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Square Root PropertyIfx2=kx2=k, andkâ¥0kâ¥0, thenx=korx=âkx=korx=âk. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Binomial Squares Pattern Ifa,ba,bare real numbers,(a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2(aâb)2=a2â2ab+b2(aâb)2=a2â2ab+b2 | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Complete a SquareTo complete the square ofx2+bxx2+bx:Step 1.Identifybb, the coefficient ofxx.Step 2.Find(12b)2(12b)2, the number to complete the square.Step 3.Add the(12b)2(12b)2tox2+bxx2+bx. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 1.Identifybb, the coefficient ofxx. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 2.Find(12b)2(12b)2, the number to complete the square. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 3.Add the(12b)2(12b)2tox2+bxx2+bx. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Quadratic FormulaThe solutions to a quadratic equation of the formax2+bx+c=0,ax2+bx+c=0,aâ0aâ0are given by the formula:x=âb±b2â4ac2ax=âb±b2â4ac2a | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Solve a Quadratic Equation Using the Quadratic FormulaTo solve a quadratic equation using the Quadratic Formula.Step 1.Write the quadratic formula in standard form. Identify thea,b,ca,b,cvalues.Step 2.Write the quadratic formula. Then substitute in the values ofa,b,c.a,b,c.Step 3.Simplify.Step 4.Check the solutions. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 1.Write the quadratic formula in standard form. Identify thea,b,ca,b,cvalues. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 2.Write the quadratic formula. Then substitute in the values ofa,b,c.a,b,c. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 3.Simplify. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 4.Check the solutions. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Using the Discriminant,b2â4acb2â4ac, to Determine the Number of Solutions of a Quadratic EquationFor a quadratic equation of the formax2+bx+c=0,ax2+bx+c=0,aâ0,aâ0,ifb2â4ac>0b2â4ac>0, the equation has 2 solutions.ifb2â4ac=0b2â4ac=0, the equation has 1 solution.ifb2â4ac<0b2â4ac<0, the equation has no re... | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
ifb2â4ac>0b2â4ac>0, the equation has 2 solutions. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
ifb2â4ac=0b2â4ac=0, the equation has 1 solution. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
ifb2â4ac<0b2â4ac<0, the equation has no real solutions. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
To identify the most appropriate method to solve a quadratic equation:Step 1.Try Factoring first. If the quadratic factors easily this method is very quick.Step 2.Try the Square Root Property next. If the equation fits the formax2=kax2=kora(xâh)2=ka(xâh)2=k, it can easily be solved by using the Square Root Property... | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 1.Try Factoring first. If the quadratic factors easily this method is very quick. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 2.Try the Square Root Property next. If the equation fits the formax2=kax2=kora(xâh)2=ka(xâh)2=k, it can easily be solved by using the Square Root Property. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 3.Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Area of a TriangleFor a triangle with base,bb, and height,hh, the area,AA, is given by the formula:A=12bhA=12bh | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Pythagorean TheoremIn any right triangle, whereaaandbbare the lengths of the legs, andccis the length of the hypothenuse,a2+b2=c2a2+b2=c2 | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Projectile motionThe height in feet,hh, of an object shot upwards into the air with initial velocity,v0v0, afterttseconds can be modeled by the formula:h=â16t2+v0th=â16t2+v0t | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
The graph of every quadratic equation is a parabola. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Parabola OrientationFor the quadratic equationy=ax2+bx+cy=ax2+bx+c, ifa>0a>0, the parabola opens upward.a<0a<0, the parabola opens downward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
a>0a>0, the parabola opens upward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
a<0a<0, the parabola opens downward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Axis of Symmetry and Vertex of a ParabolaFor a parabola with equationy=ax2+bx+cy=ax2+bx+c:The axis of symmetry of a parabola is the linex=âb2ax=âb2a.The vertex is on the axis of symmetry, so itsx-coordinate isâb2aâb2a.To find they-coordinate of the vertex we substitutex=âb2ax=âb2ainto the quadratic equation... | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
The axis of symmetry of a parabola is the linex=âb2ax=âb2a. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
The vertex is on the axis of symmetry, so itsx-coordinate isâb2aâb2a. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
To find they-coordinate of the vertex we substitutex=âb2ax=âb2ainto the quadratic equation. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Find the Intercepts of a ParabolaTo find the intercepts of a parabola with equationy=ax2+bx+cy=ax2+bx+c:y-interceptx-interceptsLetx=0and solve fory.Lety=0and solve forx.y-interceptx-interceptsLetx=0and solve fory.Lety=0and solve forx. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
To Graph a Quadratic Equation in Two VariablesStep 1.Write the quadratic equation withyyon one side.Step 2.Determine whether the parabola opens upward or downward.Step 3.Find the axis of symmetry.Step 4.Find the vertex.Step 5.Find they-intercept. Find the point symmetric to they-intercept across the axis of symmetry.St... | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 1.Write the quadratic equation withyyon one side. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 2.Determine whether the parabola opens upward or downward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 3.Find the axis of symmetry. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 4.Find the vertex. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 5.Find they-intercept. Find the point symmetric to they-intercept across the axis of symmetry. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 6.Find thex-intercepts. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Step 7.Graph the parabola. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
Minimum or Maximum Values of a Quadratic EquationThey-coordinate of the vertexof the graph of a quadratic equation is theminimumvalue of the quadratic equation if the parabola opens upward.maximumvalue of the quadratic equation if the parabola opens downward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
They-coordinate of the vertexof the graph of a quadratic equation is the | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
minimumvalue of the quadratic equation if the parabola opens upward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
maximumvalue of the quadratic equation if the parabola opens downward. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-concepts |
axis of symmetry : The axis of symmetry is the vertical line passing through the middle of the parabolay=ax2+bx+c.y=ax2+bx+c. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
completing the square : Completing the square is a method used to solve quadratic equations. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
consecutive even integers : Consecutive even integers are even integers that follow right after one another. If an even integer is represented bynn, the next consecutive even integer isn+2n+2, and the next after that isn+4n+4. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
consecutive odd integers : Consecutive odd integers are odd integers that follow right after one another. If an odd integer is represented bynn, the next consecutive odd integer isn+2n+2, and the next after that isn+4n+4. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
discriminant : In the Quadratic Formula,x=âb±b2â4ac2ax=âb±b2â4ac2athe quantityb2â4acb2â4acis called the discriminant. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
parabola : The graph of a quadratic equation in two variables is a parabola. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
quadratic equation : A quadratic equation is an equation of the formax2+bx+c=0ax2+bx+c=0, whereaâ0.aâ0. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
quadratic equation in two variables : A quadratic equation in two variables, wherea,b,andca,b,andcare real numbers andaâ0aâ0is an equation of the formy=ax2+bx+c.y=ax2+bx+c. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
Square Root Property : The Square Root Property states that, ifx2=kx2=kandkâ¥0kâ¥0, thenx=korx=âk.x=korx=âk. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
vertex : The point on the parabola that is on the axis of symmetry is called thevertexof the parabola; it is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
x-intercepts of a parabola : Thex-intercepts are the points on the parabola wherey=0.y=0. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
y-intercept of a parabola : They-intercept is the point on the parabola wherex=0.x=0. | https://openstax.org/books/elementary-algebra-2e/pages/10-key-terms |
a + b a + b | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
a ÷ b , a / b , a b , b a a ÷ b , a / b , a b , b a | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
Commutative Property When adding or multiplying, changing the order gives the same result of addition If a , b are real numbers, then a + b = b + a of multiplication If a , b are real numbers, then a · b = b · a of addition If a , b are real numbers, then a + b = b + a of multiplication If a , b are real numbers, the... | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
Associative Property When adding or multiplying, changing the grouping gives the same result. of addition If a , b , and c are real numbers, then ( a + b ) + c = a + ( b + c ) of multiplication If a , b , and c are real numbers, then ( a · b ) · c = a · ( b · c ) of addition If a , b , and c are real numbers, then ... | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
Distributive Property If a , b , and c are real numbers, then a ( b + c ) = a b + a c ( b + c ) a = b a + c a a ( b â c ) = a b â a c ( b â c ) a = b a â c a If a , b , and c are real numbers, then a ( b + c ) = a b + a c ( b + c ) a = b a + c a a ( b â c ) = a b â a c ( b â c ) a = b a â c a | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
Identity Property of addition For any real number a : a + 0 = a 0 is the additive identity 0 + a = a of multiplication For any real number a : a · 1 = a 1 is the multiplicative identity 1 · a = a of addition For any real number a : a + 0 = a 0 is the additive identity 0 + a = a of multiplication For any real number a... | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
Inverse Property of addition For any real number a , a + ( â a ) = 0 â a is the additive inverse of a A number and its o p p o s i t e add to zero. of multiplication For any real number a , a â 0 a · 1 a = 1 1 a is the multiplicative inverse of a A number and its r e c i p r o c a l multiply to one. of addition F... | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
Properties of Zero For any real number a , a · 0 = 0 0 · a = 0 For any real number a , a â 0 , 0 a = 0 For any real number a , a 0 is undefined For any real number a , a · 0 = 0 0 · a = 0 For any real number a , a â 0 , 0 a = 0 For any real number a , a 0 is undefined | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-concepts |
absolute value : The absolute value of a number is its distance from00on the number line. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
additive identity : The number 0 is the additive identity because adding 0 to any number does not change its value. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
additive inverse : The opposite of a number is its additive inverse. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
coefficient : The coefficient of a term is the constant that multiplies the variable in a term. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
complex fraction : A fraction in which the numerator or the denominator is a fraction is called a complex fraction. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
composite number : A composite number is a counting number that is not prime. It has factors other than 1 and the number itself. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
constant : A constant is a number whose value always stays the same. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
denominator : In a fraction, writtenab,ab,wherebâ0,bâ0,the denominatorbis the number of equal parts the whole has been divided into. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
divisible by a number : If a numbermis a multiple ofn, thenmis divisible byn. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
equation : An equation is two expressions connected by an equal sign. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
equivalent fractions : Equivalent fractions are fractions that have the same value. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
evaluate an expression : To evaluate an expression means to find the value of the expression when the variables are replaced by given numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
expression : An expression is a number, a variable, or a combination of numbers and variables using operation symbols. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
factors : Ifa·b=m,a·b=m,thenaandbare factors ofm. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
fraction : A fraction is writtenab,ab,wherebâ0,bâ0,andais the numerator andbis the denominator. A fraction represents parts of a whole. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
integers : The whole numbers and their opposites are called the integers. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
irrational number : An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
least common denominator : The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
least common multiple : The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
like terms : Terms that are either constants or have the same variables raised to the same powers are called like terms. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
multiple of a number : A number is a multiple ofnif it is the product of a counting number andn. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
multiplicative identity : The number 1 is the multiplicative identity because multiplying 1 by any number does not change its value. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
multiplicative inverse : The reciprocal of a number is its multiplicative inverse. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
negative numbers : Numbers less than00are negative numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
numerator : In a fraction, writtenab,ab,wherebâ0,bâ0,the numeratoraindicates how many parts are included. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
opposite : The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
order of operations : The order of operations are established guidelines for simplifying an expression. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
percent : A percent is a ratio whose denominator is 100. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
prime factorization : The prime factorization of a number is the product of prime numbers that equals the number. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
prime number : A prime number is a counting number greater than 1 whose only factors are 1 and the number itself. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
principal square root : The positive square root is called the principal square root. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
rational number : A rational number is a number of the formpq,pq,wherepandqare integers andqâ0.qâ0.Its decimal form stops or repeats. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
real number : A real number is a number that is either rational or irrational. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
reciprocal : The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
simplify an expression : To simplify an expression means to do all the math possible. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
square of a number : Ifn2=m,n2=m,thenmis the square ofn. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
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