 βˆ’ 8(...
. The length of a rectangle is 3 more than the width. If the area is 40 square inches, what are the dimensions? 40 x + 3 x We do not know the width, x. Length is 4 more, or x + 4, and area is 40. 357 x(x + 3) = 40 Multiply length by width to get area x2 + 3x = 40 Distribute 40 40 Make equation equal zero βˆ’ βˆ’ x2 + 3x βˆ’ ...
(x + 4) + 33 x + 3 Increase each side by 3. width becomes x + 3, length x + 3)(x + 7) = x(x + 4) + 33 x2 + 10x + 21 = x2 + 4x + 33 x2 x2 βˆ’ βˆ’ Area is 33 more than original, x(x + 4) + 33 Set up equation, length times width is area Subtract x2 from both sides βˆ’ βˆ’ 10x + 21 = 4x + 33 Move variables to one side 4x Subtract ...
= 14 Substitute 14 into y = βˆ’ Substitute 12 into y = βˆ’ Both are the same rectangle, variables switched! x + 26 x + 26 12 cm by 14cm Our Solution World View Note: Indian mathematical records from the 9th century demonstrate that their civilization had worked extensivly in geometry creating religious alters of various s...
side to get center 400 2x βˆ’ 400 400 200 = 80000 Area of entire field, length times width (0.72) = 57600 Area of center, multiply by 28% as decimal Β· 2x) = 57600 Area of center, length times width 80000 Β· 2x)(200 βˆ’ (400 βˆ’ 800x βˆ’ 4x2 βˆ’ βˆ’ 80000 400x + 4x2 = 57600 FOIL 1200x + 80000 = 57600 Combine like terms 57600 57600 M...
7) A rectangular piece of paper is twice as long as a square piece and 3 inches wider. The area of the rectangular piece is 108 in2. Find the dimensions of the square piece. 8) A room is one yard longer than it is wide. At 75c per sq. yd. a covering for the floor costs S31.50. Find the dimensions of the floor. 9) The ar...
A rectangular field 225 ft by 120 ft has a ring of uniform width cut around the outside edge. The ring leaves 65% of the field uncut in the center. What is the width of the ring? 20) One Saturday morning George goes out to cut his lot that is 100 ft by 120 ft. He starts cutting around the outside boundary spiraling arou...
, he paints 1 4 + 1 12 of the room each hour. So together, each hour they paint 1 12 of the room. Using a common denominator of 12 gives: 3 12 = 1 3. This means each hour, working together they complete 1 3 is completed each hour, it follows that it will take 3 hours to complete the entire room. 12 + 1 12 = 4 3 of the ...
project alone? Mike: 2x, Rachel: x, Total: 10 Clearly define variables. If Rachel is x, Mike is 2x 1 2x + 1 x = 1 10 Using reciprocals, add individal times equaling total 1(10x) 2x + 1(10x) x = 1(10x) 10 Multiply each term by LCD, 10x 5 + 10 = x Combine like terms 15 = x We have our x, we said x was Rachelβ€²s time 2(15)...
difficult to factor. We could have also chosen to complete the square or use the quadratic formula to find our solutions. βˆ’ It is important that units match as we solve problems. This means we may have to convert minutes into hours to match the other units given in the problem. Example 493. An electrician can complete a ...
ne variables, drain is negative 1 5 βˆ’ 1 7 = 1 x 1(35x) 5 βˆ’ 1(35x) 7 = 1(35x) x Using reciprocals to make equation, Subtract because they are opposite Multiply each term by LCD: 35x 7x βˆ’ 5x = 35 Reduce fractions 2x = 35 Combine like terms 2 x = 17.5 Our answer for x Divide each term by 2 2 17.5 min or 17 min 30 sec Our ...
3 4 days. If the carpenter himself could do the work alone in 5 days, how long would the assistant take to do the work alone? 11) If Sam can do a certain job in 3 days, while it takes Fred 6 days to do the same job, how long will it take them, working together, to complete the job? 12) Tim can finish a certain job in 1...
��ll the sink with just the hot-water faucet open? 20) A water tank is being filled by two inlet pipes. Pipe A can fill the tank in 4 1 2 hrs, while both pipes together can fill the tank in 2 hours. How long does it take to fill the tank using only pipe B? 21) A tank can be emptied by any one of three caps. The first can em...
the variables are multiplied together we can use the same idea of substitution that we used with linear equations. When we do so we may end up with a quadratic equation to solve. When we used substitution we solved for a variable and substitute this expression into the other equation. If we have two products we will c...
οΏ½οΏ½rst equation by x, second by x + 6 35 y = βˆ’ x and y 2 = βˆ’ 5 x + 6 35 Substitute βˆ’ x for y in the second equation βˆ’ 35 x βˆ’ 2 = 5 x + 6 Multiply each term by LCD: x(x + 6) βˆ’ 35x(x + 6) x βˆ’ 2x(x + 6) = 5x(x + 6) x + 6 Reduce fractions βˆ’ βˆ’ 35(x + 6) 35x 210 2x2 βˆ’ βˆ’ 12x = 5x Combine like terms 210 = 5x Make equation equal...
y βˆ’ βˆ’ xy = 120 1 2) = 205 (x + 2)(y 3)=120 βˆ’ xy = 65 (x 8)(y + 2) = 35 βˆ’ xy = 48 (x 6)(y + 3) = 60 βˆ’ xy = 60 (x + 5)(y + 3) = 150 12) xy = 80 5)(y + 5) = 45 (x βˆ’ 372 9.10 Quadratics - Revenue and Distance Objective: Solve revenue and distance applications of quadratic equations. A common application of quadratics comes...
3 Substitute 56 n for p in second equation 56n(n n βˆ’ 3) + 5n(n 3) = βˆ’ 3) 60n(n n βˆ’ βˆ’ 3 Multiply each term by LCD: n(n 3) βˆ’ 56(n 56n βˆ’ 3) + 5n(n βˆ’ 168 + 5n2 5n2 + 41n 60n βˆ’ βˆ’ 3) = 60n βˆ’ 15n = 60n 168 = 60n Move all terms to one side Reduce fractions Combine like terms βˆ’ 5n2 βˆ’ 19)2 19n βˆ’ 4(5)( βˆ’ 2(5) βˆ’ 60n 168 = 0 168) ...
like terms Set equation equal to zero Solve by completing the square, Separate variables and constant Divide each term by a or 4 2 1 2 Β· n2 2n = 48 Complete the square: b βˆ’ 2 2 n2 Β· = 12 = 1 Add to both sides of equation 1 2 2n + 1 = 49 βˆ’ 1)2 = 49 ( We donβ€²t want a negative solution n = 1 + 7 = 8 Factor Square root of...
equal to zero Divide each term by 2 Factor Set each factor equal to zero Solve each equation 120(r 120r βˆ’ βˆ’ 20r = 120r βˆ’ 20r = 120r βˆ’ 1200 = 120r 10) + 2r2 βˆ’ 1200 + 2r2 βˆ’ 2r2 + 100r 120r βˆ’ 2r2 20r 1200 = 0 βˆ’ βˆ’ r2 10r 600 = 0 βˆ’ βˆ’ 30)(r + 20) = 0 30 = 0 and r + 20 = 0 20 βˆ’ 20 βˆ’ 30 mph 20 r = 30 and r = βˆ’ + 30 + 30 (r βˆ’ ...
)(r 2) βˆ’ βˆ’ 30(r + 2)(r r + 2 2) = βˆ’ 30(r + 2)(r r 2 βˆ’ 2) βˆ’ Multiply each term by LCD: (r + 2)(r 2) βˆ’ 8(r + 2)(r βˆ’ 8r2 βˆ’ 2) βˆ’ 32 8r2 βˆ’ 30(r 2) = 30(r + 2) Reduce fractions 30r + 60 = 30r + 60 Multiply and distribute 30r + 28 = 30r + 60 Make equation equal zero 30r 30r βˆ’ βˆ’ βˆ’ 60 60 βˆ’ βˆ’ βˆ’ 8r2 60r 32 = 0 βˆ’ βˆ’ 2r2 15r 8 = 0 βˆ’...
? 7) A group of boys bought a boat for S450. Five boys failed to pay their share, hence each remaining boys were compelled to pay S4.50 more. How many boys were in the original group and how much had each agreed to pay? 8) The total expenses of a camping party were S72. If there had been 3 fewer persons in the party, i...
the rate of the current is 2 km/hr, how fast should he row? 15) An automobile goes to a place 72 miles away and then returns, the round trip occupying 9 hours. His speed in returning is 12 miles per hour faster than his speed in going. Find the rate of speed in both going and returning. 16) An automobile made a trip o...
0) + 3 = 0 y = (1)2 4(1) + 3 = 1 βˆ’ y = (2)2 4(23)2 4(34)2 4(4) + 3 = 16 βˆ’ βˆ’ 0 + 3 = 3 Plug 0 in for x and evaluate βˆ’ 4 + 3 = 0 Plug 1 in for x and evaluate βˆ’ 1 Plug 2 in for x and evaluate 8 + 3 = 12 + 3 = 0 Plug 3 in for x and evaluate 16 + 3 = 3 Plug 4 in for x and evaluate βˆ’ Our completed table. Plot points on graph...
we can connect the dots with a smooth curve to find our graph! Example 502. y = x2 + 4x + 3 Find the key points y = 3 y = c is the y intercept βˆ’ 0 = x2 + 4x + 3 To find x intercept we solve the equation βˆ’ 0 = (x + 3)(x + 1) x + 3 = 0 and x + 1 = 0 1 1 Our x Factor Set each factor equal to zero Solve each equation interc...
Evaluate βˆ’ 9 βˆ’ y = 3 (2, 3) Vertex as a point value of vertex βˆ’ y coordinate βˆ’ Graph the y-intercept at 9, the xintercepts at 3 and 1, and the vertex at (2, 3). Connect the dots with smooth curve in an upside-down U shape to get our parabola. βˆ’ Our Solution 382 It is important to remember the graph of all quadratics i...
) y = 5x2 + 30x + 45 18) y = 5x2 + 20x + 15 20) y = βˆ’ 5x2 + 20x 15 βˆ’ 384 Chapter 10 : Functions 10.1 Function Notation..................................................................................386 10.2 Operations on Functions........................................................................393 10.3 Inverse...
algebraic method by taking a relationship and solving it for y. If we have only one solution then it is a function. Example 505. Is 3x2 y = 5 a function? 3x2 βˆ’ 3x2 + 5 βˆ’ 1 1 βˆ’ 3x2 βˆ’ y = 1 βˆ’ y = 3x2 βˆ’ 5 βˆ’ βˆ’ βˆ’ Solve the relation for y Subtract 3x2 from both sides Divide each term by 1 βˆ’ Only one solution for y. It is a ...
0 and x 3 3 βˆ’ βˆ’ x βˆ’ 3, 2 Our Solution With fractions, zero canβ€²t be in denominator Solve by factoring Set each factor not equal to zero Solve each equation The notation in the previous example tells us that x can be any value except for 3 and 2. If x were one of those two values, the function would be undefined. βˆ’ Exam...
βˆ’ 7) = 2(3) Multiply 7 in for a in the function Substitute βˆ’ Add inside absolute values Evaluate absolute value 389 k( βˆ’ 7) = 6 Our Solution As the above examples show, the function can take many different forms, but the pattern to evaluate the function is always the same, replace the variable with what is in parenthes...
2 βˆ’ 12 25 391 Evaluate each function. 11) g(x) = 4x 4; Find g(0) βˆ’ 3x + 1 13) f (x) = | + 1; Find f (0) | 17) f (t) = 3t 19) f (t) = | βˆ’ t + 3 2) 2; Find f ( βˆ’ ; Find f (10) | 15) f (n; Find f ( 6) βˆ’ 16) f (n) = n 12) g(n) = 3 5βˆ’ n; Find g(2) βˆ’ 14) f (x) = x2 + 4; Find f ( Β· 9) βˆ’ 3; Find f (10) βˆ’ 1 βˆ’ 3a 18) f (a) βˆ’ 3; ...
h(x) = x2 + 1; Find h( x 4 ) Β· 38) h(t) = t2 + t; Find h(t2) 40) h(n) = 5n βˆ’ 1 + 1; Find h( n 2 ) 392 10.2 Functions - Operations on Functions Objective: Combine functions using sum, difference, product, quotient and composition of functions. Several functions can work together in one larger function. There are 5 commo...
multiply, or divide functions, we do so in a way that keeps the variable. If there is no number to plug into the equations we will simply use each equation, in parenthesis, and simplify the expression. Example 517. f (x) = 2x 4 βˆ’ g(x) = x2 Find (f βˆ’ βˆ’ x + 5 Write subtraction problem of functions g)(x) (2x βˆ’ 2x 4) βˆ’ f ...
3x) Replace x in f (x) and g(x) with 3x 1][(3x) + 4] Multiply our 2(3x) 1)(3x + 4) FOIL 3x 18x2 + 21x 4 Combine like terms 4 Our Solution βˆ’ βˆ’ βˆ’ The fifth operation of functions is called composition of functions. A composition of functions is a function inside of a function. The notation used for composition of function...
a function in function g(f (x)) Replace f (x) with x2 x βˆ’ g(x2 x) Replace the variable in g with (x2 x) βˆ’ x) + 3 Here the parenthesis donβ€²t change the expression x + 3 Our Solution βˆ’ (x2 x2 βˆ’ βˆ’ World View Note: The term β€œfunction” came from Gottfried Wihelm Leibniz, a German mathematician from the late 17th century. 3...
) = βˆ’ h(x) = 2x 1 βˆ’ Find g(5) + h(5) βˆ’ 1) 8) g(x) = 3x + 1 f (x) = x3 + 3x2 f (2) Find g(2) Β· 10) f (n) = n 5 g(n) = 4n + 2 Find (f + g)( βˆ’ 12) g(a) = 3a 2 h(a) = 4a 2 Find (g + h) ( βˆ’ βˆ’ 8) βˆ’ 10) βˆ’ 14) g(x) = x2 2 h(x) = 2x + 5 Find g( βˆ’ βˆ’ 16) g(n) = n2 h(n) = 2n Find (g βˆ’ 18) g(x) = 2x h(x) = x3 Find (g βˆ’ 6) + h( 6) βˆ’...
f (n) = βˆ’ g(n) = 2n + 1 Find (f βˆ’ 3n2 + 1 g)( n 3 ) βˆ’ 4x + 1 41) f (x) = g(x) = 4x + 3 Find (f g)(9) βˆ’ β—¦ 43) h(a) = 3a + 3 g(a) = a + 1 Find (h g)(5) β—¦ 45) g(x) = x + 4 h(x) = x2 1 βˆ’ h)(10) Find (g β—¦ 24) f (x) = 4x g(x) = 3x2 Find (f + g)(x) βˆ’ βˆ’ 4 5 26) f (x) = 2x + 4 g(x) = 4x 5 g(x) Find f (x) βˆ’ βˆ’ 28) g(t) = t3 + 3t...
(x) = 2x 4 βˆ’ h(x) = 2x3 + 4x2 Find (g h)(3) β—¦ 51) g(x) = x2 5x h(x) = 4x + 4 Find (g h)(x) βˆ’ 53) f (a) = βˆ’ g(a) = 4a Find (f 2a + 2 g)(a) β—¦ β—¦ 55) g(x) = 4x + 4 f (x) = x3 1 βˆ’ f )(x) Find (g β—¦ 57) g(x) = x + 5 3 βˆ’ f )(x) βˆ’ f (x) = 2x Find (g β—¦ 59) f (t) = 4t + 3 g(t) = βˆ’ Find (f β—¦ 4t 2 βˆ’ g)(t) 48) g(x) = 3x + 4 h(x) = x...