Factoring and Quadratic Equations
Then
Now
Why?
In Chapter 7, you multiplied monomials and polynomials.
In Chapter 8, you will:
ARCHITECTURE Quadratic equations can be used to model the shape of architectural structures such as the tallest memorial in the United States, the Gateway Arch in St. Louis, Missouri.
Factor monomials. Factor trinomials. Factor differences of squares. Solve quadratic equations.
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Tennessee Curriculum Standards CLE 3102.3.8, SPI 3102.3.3
Get Ready for the Chapter Diagnose Readiness
1

You have two options for checking prerequisite skills.
Textbook Option Take the Quick Check below. Refer to the Quick Review for help.
QuickCheck Rewrite each expression using the Distributive Property. Then simplify. (Lesson 14) 1. a(a + 5) 2
3. n (n  3n + 2)
QuickReview Example 1
2. 2(3 + x)
Rewrite 6x (3x  5x  5x 2 + x 3) using the Distributive Property. Then simplify.
4. 6(x 2  5x + 6)
6x (3x  5x  5x 2 + x 3)
5. FINANCIAL LITERACY Five friends will pay $9 per ticket, $3 per drink, and $6 per popcorn at the movies. Write an expression that could be used to determine the cost for them to go to the movies. Find each product. (Lesson 77)
= 6x (3x ) + 6x (5x ) + 6x (5x 2) + 6x (x 3) = 18x 2  30x 2  30x 3 + 6x 4 = 48x 2  30x 3 + 6x 4
Example 2
6. (x + 2)(x  5)
Find (x + 3)(2x  1).
7. (x + 4)(x  1)
(x + 3)(2x  1)
Original expression
8. (2a  3)(5a + 4)
= x (2x ) + x (1) + 3(2x ) + 3(1)
FOIL method
9. (3x  4)(x + 5)
= 2x 2  x + 6x  3
Multiply.
10. (x + 4)(x + 7)
2
= 2x + 5x  3
Combine like terms.
11. (6a  2b)(9a + b) 12. TABLECLOTH The dimensions of a tablecloth are represented by a width of 2x + 3 and a length of x + 1. Find an expression for the area of the tablecloth. Find each product. (Lesson 78)
Example 3
13. (3  a) 2
Find (y + 8) 2.
14. (x + 5) 2
(a + b) 2 = a 2 + 2ab + b 2
15. (3x  2y ) 2
( y + 8) 2 = (y ) 2 + 2(y )(8) + 8 2
16. (2x + 5y )(2x  5y )
2
= y + 16y + 64
Square of a sum a = y, b = 8 Simplify.
17. PHOTOGRAPHY A photo is x + 6 inches by x  6 inches. What is the area of the photo?
2
Online Option Take an online selfcheck Chapter Readiness Quiz at connectED.mcgrawhill.com. 469
Get Started on the Chapter You will learn several new concepts, skills, and vocabulary terms as you study Chapter 8. To get ready, identify important terms and organize your resources. You may wish to refer to Chapter 0 to review prerequisite skills.
StudyOrganizer
NewVocabulary
Factoring and Quadratic Equations Make this Foldable to help you organize your Chapter 8 notes about factoring and quadratic equations. Begin with four sheets of grid paper.
1
Fold in half along the width. On the first two sheets, cut 5 centimeters along the fold at the ends. On the second two sheets cut in the center, stopping 5 centimeters from the ends. 'JSTU4IFFUT
4FDPOE4IFFUT
English
Español
factored form
p. 471
forma reducida
greatest common factor (GCF)
p. 471
máximo común divisor (MCD)
factoring
p. 476
factorización
factoring by grouping
p. 477
factorización por agrupamiento
Zero Product Property
p. 478
propiedad del producto de cero
quadratic equation
p. 488
ecuación cuadrática
prime polynomial
p. 495
polinomio primo
difference of two squares
p. 499
diferencia de cuadrados
perfect square trinomial
p. 505
trinomio cuadrado perfecto
Square Root Property
p. 508
Propiedad de la raíz cuadrada
ReviewVocabulary
2
Insert the first sheets through the second sheets and align the folds. Label the front Chapter 8, Factoring and Quadratic Equations. Label the pages with lesson numbers and the last page with vocabulary.
absolute value p. P11 valor absoluto the absolute value of any number n is the distance the number is from zero on a number line and is written ⎪n⎥ 2 units
2 1
0
1
2
The absolute value of 2 is 2 because it is 2 units from 0.
perfect square p. P7 cuadrado perfecto a number with a square root that is a rational number prime number p. 861 numero primo a whole number, greater than 1, with the only factor being 1 and itself
470  Chapter 8  Factoring and Quadratic Equations
Monomials and Factoring Then
Now
Why?
You multiplied monomials and divided a polynomial by a monomial.
1 2
Susie is making beaded bracelets for extra money. She has 60 gemstone beads and 15 glass beads. She wants each bracelet to have only one type of bead and all of the bracelets to have the same number of beads. Susie needs to determine the greatest common factor of 60 and 15.
(Lesson 71 and 72)
NewVocabulary factored form greatest common factor (GCF)
Factor monomials. Find the greatest common factors of monomials.
1 Factor Monomials
Factoring a monomial is similar to factoring a whole number. A monomial is in factored form when it is expressed as the product of prime numbers and variables, and no variable has an exponent greater than 1.
Example 1 Monomial in Factored Form Factor 20x 3y 2 completely. Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. SPI 3102.3.3 Factor polynomials.
20x 3y 2 = 1 · 20x 3y 2 Express 20 as 1 · 20. = 1 · 2 · 10 · x · x · x · y · y 20 = 2 · 10, x 3 = x · x · x, and y 2 = y · y =1·2·2·5·x·x·x·y·y 10 = 2 · 5 3 2 Thus, 20x y in factored form is 1 · 2 · 2 · 5 · x · x · x · y · y.
GuidedPractice Factor each monomial completely. 1A. 34x 4y 3
1B. 52a 2b
2 Greatest Common Factor
Two or more whole numbers may have some common prime factors. The product of the common prime factors is called their greatest common factor. The greatest common factor (GCF) is the greatest number that is a factor of both original numbers. The GCF of two or more monomials can be found in a similar way.
Example 2 GCF of a Set of Monomials Find the GCF of 12a 2b 2c and 18ab 3. 12a 2b 2c = 2 · 2 · 3 · a · a · b · b · c
Factor each number, and write all powers of variables as products.
18ab 3
Circle the common prime factors.
= 2 · 3 ·3· a · b · b ·b
The GCF of 12a 2b 2c and 18ab 2 is 2 · 3 · a · b · b or 6ab 2.
GuidedPractice Find the GCF of each pair of monomials. 2A. 6xy 3, 18yz
2B. 11a 2b, 21ab 2
2C. 30q 3r 2t, 50q 2rt connectED.mcgrawhill.com
471
RealWorld Example 3 Find a GCF FLOWERS A florist has 20 roses and 30 tulips to make bouquets. What is the greatest number of identical bouquets she can make without having any flowers left over? How many of each kind of flower will be in each bouquet? Find the GCF of 20 and 30. 20 = 2 2 · 5
Write the prime factorization of each number.
30 = 2 · 3 · 5
The common prime factors are 2 and 5 or 10.
The GCF of 20 and 30 is 10. So, the florist can make 10 bouquets. Since 2 × 10 = 20 and 3 × 10 = 30, each bouquet will have 2 roses and 3 tulips.
GuidedPractice 3. What is the greatest possible value for the widths of two rectangles if their areas are 84 square inches and 70 square inches, respectively, and the length and width are whole numbers?
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R12.
Factor each monomial completely. 1. 12g 2h 4
2. 38rp 2t 2
3. 17x 3y 2z
4. 23ab 3
Examples 2–3 Find the GCF of each pair of monomials. 5. 24cd 3, 48c 2d
6. 7gh, 11mp
7. 8x 2y 5, 31xy 3
8. 10ab, 25a
9. GEOMETRY The areas of two rectangles are 15 square inches and 16 square inches. The length and width of both figures are whole numbers. If the rectangles have the same width, what is the greatest possible value for their widths?
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Factor each monomial completely. 10. 95xy 2
11 35a 3c 2
12. 42g 3h 3
13. 81n 5p
14. 100q 4r
15. 121abc 3
Examples 2–3 Find the GCF of each set of monomials. 16. 25x 3, 45x 4, 65x 2
17. 26z 2, 32z, 44z 4
18. 30gh 2, 42g 2h, 66g
19. 12qr, 8r 2, 16rt
20. 42a 2b, 6a 2, 18a 3
21. 15r 2t, 35t 2, 70rt
22. BAKING Delsin wants to package the same number of cookies in each bag, and each bag should have every type of cookie. If he puts the greatest possible number of cookies in each bag, how many bags can he make?
472  Lesson 81  Monomials and Factoring
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23. GEOMETRY The area of a triangle is 28 square inches. What are possible wholenumber dimensions for the base and height of the triangle? 24. MUSIC In what ways can Clara organize her 36 CDs so that she has the same number of CDs on each shelf, at least 4 per shelf, and at least 2 shelves of CDs? 25 MOVIES In what ways can Shannon arrange her 80 DVDs so that she has at least 4 shelves of DVDS, the same number on each shelf, and at least 5 on each shelf? 26. VOLUNTEER Denzell is donating packages of school supplies to an elementary school where he volunteers. He bought 200 pencils, 150 glue sticks, and 120 folders. How many packages can Denzell make using an equal number of each item? How many items of each type will each package contain? 27. NUMBER THEORY Twin primes are two consecutive odd numbers that are prime. The first two pairs of twin primes are 3 and 5 and 5 and 7. List the next five pairs.
C
28.
MULTIPLE REPRESENTATIONS In this problem, you will investigate a method of factoring a number. a. Analytical Copy the ladder diagram shown at the right six times and record six whole numbers, two of which are prime, in the top right portion of the diagrams. b. Analytical Choose a prime factor of one of your numbers. Record the factor on the left of the number in the diagram. Divide the two numbers. Keep dividing by prime factors until the quotient is 1. Add to or subtract boxes from the diagram as necessary. Repeat this process with all of your numbers. c. Verbal What is the prime factorization of your six numbers?
H.O.T. Problems
3 12 2 4 2 2 1 So, the prime factorization 2 of 12 is 2 ·3.
Use HigherOrder Thinking Skills
29. CHALLENGE Find the least pair of numbers that satisfies the following conditions. The GCF of the numbers is 11. One number is even and the other number is odd. One number is not a multiple of the other. 30. REASONING The least common multiple (LCM) of two or more numbers is the least number that is a multiple of each number. Compare and contrast the GCF and LCM of two or more numbers. 31. REASONING Determine whether the following statement is true or false. Provide an example or counterexample. Two monomials always have a greatest common factor that is not equal to 1. 32. CHALLENGE Two or more integers or monomials with a GCF of 1 are said to be relatively prime. Copy and complete the chart to determine which pairs of monomials are relatively prime.
Monomial
Prime Factorization
15a 2bc 3 6b 3c 3d 12cd 2f
33. OPEN ENDED Name three monomials with a GCF of 6y 3. Explain your answer.
22d 3fg 2 30f 2gh 2
34. WRITING IN MATH Define prime factorization in your own words. Explain how to find the prime factorization of a monomial, and how a prime factorization helps you determine the GCF of two or more monomials. connectED.mcgrawhill.com
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SPI 3102.5.1, SPI 3102.3.5, SPI 3102.3.8
Standardized Test Practice 35. Abigail surveyed 320 of her classmates about what type of movie they prefer. The results of the survey are shown below. What percent of her classmates enjoyed action movies? Type of Movie
Number of Responses
comedy
160
drama
25
science ﬁction
55
action
80
A 25% B 50%
y
A y = 2x + 4 B y = 2x  5 1 C y=_ x6 2
1 x+3 D y = _
x
0
2
C 75% D 95%
36. What is the value of c in the equation 4c  27 = 19 + 2c? F G H J
37. Which equation best represents a line parallel to the line shown below?
4 4 23 46
38. SHORT RESPONSE The table shows a fiveday forecast indicating high (H) and low (L) temperatures. Organize the temperatures in a matrix. Mon
Tue
Wed
Thurs
Fri
H
92
87
85
88
90
L
68
64
62
65
66
Spiral Review Find each product. (Lesson 78) 39. (a  4)2
40. (c + 6)2
41. (z  5)2
42. (n  3)(n + 3)
43. (y + 2)2
44. (d  7)(d + 7)
45. (2m  3)(m + 4)
46. (h  2)(3h  5)
47. (t + 2)(t + 9)
48. (8r  1)(r  6)
49. (p + 3q)(p + 3q)
50. (n  4)(n + 2)(n + 1)
Find each product. (Lesson 77)
Write an augmented matrix to solve each system of equations. (Lesson 67) 51. y = 2x + 3 y = 4x  1
52. 8x + 2y = 13 4x + y = 11
1 53. x + _ y=5 3 2x + 3y = 1
54. FINANCIAL LITERACY Suppose you have already saved $50 toward the cost of a new television. You plan to save $5 more each week. Write and graph an equation for the total amount T that you will have w weeks from now. (Lesson 41)
Skills Review Use the Distributive Property to rewrite each expression. (Lesson 14) 55. 2(4x  7)
1 56. _ d(2d + 6)
57. h(6h  1)
58. 9m  9p
59. 5y  10
60. 3z  6x
2
474  Lesson 81  Monomials and Factoring
Algebra Lab
Factoring Using the Distributive Property When two or more numbers are multiplied, these numbers are factors of the product. Sometimes you know the product of binomials and are asked to find the factors. This is called factoring. You can use algebra tiles and a product mat to factor binomials.
Tennessee Curriculum Standards SPI 3102.3.3 Factor polynomials. Also addresses CLE 3102.3.2.
Activity 1 Use algebra tiles to factor 2x  8. Step 1
Step 2
Model 2x  8.
Y
Y
1
1
1
1
1
1
1
1
Arrange the tiles into a rectangle. The total area of the rectangle represents the product, and its length and width represent the factors. x4 1 1 1 1 1 1 1 1
Y Y
2
The rectangle has a width of 2 and a length of x  4. Therefore, 2x  8 = 2(x  4).
Activity 2 Use algebra tiles to factor x 2 + 3x. Step 1
Model x 2 + 3x.
Step 2
Arrange the tiles into a rectangle. x+3
Y
2
Y
Y
Y x
Y
2
Y Y Y
The rectangle has a width of x and a length of x + 3. Therefore, x 2 + 3x = x(x + 3).
Model and Analyze Use algebra tiles to factor each binomial. 1. 4x + 12
2. 4x  6
3. 3x 2 + 4x
4. 10  2x
Determine whether each binomial can be factored. Justify your answer with a drawing. 5. 6x  9
6. 5x  4
7. 4x 2 + 7
8. x 2 + 3x
9. WRITING IN MATH Write a paragraph that explains how you can use algebra tiles to determine whether a binomial can be factored. Include an example of one binomial that can be factored and one that cannot. connectED.mcgrawhill.com
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Using the Distributive Property Then
Now
Why?
You found the GCF of a set of monomials.
1
Use the Distributive Property to factor polynomials.
2
Solve equations of the form ax 2 + bx = 0.
The cost of rent for Mr. Cole’s store is determined by the square footage of the space. The area of the store can be modeled by the equation A = 1.6w 2 + 6w, where w is the width of the store in feet. We can use factoring and the Zero Product Property to find possible dimensions of the store.
(Lesson 81)
NewVocabulary factoring factoring by grouping Zero Product Property
1 Use the Distributive Property to Factor
In Chapter 7, the Distributive Property was used to multiply a monomial by a polynomial.
You can work backward to express a polynomial as a product of a monomial factor and a polynomial factor. 1.6w 2 + 6w = 1.6w(w) + 6(w) = w(1.6w + 6)
Tennessee Curriculum Standards CLE 3102.3.2 Understand and apply properties in order to perform operations with, evaluate, simplify, and factor expressions and polynomials. ✔ 3102.3.8 Find the GCF of the terms in a polynomial. SPI 3102.3.3 Factor polynomials.
So, 5z(4z + 7) is the factored form of 20z 2 + 35z. Factoring a polynomial involves finding the completely factored form.
Example 1 Use the Distributive Property Use the Distributive Property to factor each polynomial. a. 27y 2 + 18y Find the GCF of each term. 27y 2 = 3 · 3 · 3 · y · y 18y = 2 · 3 · 3 · y
Factor each term. Circle common factors.
GCF = 3 · 3 · y or 9y Write each term as the product of the GCF and the remaining factors. Use the Distributive Property to factor out the GCF. Rewrite each term using the GCF. 27y 2 + 18y = 9y(3y) + 9y(2) = 9y(3y + 2) Distributive Property b. 4a 2b  8ab 2 + 2ab 4a 2b = 1 · 2 · 2 · a · a · b
Factor each term.
8ab 2
Circle common factors.
= 1 · 2 · 2 · 2 · a · b · b
2ab = 2 · a · b GCF = 2 · a · b or 2ab 4a 2b  8ab 2 + 2ab = 2ab(2a)  2ab(4b) + 2ab(1) = 2ab(2a  4b + 1)
Rewrite each term using the GCF. Distributive Property
GuidedPractice 1A. 15w  3v
476  Lesson 82
1B. 7u 2t 2 + 21ut 2  ut
Using the Distributive Property to factor polynomials with four or more terms is called factoring by grouping because terms are put into groups and then factored. The Distributive Property is then applied to a common binomial factor.
KeyConcept Factoring by Grouping Words
A polynomial can be factored by grouping only if all of the following conditions exist. • There are four or more terms. • Terms have common factors that can be grouped together. • There are two common factors that are identical or additive inverses of each other.
Symbols
ax + bx + ay + by = (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (x + y)(a + b)
Example 2 Factor by Grouping Factor 4qr + 8r + 3q + 6. 4qr + 8r + 3q + 6 = (4qr + 8r) + (3q + 6) = 4r(q + 2) + 3(q + 2)
Original expression Group terms with common factors. Factor the GCF from each group.
Notice that (q + 2) is common in both groups, so it becomes the GCF. Distributive Property = (4r + 3)(q + 2)
GuidedPractice Factor each polynomial. 2A. rn + 5n  r  5
2B. 3np + 15p  4n  20
It can be helpful to recognize when binomials are additive inverses of each other. For example 6  a = 1(a  6).
StudyTip Check To check your factored answers, multiply your factors out. You should get your original expression as a result.
Example 3 Factor by Grouping with Additive Inverses Factor 2mk  12m + 42  7k. 2mk  12m + 42  7k = (2mk  12m) + (42  7k) = 2m(k  6) + 7(6  k) = 2m(k  6) + 7[(1)(k  6)] = 2m(k  6)  7(k  6) = (2m  7)(k  6)
Group terms with common factors. Factor the GCF from each group. 6  k = 1(k  6) Associative Property Distributive Property
GuidedPractice Factor each polynomial. 3A. c  2cd + 8d  4
3B. 3p  2p 2  18p + 27
connectED.mcgrawhill.com
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2 Solve Equations by Factoring
Some equations can be solved by factoring.
Consider the following.
3(0) = 0
0(2  2) = 0
312(0) = 0
0(0.25) = 0
Notice that in each case, at least one of the factors is 0. These examples are demonstrations of the Zero Product Property.
KeyConcept Zero Product Property Words
If the product of two factors is 0, then at least one of the factors must be 0.
Symbols
For any real numbers a and b, if ab = 0, then a = 0, b = 0, or both a and b equal zero.
Recall from Lesson 32 that a solution or root of an equation is any value that makes the equation true.
Example 4 Solve Equations
WatchOut!
Solve each equation. Check your solutions.
Unknown Value It may be tempting to solve an equation by dividing each side by the variable. However, the variable has an unknown value, so you may be dividing by 0, which is undefined.
a. (2d + 6)(3d  15) = 0 (2d + 6)(3d  15) = 0
Original equation
2d + 6 = 0
Zero Product Property
or 3d  15 = 0
2d = 6
3d = 15
d = 3
d=5
Solve each equation. Divide.
The roots are 3 and 5. CHECK Substitute 3 and 5 for d in the original equation. (2d + 6)(3d  15) = 0
(2d + 6)(3d  15) = 0
[2(3) + 6][3(3)  15] 0
[2(5) + 6][3(5)  15] 0
(6 + 6)(9  15) 0
(10 + 6)(15  15) 0
(0)(24) 0
16(0) 0
0=0
0=0
b. c 2 = 3c c 2 = 3c
Original equation
c 2  3c = 0
Subtract 3c from each side to get 0 on one side of the equation.
c(c  3) = 0 c=0
or
Factor by using the GCF to get the form ab = 0.
c3=0 c=3
The roots are 0 and 3.
Zero Product Property Solve each equation. Check by substituting 0 and 3 for c.
GuidedPractice 4A. 3n(n + 2) = 0
478  Lesson 82  Using the Distributive Property
4B. 8b 2  40b = 0
4C. x 2 = 10x
RealWorld Example 5 Use Factoring AGILITY Penny is a Fox Terrier who competes with her trainer in the agility course. Within the course, Penny must leap over a hurdle. Penny’s jump can be modeled by the equation h = 16t 2 + 20t, where h is the height of the leap in inches at t seconds. Find the values of t when h = 0. h = 16t 2 + 20t
Original equation
0 = 16t 2 + 20t
Substitution, h = 0
0 = 4t(4t + 5)
Factor by using the GCF.
4t = 0 or
Zero Product Property
4t + 5 = 0
t=0
RealWorldLink Dog agility tests a person’s skills as a trainer and handler. Competitors race through an obstacle course that includes hurdles, tunnels, a seesaw, and line poles. Source: United States Dog Agility Association
4t = 5
Solve each equation.
5 t=_ or 1.25 4
Penny’s height is 0 inches at 0 seconds and 1.25 seconds into the jump.
GuidedPractice 5. KANGAROOS The hop of a kangaroo can be modeled by h = 24t  16t 2 where h represents the height of the hop in meters and t is the time in seconds. Find the values of t when h = 0.
Check Your Understanding Example 1
Divide each side by 4.
= StepbyStep Solutions begin on page R12.
Use the Distributive Property to factor each polynomial. 1. 21b  15a
2. 14c 2 + 2c
3. 10g 2h 2 + 9gh 2  g 2h
4. 12jk 2 + 6j 2k + 2j 2k
2
Examples 2–3 Factor each polynomial.
Example 4
5 np + 2n + 8p + 16
6. xy  7x + 7y  49
7. 3bc  2b  10 + 15c
8. 9fg  45f  7g + 35
Solve each equation. Check your solutions. 9. 3k(k + 10) = 0 11. 20p 2  15p = 0
Example 5
10. (4m + 2)(3m  9) = 0 12. r 2 = 14r
13. SPIDERS Jumping spiders can commonly be found in homes and barns throughout the United States. A jumping spider’s jump can be modeled by the equation h = 33.3t  16t 2, where t represents the time in seconds and h is the height in feet. a. When is the spider’s height at 0 feet? b. What is the spider’s height after 1 second? after 2 seconds? 14. ROCKETS At a Fourth of July celebration, a rocket is launched straight up with an initial velocity of 125 feet per second. The height h of the rocket in feet above sea level is modeled by the formula h = 125t  16t 2, where t is the time in seconds after the rocket is launched. a. What is the height of the rocket when it returns to the ground? b. Let h = 0 in the equation and solve for t. c. How many seconds will it take for the rocket to return to the ground? connectED.mcgrawhill.com
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Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Use the Distributive Property to factor each polynomial. 15. 16t  40y
16. 30v + 50x
17. 2k 2 + 4k
18. 5z 2 + 10z
19. 4a 2b 2 + 2a 2b  10ab 2
20. 5c 2v  15c 2v 2 + 5c 2v 3
Examples 2–3 Factor each polynomial. 21 fg  5g + 4f  20
22. a 2  4a  24 + 6a
23. hj  2h + 5j  10
24. xy  2x  2 + y
25. 45pq  27q  50p + 30
26. 24ty  18t + 4y  3
27. 3dt  21d + 35  5t
28. 8r 2 + 12r
29. 21th  3t  35h + 5
30. vp + 12v + 8p + 96
31. 5br  25b + 2r  10
32. 2nu  8u + 3n  12
33. 5gf 2 + g 2f + 15gf
34. rp  9r + 9p  81
2
Example 4
Example 5
2 2
35. 27cd  18c d + 3cd
36. 18r 3t 2 + 12r 2t 2  6r 2t
37. 48tu  90t + 32u  60
38. 16gh + 24g  2h  3
Solve each equation. Check your solutions. 39. 3b(9b  27) = 0
40. 2n(3n + 3) = 0
41. (8z + 4)(5z + 10) = 0
42. (7x + 3)(2x  6) = 0
43. b 2 = 3b
44. a 2 = 4a
45. GEOMETRY Use the drawing at the right. a. Write an expression in factored form to represent the area of the blue section.
3 3
3
a
b. Write an expression in factored form to represent the area of the region formed by the outer edge.
b 3
c. Write an expression in factored form to represent the orange region.
B
46. FIREWORKS A teninch fireworks shell is fired from ground level. The height of the shell in feet is given by the formula h = 263t  16t 2, where t is the time in seconds after launch. a. Write the expression that represents the height in factored form. b. At what time will the height be 0? Is this answer practical? Explain. c. What is the height of the shell 8 seconds and 10 seconds after being fired? d. At 10 seconds, is the shell rising or falling? 47. ARCHITECTURE The frame of a doorway is an arch that can be modeled by the graph of the equation y = 3x 2 + 12x, where x and y are measured in feet. On a coordinate plane, the floor is represented by the xaxis. a. Make a table of values for the height of the arch if x = 0, 1, 2, 3, and 4 feet. b. Plot the points from the table on a coordinate plane and connect the points to form a smooth curve to represent the arch. c. How high is the doorway?
480  Lesson 82  Using the Distributive Property
48. RIDES Suppose the height of a rider after being dropped can be modeled by h = 16t 2  96t + 160, where h is the height in feet and t is time in seconds. a. Write an expression to represent the height in factored form. b. From what height is the rider initially dropped? c. At what height will the rider be after 3 seconds of falling? Is this possible? Explain.
C
49 ARCHERY The height h in feet of an arrow can be modeled by the equation h = 64t  16t 2, where t is time in seconds. Ignoring the height of the archer, how long after the arrow is released does it hit the ground? 50. TENNIS A tennis player hits a tennis ball upward with an initial velocity of 80 feet per second. The height h in feet of the tennis ball can be modeled by the equation h = 80t  16t 2, where t is time in seconds. Ignoring the height of the tennis player, how long does it take the ball to hit the ground? ?
51.
?
MULTIPLE REPRESENTATIONS In this problem, you will 2 x ? ? explore the box method of factoring. To factor x 2 + x  6, write the first term in the top lefthand corner of the box, and then write the last term in the lower righthand corner. ? ? 6 a. Analytical Determine which two factors have a product of 6 and a sum of 1. b. Symbolic Write each factor in an empty square in the box. Include the positive or negative sign and variable. c. Analytical Find the factor for each row and column of the box. What are the factors of x 2 + x  6? d. Verbal Describe how you would use the box method to factor x 2  3x  40.
H.O.T. Problems
Use HigherOrder Thinking Skills
52. ERROR ANALYSIS Hernando and Rachel are solving 2m 2 = 4m. Is either of them correct? Explain your reasoning.
Hernando 2m 2 = 4m 2 2m 2 _ _ = 4m
m
2m
2m = 2 m=1
Rachel 2m 2 = 4m 2m 2 – 4m = 0 2m(m – 2) = 0 2m = 0 or m – 2 = 0 m = 0 or 2
53. CHALLENGE Given the equation (ax + b)(ax  b) = 0, solve for x. What do we know about the values of a and b? 54. OPEN ENDED Write a fourterm polynomial that can be factored by grouping. Then factor the polynomial. 55. REASONING Given the equation c = a 2  ab, for what values of a and b does c = 0? 56. WRITING IN MATH Explain how to solve a quadratic equation by using the Zero Product Property. connectED.mcgrawhill.com
481
SPI 3102.3.3, SPI 3102.5.5, SPI 3108.4.7, SPI 3108.4.6
Standardized Test Practice 57. Which is a factor of 6z 2  3z  2 + 4z? A 2z + 1 B 3z  2
C z+2 D 2z  1
58. PROBABILITY Hailey has 10 blocks: 2 red, 4 blue, 3 yellow, and 1 green. What is the probability that a randomly chosen block will be either red or yellow? 3 F _ 10 1 G _ 5
1 H _ 2 7 J _ 10
59. GRIDDED RESPONSE Cho is making a 140inch by 160inch quilt with quilt squares that measure 8 inches on each side. How many will be needed to make the quilt? 60. GEOMETRY The area of the right triangle shown below is 5h square centimeters. What is the height of the triangle? A B C D
2 cm 5 cm 8 cm 10 cm
2h
h
Spiral Review Find the GCF of each set of monomials. (Lesson 81) 61. 15, 25
62. 40, 100
63. 16x, 24x 2
64. 30a 2, 50ab 2
65. 8c 2d 3, 16c 3d
66. 4y, 18y 2, 6y 3
67. GENETICS Brown genes B are dominant over blue genes b. A person with genes BB or Bb has brown eyes. Someone with genes bb has blue eyes. Elisa has brown eyes with Bb genes, and Bob has blue eyes. Write an expression for the possible eye coloring of Elisa and Bob’s children. Determine the probability that their child would have blue eyes. (Lesson 78) Simplify. (Lesson 71) 68. (ab 4)(ab 2) 2
71. (9xy 7)
69. (p 5r 4)(p 2r)
70. (7c 3d 4)(4cd 3)
4 2 72. ⎣⎡(3 2) ⎤⎦
3 2 73. ⎣⎡(4 2) ⎤⎦
74. BASKETBALL In basketball, a free throw is 1 point and a field goal is either 2 or 3 points. In a season, Tim Duncan of the San Antonio Spurs scored a total of 1342 points. The total number of 2point field goals and 3point field goals was 517, and he made 305 of the 455 free throws that he attempted. Find the number of 2point field goals and 3point field goals Duncan made that season. (Lesson 64) Solve each inequality. Check your solution. (Lesson 53) 75. 3y  4 > 37
76. 5q + 9 > 24
77. 2k + 12 < 30
78. 5q + 7 ≤ 3(q + 1)
z 79. _ + 7 ≥ 5 4
80. 8c  (c  5) > c + 17
81. (a + 2)(a + 5)
82. (d + 4)(d + 10)
83. (z  1)(z  8)
84. (c + 9)(c  3)
85. (x  7)(x  6)
86. (g  2)(g + 11)
Skills Review Find each product. (Lesson 77)
482  Lesson 82  Using the Distributive Property
Algebra Lab
Factoring Trinomials You can use algebra tiles to factor trinomials. If a polynomial represents the area of a rectangle formed by algebra tiles, then the rectangle’s length and width are factors of the area. If a rectangle cannot be formed to represent the trinomial, then the trinomial is not factorable.
Tennessee Curriculum Standards SPI 3102.3.3 Factor polynomials.
Activity 1 Factor x 2 + bx + c Use algebra tiles to factor x 2 + 4x + 3. 2
Step 1 Model x + 4x + 3.
1
Y
2
Y
Y
Y
Y
1 1
Step 2 Place the x 2tile at the corner of the product mat. Arrange the 1tiles into a rectangular array. Because 3 is prime, the 3 tiles can be arranged in a rectangle in one way, a 1by3 rectangle.
Y
2
1
1
x+3
Step 3 Complete the rectangle with the xtiles. The rectangle has a width of x + 1 and a length of x + 3.
1
Y
x+1
Therefore, x 2 + 4x + 3 = (x + 1)(x + 3).
2
Y Y Y
Y
1 1
1
Activity 2 Factor x 2 + bx + c Use algebra tiles to factor x 2 + 8x + 12. Step 1 Model x 2 + 8x + 12.
Y
Step 2 Place the x 2tile at the corner of the product mat. Arrange the 1tiles into a rectangular array. Since 12 = 3 × 4, try a 3by4 rectangle. Try to complete the rectangle. Notice that there is an extra xtile.
Therefore, x 2 + 8x + 12 = (x + 2)(x + 6).
Y Y
Y
Y
Y
Y
Y
Y
Y
2
1
1
1
1
1
1
1
1
1
1
1
1
Y 1 1 1
1 1 1
1 1 1
Y Y Y
1 1 1
2
Y Y Y Y Y 1 1 1
1 1 1
1 1 1
1 1 1
x+6
Step 3 Arrange the 1tiles into a 2by6 rectangular array. This time you can complete the rectangle with the xtiles. The rectangle has a width of x + 2 and a length of x + 6.
2
x+2
Y Y Y
2
Y Y Y Y Y Y 1 1
1 1
1 1
1 1
1 1
1 1
(continued on the next page) connectED.mcgrawhill.com
483
Algebra Lab
Factoring Trinomials Continued Activity 3 Factor x 2  bx + c Use algebra tiles to factor x 2  5x + 6. Y
Step 1 Model x 2  5x + 6. Step 2 Place the x 2tile at the corner of the product mat. Arrange the 1tiles into a 2by3 rectangular array as shown.
2
Y Y Y Y Y
1
1
1
1
1
1
x3
Y
2
Y
2
x2
Step 3 Complete the rectangle with the xtiles. The rectangle has a width of x  2 and a length of x  3. Therefore, x 2  5x + 6 = (x  2)(x  3).
1 1
1 1
YY YY Y
Y Y
1 1
1 1
1 1
1 1
Activity 4 Factor x 2  bx  c Use algebra tiles to factor x 2  4x  5. Y
Step 1 Model x 2  4x  5.
1 Y Y Y Y 1 1
2
1 1
Step 2 Place the x 2tile at the corner of the product mat. Arrange the 1tiles into a 1by5 rectangular array as shown.
Y
2
1 1 1 1 1
Step 3 Place the xtile as shown. Recall that you can add zero pairs without changing the value of the polynomial. In this case, add a zero pair of xtiles. The rectangle has a width of x + 1 and a length of x  5.
x5
Y
2
YY YY YY Y
x+1
1 1 1 1 1
Y
2
Y
Y Y Y YY Y 1 1 1 1 1 zero pair
Therefore, x 2  4x  5 = (x + 1)(x  5).
Model and Analyze Use algebra tiles to factor each trinomial. 1. x 2 + 3x + 2
2. x 2 + 6x + 8
3. x 2 + 3x  4
4. x 2  7x + 12
5. x 2 + 7x + 10
6. x 2  2x + 1
7. x 2 + x  12
8. x 2  8x + 15
Tell whether each trinomial can be factored. Justify your answer with a drawing. 9. x 2 + 3x + 6
10. x 2  5x  6
11. x 2  x  4
13. WRITING IN MATH How can you use algebra tiles to determine whether a trinomial can be factored?
484  Explore 83  Algebra Lab: Factoring Trinomials
12. x 2  4
Quadratic Equations: x 2 + bx + c = 0 Then
Now
Why?
You multiplied binomials by using the FOIL method.
1
Factor trinomials of the form x 2 + bx + c.
2
Solve equations of the form x 2 + bx + c = 0.
Diana is having a rectangular inground swimming pool installed and she wants to include a 24foot fence around the pool. The pool requires a space of 36 square feet. What dimensions should the pool have?
(Lesson 77)
To solve this problem, the landscape architect needs to find two numbers that have a product of 36 and a sum of 12, half the perimeter of the pool.
NewVocabulary quadratic equation
1
Factor x 2 + bx + c In Lesson 77, you learned how to multiply two
binomials by using the FOIL method. Each of the binomials was a factor of the product. The pattern for multiplying two binomials can be used to factor certain types of trinomials. (x + 3)(x + 4) = x 2 + 4x + 3x + 3 · 4
Tennessee Curriculum Standards
= x 2 + (4 + 3)x + 3 · 4
CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. SPI 3102.3.3 Factor polynomials. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function. Also addresses CLE 3102.3.2, ✓3103.3.9, ✓3103.3.30, and ✓3102.3.32.
= x + 7x + 12
2
Use the FOIL method. Distributive Property Simplify.
Notice that the coefficient of the middle term, 7x, is the sum of 3 and 4, and the last term, 12, is the product of 3 and 4. Observe the following pattern in this multiplication. (x + 3)(x + 4) = x 2 + (4 + 3)x + (3 · 4) (x + m)(x + p) = x 2 + (p + m)x + mp = x 2 + (m + p)x + mp x2 +
bx
+ c
Let 3 = m and 4 = p. Commutative (+) b = m + p and c = mp
Notice that the coefficient of the middle term is the sum of m and p, and the last term is the product of m and p. This pattern can be used to factor trinomials of the form x 2 + bx + c.
KeyConcept Factoring x 2 + bx + c Words
To factor trinomials in the form x 2 + bx + c, find two integers, m and p, with a sum of b and a product of c. Then write x 2 + bx + c as (x + m)(x + p).
Symbols
x 2 + bx + c = (x + m)(x + p) when m + p = b and mp = c
Example
x 2 + 6x + 8 = (x + 2)(x + 4), because 2 + 4 = 6 and 2 · 4 = 8.
When c is positive, its factors have the same signs. Both of the factors are positive or negative based upon the sign of b. If b is positive, the factors are positive. If b is negative, the factors are negative. connectED.mcgrawhill.com
485
Example 1 b and c Are Positive
ProblemSolvingTip Guess and Check When factoring a trinomial, make an educated guess, check for reasonableness, and then adjust the guess until the correct answer is found.
Factor x 2 + 9x + 20. In this trinomial, b = 9 and c = 20. Since c is positive and b is positive, you need to find two positive factors with a sum of 9 and a product of 20. Make an organized list of the factors of 20, and look for the pair of factors with a sum of 9. Factors of 20 1, 20 2, 10 4, 5
Sum of Factors 21 12 9
The correct factors are 4 and 5.
2
x + 9x + 20 = (x + m)(x + p) = (x + 4)(x + 5)
Write the pattern. m = 4 and p = 5
CHECK You can check this result by multiplying the two factors. The product should be equal to the original expression. (x + 4)(x + 5) = x 2 + 5x + 4x + 20 = x 2 + 9x + 20
FOIL Method Simplify.
GuidedPractice Factor each polynomial. 1B. 9 + 10t + t 2
1A. d 2 + 11x + 24
When factoring a trinomial in which b is negative and c is positive, use what you know about the product of binomials to narrow the list of possible factors.
Example 2 b Is Negative and c Is Positive Factor x 2  8x + 12. Confirm your answer using a graphing calculator. In this trinomial, b = 8 and c = 12. Since c is positive and b is negative, you need to find two negative factors with a sum of 8 and a product of 12. Factors of 12
Sum of Factors
StudyTip
1, 12
13
Finding Factors Once the correct factors are found, it is not necessary to test any other factors. In Example 2, 2 and 6 are the correct factors, so 3 and 4 do not need to be tested.
2, 6
8
3, 4
7
2
x  8x + 12 = (x + m)(x + p) = (x  2)(x  6)
The correct factors are 2 and 6. Write the pattern. m = 2 and p = 6
CHECK Graph y = x 2  8x + 12 and y = (x  2)(x  6) on the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly.
GuidedPractice
[10, 10] scl: 1 by [10, 10] scl: 1
Factor each polynomial. 2A. 21  22m + m 2
486  Lesson 83  Quadratic Equations: x 2 + bx + c = 0
2B. w 2  11w + 28
ReviewVocabulary absolute value the distance a number is from zero on a number line, written n (Lesson 02)
When c is negative, its factors have opposite signs. To determine which factor is positive and which is negative, look at the sign of b. The factor with the greater absolute value has the same sign as b.
Example 3 c is Negative Factor each polynomial. Confirm your answers using a graphing calculator. a. x 2 + 2x  15 In this trinomial, b = 2 and c = 15. Since c is negative, the factors m and p have opposite signs. So either m or p is negative, but not both. Since b is positive, the factor with the greater absolute value is also positive. List the factors of 15, where one factor of each pair is negative. Look for the pair of factors with a sum of 2. Factors of 15 1, 15
Sum of Factors 14
3, 5
2
The correct factors are 3 and 5.
2
x + 2x  15 = (x + m)(x + p) = (x  3)(x + 5) 2
Write the pattern. m = 3 and p = 5
CHECK (x  3)(x + 5) = x + 5x  3x  15 = x 2 + 2x  15
FOIL Method Simplify.
b. x 2  7x  18 In this trinomial, b = 7 and c = 18. Either m or p is negative, but not both. Since b is negative, the factor with the greater absolute value is also negative. List the factors of 18, where one factor of each pair is negative. Look for the pair of factors with a sum of 7. Factors of 18
Sum of Factors
1, 18
17
2, 9
7
3, 6
3
2
The correct factors are 2 and 9.
x  7x  18 = (x + m)(x + p) = (x + 2)(x  9)
Write the pattern. m = 2 and p = 9
CHECK Graph y = x 2  7x  18 and y = (x + 2)(x  9) on the same screen.
[10, 15] scl: 1 by [40, 20] scl: 1
The graphs coincide. Therefore, the trinomial has been factored correctly.
GuidedPractice 3A. y 2 + 13y  48
3B. r 2  2r  24 connectED.mcgrawhill.com
487
2 Solve Equations by Factoring
A quadratic equation can be written in the standard form ax 2 + bx + c = 0, where a ≠ 0. Some equations of the form x 2 + bx + c = 0 can be solved by factoring and then using the Zero Product Property.
Example 4 Solve an Equation by Factoring
StudyTip Solving an Equation By Factoring Remember to get 0 on one side of the equation before factoring.
Solve x 2 + 6x = 27. Check your solutions. Original equation x 2 + 6x = 27 2 x + 6x  27 = 0 Subtract 27 from each side. (x  3)(x + 9) = 0 Factor. x  3 = 0 or x + 9 = 0 Zero Product Property x=3 x = 9 Solve each equation. The roots are 3 and 9. CHECK Substitute 3 and 9 for x in the original equation. x 2 + 6x = 27 (3) + 6(3) 27 9 + 18 27 27 = 27
x 2 + 6x = 27 (9) + 6(9) 27 81  54 27 27 = 27
2
2
GuidedPractice Solve each equation. Check your solutions. 4A. z 2  3z = 70
4B. x 2 + 3x  18 = 0
Factoring can be useful when solving realworld problems.
RealWorld Example 5 Solve a Problem by Factoring DESIGN Ling is designing a poster. The top of the poster is 4 inches long and the rest of the poster is 2 inches longer than the width. If the poster requires 616 square inches of poster board, find the width w of the poster. Understand You want to find the width of the poster.
4
w+2
Plan Since the poster is a rectangle, width · length = area.
RealWorldLink A company that produces event signs recommends foamcore boards for event signs that will be used only once. For signs used more than once, use a stronger type of foamcore board. Source: MegaPrint Inc.
Solve Let w = the width of the poster. w The length is w + 2 + 4 or w + 6. w(w + 6) = 616 Write the equation. w 2 + 6w = 616 Multiply. w 2 + 6w  616 = 0 Subtract 616 from each side. (w + 28)(w  22) = 0 Factor. w + 28 = 0 or w  22 = 0 Zero Product Property w = 28 w = 22 Solve each equation. Since dimensions cannot be negative, the width is 22 inches. Check If the width is 22 inches, then the area of the poster is 22 · (22 + 6) or 616 square inches, which is the amount the poster requires.
GuidedPractice 5. GEOMETRY The height of a parallelogram is 18 centimeters less than its base. If the area is 175 square centimeters, what is its height?
488  Lesson 83  Quadratic Equations: x 2 + bx + c = 0
Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–3 Factor each polynomial. Confirm your answers using a graphing calculator.
Example 4
1. x 2 + 14x + 24
2. y 2  7y  30
3. n 2 + 4n  21
4. m 2  15m + 50
Solve each equation. Check your solutions. 5. x 2  4x  21 = 0
6. n 2  3n + 2 = 0
7. x 2  15x + 54 = 0
8. x 2 + 12x = 32
9. x 2  x  72 = 0 Example 5
10. x 2  10x = 24
11. FRAMING Tina bought a frame for a photo, but the photo is too big for the frame. Tina needs to reduce the width and length of the photo by the same amount. The area of the photo should be reduced to half the original area. If the original photo is 12 inches by 16 inches, what will be the dimensions of the smaller photo?
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–3 Factor each polynomial. Confirm your answers using a graphing calculator.
Example 4
Example 5
12. x 2 + 17x + 42
13. y 2  17y + 72
14. a 2 + 8a  48
15. n 2  2n  35
16. 44 + 15h + h 2
17. 40  22x + x 2
18. 24  10x + x 2
19. –42  m + m 2
Solve each equation. Check your solutions. 20. x 2  7x + 12 = 0
21 y 2 + y = 20
22. x 2  6x = 27
23. a 2 + 11a = 18
24. c 2 + 10c + 9 = 0
25. x 2  18x = 32
26. n 2  120 = 7n
27. d 2 + 56 = 18d
28. y 2  90 = 13y
29. h 2 + 48 = 16h
30. GEOMETRY A triangle has an area of 36 square feet. If the height of the triangle is 6 feet more than its base, what are its height and base? 31. GEOMETRY A rectangle has an area represented by x 2  4x  12 square feet. If the length is x + 2 feet, what is the width of the rectangle?
B
32. SOCCER The width of a high school soccer field is 45 yards shorter than its length. a. Define a variable, and write an expression for the area of the field. b. The area of the field is 9000 square yards. Find the dimensions.
– 45
Factor each polynomial. 33. q 2 + 11qr + 18r 2
34. x 2  14xy  51y 2
35. x 2  6xy + 5y 2
36. a 2 + 10ab  39b 2 connectED.mcgrawhill.com
489
37 SWIMMING The length of a rectangular swimming pool is 20 feet greater than its width. The area of the pool is 525 square feet. a. Define a variable and write an equation for the area of the pool. b. Solve the equation. c. Interpret the solutions. Do both solutions make sense? Explain. GEOMETRY Find an expression for the perimeter of a rectangle with the given area. 38. A = x 2 + 24x  81
C
40.
39. A = x 2 + 13x  90
MULTIPLE REPRESENTATIONS In this problem, you will explore factoring when the leading coefficient is not 1. a. Tabular Copy and complete the table below. Product of Two Binomials
ax 2 + mx + px + c
ax 2 + bx + c
m×p
a×c
(2x + 3)(x + 4)
2x 2 + 8x + 3x + 12
2x 2 + 11x + 12
24
24
(x + 1)(3x + 5) (2x  1)(4x + 1) (3x + 5)(4x  2)
b. Analytical How are m and p related to a and c? c. Analytical How are m and p related to b? d. Verbal Describe a process you can use for factoring a polynomial of the form ax 2 + bx + c.
H.O.T. Problems
Use HigherOrder Thinking Skills
41. ERROR ANALYSIS Jerome and Charles have factored x 2 + 6x  16. Is either of them correct? Explain your reasoning.
Charles
Jerome 2
x 2 + 6x – 16 = (x + 2)(x – 8)
x + 6x – 16 = (x – 2)(x + 8)
CHALLENGE Find all values of k so that each polynomial can be factored using integers. 42. x 2 + kx  19
43. x 2 + kx + 14
44. x 2  8x + k, k > 0
45. x 2  5x + k, k > 0
46. REASONING For any factorable trinomial, x 2 + bx + c, will the absolute value of b sometimes, always, or never be less than the absolute value of c? Explain. 47. OPEN ENDED Give an example of a trinomial that can be factored using the factoring techniques presented in this lesson. Then factor the trinomial. 48. CHALLENGE Factor (4y  5)2 + 3(4y  5)  70. 49.
E
WRITING IN MATH Explain how to factor trinomials of the form x 2 + bx + c and how to determine the signs of the factors of c.
490  Lesson 83  Quadratic Equations: x 2 + bx + c = 0
SPI 3102.3.5, SPI 3102.3.2, SPI 3102.1.2
Standardized Test Practice 50. Which inequality is shown in the graph below?
52. GEOMETRY Which expression represents the length of the rectangle?
y
3 A y ≤ _ x+3
4 _ B y < 3x + 3 4 _ C y > 3x + 3 4 _ D y ≥ 3x + 3 4
2
"=x 3x18
0
F G H J
x
51. SHORT RESPONSE Olivia must earn more than $254 from selling candy bars in order to go on a trip with the National Honor Society. If each candy bar is sold for $1.25, what is the fewest candy bars she must sell?
x+3
x+5 x+6 x6 x5
53. The difference of 21 and a number n is 6. Which equation shows the relationship? A 21  n = 6 B 21 + n = 6
C 21n = 6 D 6n = 21
Spiral Review Factor each polynomial. (Lesson 82) 54. 10a 2 + 40a
55. 11x + 44x 2y
56. 2m 3p 2  16mp 2 + 8mp
57. 2ax + 6xc + ba + 3bc
58. 8ac  2ad + 4bc  bd
59. x 2  xy  xy + y 2
60. FLOORING Emma is replacing her dining room floor, which is 10 feet by 12 feet. The flooring comes in pieces 1 foot by 1 foot, 2 foot by 2 foot, 3 foot by 3 foot, and 2 foot by 3 foot. Without cutting the pieces, which of the four sizes of flooring can Emma use? Explain. (Lesson 81) Perform the indicated matrix operations. If an operation cannot be performed, write impossible. (Lesson 66) ⎡7 3 4 2⎤ 1 ⎤ ⎡ 7 ⎤ ⎡ ⎤ ⎡ 0 8 10 4 2 9 61. ⎢ + 62. 9 2 4 6 3 3 ⎢ 2 3 ⎦ ⎣ 2 5 7 ⎦ ⎣ 1 ⎣ 4 8 7 ⎦ ⎣ 11 16 5⎦
⎢
⎢
⎡ 25 ⎤ ⎡ 63. 8 + ⎢⎣ 9 ⎣23 ⎦
⎢
21
2
⎤ 7 ⎦
⎡5
64. 3
⎢
4 ⎣1
3 ⎤ 0 8⎦
65. LANDSCAPING Kendrick is planning a circular flower garden with a low fence around the border. He has 38 feet of fence. What is the radius of the largest garden he can make? (Hint: C = 2πr) (Lesson 52)
Skills Review Factor each polynomial. (Lesson 82) 66. 6mx  4m + 3rx  2r
67. 3ax  6bx + 8b  4a
68. 2d 2g + 2fg + 4d 2h + 4fh connectED.mcgrawhill.com
491
MidChapter Quiz
Tennessee Curriculum Standards
Lessons 81 through 83
SPI 3102.3.2, SPI 3102.3.10
Factor each monomial completely. (Lesson 81)
Factor each polynomial. (Lesson 82)
1. 16x 3y 2
13. 5h + 40g
2. 35ab 4
14. 3x 2 + 6x + x + 2
3. 20m 5n 2
15. 5a 2  25a  a + 5
4. 13xy 3 Solve each equation. Check your solutions. (Lesson 82) 5. ROOM DESIGN The area of a rectangular room is 120 square feet. What are the possible wholenumber dimensions for the length and width of the room?
16. 2x(x  5) = 0 17. 6p 2  3p = 0 18. a 2 = 15a
(Lesson 81)
Find the GCF of each set of monomials. (Lesson 81) 2
6. 10a, 20a , 25a
19. ARCHITECTURE The curve of the archway under a bridge can be modeled by the equation
7. 13c, 25d
1 2 y = _ x + 6x, where x and y are
8. 21ab, 35a, 56ab 3
measured in feet. Copy and complete the table for each value of x. (Lesson 82)
5
x
y
0 10 15 20 30
9. FASHION A sales clerk is organizing 24 pairs of shoes for a sales display. In what ways can she organize the shoes so that she has the same number of shoes on each shelf, at least 4 pairs of shoes per shelf, and at least 2 shelves of shoes? Use the Distributive Property to factor each polynomial.
Factor each polynomial. Confirm your answers using a graphing calculator. (Lesson 83) 20. x 2  4x  21 21. x 2  10x + 24 22. x 2 + 4x  21
(Lesson 82)
10. 3xy  9x Solve each equation. Check your solutions. (Lesson 83)
11. 6ab + 12ab 2 + 18b
23. x 2  5x = 14 12. MULTIPLE CHOICE The area of the rectangle is 3x 2 + 6x  12 square units. What is the width of the rectangle? (Lesson 82)
2
x + 2x  4
24. x 2  3x  18 = 0 25. 24 + x 2 = 10x 26. MULTIPLE CHOICE A rectangle has a length that is 2 inches longer than its width. The area of the rectangle is 48 square inches. What is the length of the rectangle? (Lesson 83)
A 2 units
F 48 in.
B 3 units
G 8 in.
C 4 units
H 6 in.
D 6 units
J 2 in.
492  Chapter 8  MidChapter Quiz
Quadratic Equations: ax 2 + bx + c = 0 Then
Now
Why?
You factored trinomials of the form x 2 + bx + c.
1
Factor trinomials of the form ax 2 + bx + c.
The path of a rider on the amusement park ride shown at the right can be modeled by 16t 2  5t + 120.
2
Solve equations of the form ax 2 + bx + c = 0.
Factoring this expression can help the ride operators determine how long a rider rides on the initial swing.
(Lesson 83)
NewVocabulary prime polynomial
1
Factor ax 2 + bx + c In the last lesson, you factored quadratic expressions of
the form ax 2 + bx + c, where a = 1. In this lesson, you will apply the factoring methods to quadratic expressions in which a is not 1. The dimensions of the rectangle formed by the algebra tiles are the factors of 2x 2 + 5x + 3. The factors of 2x 2 + 5x + 3 are x + 1 and 2x + 3. You can also use the method of factoring by grouping to solve this expression.
Tennessee Curriculum Standards CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. SPI 3102.3.3 Factor polynomials. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function. Also addresses CLE 3102.3.2, ✓3103.3.9, and ✓3103.3.30.
Step 1 Apply the pattern: 2x 2 + 5x + 3 = 2x 2 + mx + px + 3. Step 2 Find two numbers that have a product of 2 · 3 or 6 and a sum of 5. Factors of 6 1, 6 2, 3
Sum of Factors 7 5
Step 3 Use grouping to find the factors. 2x 2 + 5x + 3 = 2x 2 + mx + px + 3 2
Write the pattern.
= 2x + 2x + 3x + 3
m = 2 and p = 3
= (2x 2 + 2x) + (3x + 3)
Group terms with common factors.
= 2x(x + 1) + 3(x + 1)
Factor the GCF.
= (2x + 3)(x + 1)
x + 1 is the common factor.
Therefore, 2x 2 + 5x + 3 = (2x + 3)(x + 1).
KeyConcept Factoring ax 2 + bx + c Words
To factor trinomials of the form ax 2 + bx + c, find two integers, m and p, with a sum of b and a product of ac. Then write ax 2 + bx + c as ax 2 + mx + px + c, and factor by grouping.
Example
5x 2  13x + 6 = 5x 2  10x  3x + 6 = 5x(x  2) + (3)(x  2) = (5x  3)(x  2)
m = 10 and p = 3
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Example 1 Factor ax 2 + bx + c Factor each trinomial. a. 7x 2 + 29x + 4 In this trinomial, a = 7, b = 29, and c = 4. You need to find two numbers with a sum of 29 and a product of 7 · 4 or 28. Make a list of the factors of 28 and look for the pair of factors with the sum of 29. Factors of 28 1, 28 2
Sum of Factors 29
The correct factors are 1 and 28.
2
7x + 29x + 4 = 7x + mx + px + 4 = 7x 2 + 1x + 28x + 4 = (7x 2 + 1x) + (28x + 4) = x(7x + 1) + 4(7x + 1) = (x + 4)(7x + 1)
StudyTip Greatest Common Factor Always look for a GCF of the terms of a polynomial before you factor.
Write the pattern. m = 1 and p = 28 Group terms with common factors. Factor the GCF. 7x + 1 is the common factor.
b. 3x 2 + 15x + 18 The GCF of the terms 3x 2, 15x, and 18 is 3. Factor this first. 3x 2 + 15x + 18 = 3(x 2 + 5x + 6) = 3(x + 3)(x + 2)
Distributive Property Find two factors of 6 with a sum of 5.
GuidedPractice 1A. 5x 2 + 13x + 6
1B. 6x 2 + 22x  8
Sometimes the coefficient of the xterm is negative.
Example 2 Factor ax 2  bx + c Factor 3x 2  17x + 20.
RealWorldCareer Urban Planner Urban planners design the layout of an area. They take into consideration the available land and geographical and environmental factors to design an area that benefits the community the most. City planners have a bachelor’s degree in planning and almost half have a master’s degree.
In this trinomial, a = 3, b = 17, and c = 20. Since b is negative, m + p will be negative. Since c is positive, mp will be positive. To determine m and p, list the negative factors of ac or 60. The sum of m and p should be 17. Factors of 60
Sum of Factors
2, 30
32
3, 20
23
4, 15
19
5, 12
17
The correct factors are 5 and 12.
3x 2  17x + 20 = 3x 2  12x  5x + 20 = (3x 2  12x) + (5x + 20) = 3x(x  4) + (5)(x  4) = (3x  5)(x  4)
m = 12 and p = 5 Group terms with common factors. Factor the GCF. Distributive Property
GuidedPractice 2A. 2n 2  n  1
494  Lesson 84  Quadratic Equations: ax 2 + bx + c = 0
2B. 10y 2  35y + 30
A polynomial that cannot be written as a product of two polynomials with integral coefficients is called a prime polynomial.
Example 3 Determine Whether a Polynomial is Prime Factor 4x 2  3x + 5, if possible. If the polynomial cannot be factored using integers, write prime. Factors Sum of In this trinomial, a = 4, b = 3, and c = 5. of 20 Factors Since b is negative, m + p is negative. Since 20, 1 21 c is positive, mp is positive. So, m and p are 4, 5 9 both negative. Next, list the factors of 20. 2, 10 12 Look for the pair with a sum of 3. There are no factors with a sum of 3. So the quadratic expression cannot be factored using integers. Therefore, 4x 2  3x + 5 is prime.
GuidedPractice Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime. 3A. 4r 2  r + 7
3B. 2x 2 + 3x  5
2 Solve Equations by Factoring
A model for the height of a projectile is given by h = 16t 2 + vt + h 0, where h is the height in feet, t is the time in seconds, v is the initial velocity in feet per second, and h 0 is the initial height in feet. Equations of the form ax 2 + bx + c = 0 can be solved by factoring and by using the Zero Product Property.
RealWorld Example 4 Solve Equations by Factoring WILDLIFE Suppose a cheetah pouncing on an antelope leaps with an initial velocity of 19 feet per second. How long is the cheetah in the air if it lands on the antelope’s hind quarter, 3 feet from the ground?
RealWorldLink Cheetahs are the fastest land animals in the world, reaching speeds of up to 70 mph. It can accelerate from 0 to 40 mph in 3 strides. It takes just seconds for the cheetah to reach the full speed of 70 mph. Source: Cheetah Conservation Fund
h = 16t 2 + vt + h 0 3 = 16t 2 + 19t + 0 0 = 16t 2 + 19t  3 0 = 16t 2  19t + 3 0 = (16t  3)(t  1) 16t  3 = 0 or t  1 = 0 16t = 3 t=1 3 _ t=
Equation for height h = 3, v = 19, and h 0 = 0 Subtract 3 from each side. Multiply each side by 1. Factor 16t 2  19t + 3. Zero Product Property Solve each equation.
16
3 3 The solutions are _ and 1 seconds. It takes the cheetah _ second to reach a 16
16
height of 3 feet on his way up. It takes the cheetah 1 second to reach a height of 3 feet on his way down. So, the cheetah is in the air 1 second before he catches the antelope.
WatchOut!
GuidedPractice
Keep the 1 Do not forget to carry the 1 that was factored out through the rest of the steps or multiply both sides by 1.
4. PHYSICAL SCIENCE A person throws a ball upward from a 506foot tall building. The ball’s height h in feet after t seconds is given by the equation h = 16t 2 + 48t + 506. The ball lands on a balcony that is 218 feet above the ground. How many seconds was it in the air? connectED.mcgrawhill.com
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Check Your Understanding
= StepbyStep Solutions begin on page R12.
Examples 1–3 Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime.
Example 4
1. 3x 2 + 17x + 10
2. 2x 2 + 22x + 56
3. 5x 2  3x + 4
4. 3x 2  11x  20
Solve each equation. Confirm your answers using a graphing calculator. 5. 2x 2 + 9x + 9 = 0
6. 3x 2 + 17x + 20 = 0
7. 3x 2  10x + 8 = 0
8. 2x 2  17x + 30 = 0
9. DISCUS Ken throws the discus at a school meet. a. What is the initial height of the discus?
I=U+U+
b. After how many seconds does the discus hit the ground?
Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–3 Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime.
Example 4
10. 5x 2 + 34x + 24
11 2x 2 + 19x + 24
12. 4x 2+ 22x + 10
13. 4x 2 + 38x + 70
14. 2x 2  3x  9
15. 4x 2  13x + 10
16. 2x 2 + 3x + 6
17. 5x 2 + 3x + 4
18. 12x 2 + 69x + 45
19. 4x 2  5x + 7
20. 5x 2 + 23x + 24
21. 3x 2  8x + 15
22. SHOT PUT An athlete throws a shot put with an initial velocity of 29 feet per second and from an initial height of 6 feet. a. Write an equation that models the height of the shot put in feet with respect to time in seconds. b. After how many seconds will the shot put hit the ground? Solve each equation. Confirm your answers using a graphing calculator.
B
23. 2x 2 + 9x  18 = 0
24. 4x 2 + 17x + 15 = 0
25. 3x 2 + 26x = 16
26. 2x 2 + 13x = 15
27. 3x 2 + 5x = 2
28. 4x 2 + 19x = 30
29. BASKETBALL When Jerald shoots a free throw, the ball is 6 feet from the floor and has an initial velocity of 20 feet per second. The hoop is 10 feet from the floor. a. Use the vertical motion model to determine an equation that models Jerald’s free throw. b. How long is the basketball in the air before it reaches the hoop? c. Raymond shoots a free throw that is 5 foot 9 inches from the floor with the same initial velocity. Will the ball be in the air more or less time? Explain. 30. DIVING Ben dives from a 10foot platform. The equation h = 16t 2 + 27t + 10 models the dive. How long will it take Ben to reach the water?
496  Lesson 84  Quadratic Equations: ax 2 + bx + c = 0
31 NUMBER THEORY Six times the square of a number x plus 11 times the number equals 2. What are possible values of x? Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime. 32. 6x 2  23x  20
33. 4x 2  15x  14
34. 5x 2 + 18x + 8
35. 6x 2 + 31x  35
36. 4x 2 + 5x  12
37. 12x 2 + x + 20
38. URBAN PLANNING The city has commissioned the building of a new park. The area of the park can be expressed as 660x 2 + 524x + 85. Factor this expression to find binomials with integer coefficients that represent possible dimensions of the park. If x = 8, what is the perimeter of the park?
C
39.
MULTIPLE REPRESENTATIONS In this problem, you will explore factoring a special type of polynomial. a. Geometric Draw a square and label the sides a. Within this square, draw a smaller square that shares a vertex with the first square. Label the sides b. What are the areas of the two squares? b. Geometric Cut and remove the small square. What is the area of the remaining region? c. Analytical Draw a diagonal line between the inside corner and outside corner of the figure, and cut along this line to make two congruent pieces. Then rearrange the two pieces to form a rectangle. What are the dimensions? d. Analytical Write the area of the rectangle as the product of two binomials. e. Verbal Complete this statement: a 2  b 2 = … Why is this statement true?
H.O.T. Problems
Use HigherOrder Thinking Skills
40. ERROR ANALYSIS Zachary and Samantha are solving 6x 2  x = 12. Is either of them correct? Explain your reasoning.
Zachary 2
6x – x = 12 x(6x – 1) = 12 x = 12 or 6x – 1 = 12 6x = 13 13 x=_ 6
Samantha 6x 2 – x = 12 6x 2 – x – 12 = 0 (2x – 3)(3x + 4) = 0 2x – 3 = 0 or 3x + 4 = 0 x = _32 x =  _43
41. REASONING A square has an area of 9x 2 + 30xy + 25y 2 square inches. The dimensions are binomials with positive integer coefficients. What is the perimeter of the square? Explain. 42. CHALLENGE Find all values of k so that 2x 2 + kx + 12 can be factored as two binomials using integers. 3 1 43. OPEN ENDED Write a quadratic equation with integer coefficients that has _ and _ 2 5 as solutions. Explain your reasoning.
44. WRITING IN MATH Explain how to determine which values should be chosen for m and n when factoring a polynomial of the form ax 2 + bx + c. connectED.mcgrawhill.com
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SPI 3102.3.9, SPI 3102.1.3, SPI 3102.3.10, SPI 3102.3.5
Standardized Test Practice 45. Gridded Response Savannah has two sisters. One sister is 8 years older than her and the other sister is 2 years younger than her. The product of Savannah’s sisters’ ages is 56. How old is Savannah?
47. What is the solution set of x 2 + 2x  24 = 0?
3 5 2 2 3 5 46. What is the product of _ a b and _ a b ?
48. Which is the solution set of x ≥ 2?
2 8 7 A _ a b
3
5
F {4, 6} G {3, 8}
5 2 2 3 B _ a b 5 2 8 3 C _ a b 5 2 2 7 D _ a b 5
H {3, 8} J {4, 6}
A 6 5 4 3 21 0 1 2 3 4
B C
6 5 4 3 21 0 1 2 3 4 6 5 4 3 21 0 1 2 3 4
D 6 5 4 3 21 0 1 2 3 4
Spiral Review Factor each polynomial. (Lesson 83) 49. x 2  9x + 14
50. n 2  8n + 15
51. x 2  5x  24
52. z 2 + 15z + 36
53. r 2 + 3r  40
54. v 2 + 16v + 63
Solve each equation. Check your solutions. (Lesson 82) 55. a(a  9) = 0
56. (2y + 6)(y  1) = 0
57. 10x 2  20x = 0
58. 8b 2  12b = 0
59. 15a 2 = 60a
60. 33x 2 = 22x
61. ART A painter has 32 units of yellow dye and 54 units of blue dye to make two shades of green. The units needed to make a gallon of light green and a gallon of dark green are shown. Make a graph showing the numbers of gallons of the two greens she can make, and list three possible solutions. (Lesson 68)
Color
Units of Yellow Dye
Units of Blue Dye
light green
4
1
dark green
1
6
Solve each compound inequality. Then graph the solution set. (Lesson 54) 62. k + 2 > 12 and k + 2 ≤ 18
63. d  4 > 3 or d  4 ≤ 1
64. 3 < 2x  3 < 15
65. 3t  7 ≥ 5 and 2t + 6 ≤ 12
66. h  10 < 21 or h + 3 < 2
67. 4 < 2y  2 < 10
68. FINANCIAL LITERACY A home security company provides security systems for $5 per week, plus an installation fee. The total cost for installation and 12 weeks of service is $210. Write the pointslope form of an equation to find the total fee y for any number of weeks x. What is the installation fee? (Lesson 43)
Skills Review Find the principal square root of each number. (Lesson 02) 69. 16
70. 36
71. 64
72. 81
73. 121
74. 100
498  Lesson 84  Quadratic Equations: ax 2 + bx + c = 0
Quadratic Equations: Differences of Squares Then
Now
Why?
You factored trinomials into two binomials.
1
Factor binomials that are the difference of squares.
2
Use the difference of squares to solve equations.
Computer graphics designers use a combination of art and mathematics skills to design images and videos. They use equations to form shapes and lines on computers. Factoring can help to determine the dimensions and shapes of the figures.
(Lesson 83, 84)
NewVocabulary difference of two squares
1 Factor Differences of Squares
Recall that in Lesson 78, you learned about the product of the sum and difference of two quantities. This resulting product is referred to as the difference of two squares. So, the factored form of the difference of squares is called the product of the sum and difference of the two quantities.
KeyConcept Difference of Squares Tennessee Curriculum Standards CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. SPI 3102.3.3 Factor polynomials. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function. Also addresses CLE 3102.3.2, ✓3103.3.9, and ✓3103.3.30.
Symbols
a 2  b 2 = (a + b)(a  b) or (a  b)(a + b)
Examples
x 2  25 = (x + 5)(x  5) or (x  5)(x + 5) t 2  64 = (t + 8)(t  8) or (t  8)(t + 8)
Example 1 Factor Differences of Squares Factor each polynomial. a. 16h 2  9a 2 16h 2  9a 2 = (4h) 2  (3a) 2 = (4h + 3a)(4h  3a)
Write in the form of a 2  b 2.
b. 121  4b 2 121  4b 2 = (11)2  (2b)2 = (11  2b)(11 + 2b)
Write in the form of a 2  b 2.
Factor the difference of squares.
Factor the difference of squares.
c. 27g 3  3g Because the terms have a common factor, factor out the GCF first. Then proceed with other factoring techniques. 27g 3  3g = 3g(9g 2  1) = 3g[(3g)2  (1)2] = 3g(3g  1)(3g + 1)
Factor out the GCF of 3g. Write in the form a 2  b 2. Factor the difference of squares.
GuidedPractice 1A. 81  c 2
1B. 64g 2  h 2
1C. 9x 3  4x
1D. 4y 3 + 9y connectED.mcgrawhill.com
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WatchOut! Sum of Squares The sum of squares, a 2 + b 2, does not factor into (a + b)(a + b). The sum of squares is a prime polynomial and cannot be factored.
To factor a polynomial completely, a technique may need to be applied more than once. This also applies to the difference of squares pattern.
Example 2 Apply a Technique More than Once Factor each polynomial. a. b 4  16 Write b 4  16 in a 2  b 2 form. b 4  16 = (b 2)2  (4)2 = (b 2 + 4)(b 2  4) Factor the difference of squares. 2 Notice that the factor b  4 is also the difference of squares. Write b 2  4 in a 2  b 2 form. = (b 2 + 4)(b 2  2 2) = (b 2 + 4)(b + 2)(b  2) Factor the difference of squares. b. 625  x 4 625  x 4 = (25) 2  (x 2)2 = (25 + x 2)(25  x 2) = (25 + x 2)(5 2  x 2) = (25 + x 2)(5  x)(5 + x)
Write 625  x 4 in a 2  b 2 form. Factor the difference of squares. Write 25  x 2 in a 2  b 2 form. Factor the difference of squares.
GuidedPractice Factor each polynomial. 2A. y 4  1
2B. 4a 4  b 4
2C. 81  x 4
2D. 16y 4  1
Sometimes more than one factoring technique needs to be applied to ensure that a polynomial is factored completely.
Example 3 Apply Different Techniques Factor each polynomial. a. 5x 5  45x Factor out GCF. 5x 5  45x = 5x(x 4  9) 2 2 2 = 5x[(x )  (3) ] Write x 4  9 in the form a 2  b 2. = 5x(x 2  3)(x 2 + 3) Factor the difference of squares. 2 Notice that the factor x  3 is not the difference of squares because 3 is not a perfect square. b. 7x 3 + 21x 2  7x  21 Original expression 7x 3 + 21x 2  7x  21 = 7(x 3 + 3x 2  x  3) Factor out GCF. 3 2 = 7[(x + 3x )  (x + 3)] Group terms with common factors. 2 = 7[x (x + 3)  1(x + 3)] Factor each grouping. 2 = 7(x + 3)(x  1) x + 3 is the common factor. = 7(x + 3)(x + 1)(x  1) Factor the difference of squares.
GuidedPractice Factor each polynomial. 3A. 2y 4  50
3B. 6x 4  96
3C. 2m 3 + m 2  50m  25
3D. r 3 + 6r 2 + 11r + 66
500  Lesson 85  Quadratic Equations: Differences of Squares
2 Solve Equations by Factoring
After factoring, you can apply the Zero Product Property to an equation that is written as the product of factors set equal to 0. SPI 3102.3.10
Test Example 4
_
In the equation y = x 2  9 , which is a value of x when y = 0?
TestTakingTip
16
9 A _ 4
Use Another Method Another method that can be used to solve this equation is to substitute each answer choice into the equation.
3 C _
B 0
9 D _
4
4
Read the Test Item Replace y with 0 and then solve. Solve the Test Item 9 y = x2  _
16 9 0 = x2  _ 16 3 2 0 = x2  _ 4 3 3 0= x+_ x_ 4 4 3 3 _ 0=x+ or 0 = x  _ 4 4 3 3 _ _ x=x= 4 4
() ( )(
)
Original equation Replace y with 0. Write in the form a 2  b 2. Factor the difference of squares. Zero Product Property
The correct answer is C.
GuidedPractice 4. Which are the solutions of 18x 3 = 50x? 5 F 0, _ 3
5 _ G _ ,5
5 _ H _ , 5, 0
3 3
Check Your Understanding
3 3
5 _ J _ , 5, 1 3 3
= StepbyStep Solutions begin on page R12.
Examples 1–3 Factor each polynomial. 1. x 2  9
2. 4a 2  25
3. 9m 2  144
4. 2p 3  162p
5. u 4  81
6. 2d 4  32f 4
7 20r 4  45n 4
8. 256n 4  c 4
9. 2c 3 + 3c 2  2c  3 11. 3t 3 + 2t 2  48t  32 Example 4
10. f 3  4f 2  9f + 36 12. w 3  3w 2  9w + 27
EXTENDED RESPONSE After an accident, skid marks may result from sudden breaking. The formula 1 s 2 = d approximates a vehicle’s speed s in miles per hour given the
_ 24
length d in feet of the skid marks on dry concrete. 13. If skid marks on dry concrete are 54 feet long, how fast was the car traveling when the brakes were applied? 14. If the skid marks on dry concrete are 150 feet long, how fast was the car traveling when the brakes were applied? connectED.mcgrawhill.com
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Practice and Problem Solving
Extra Practice begins on page 815.
Examples 1–3 Factor each polynomial.
Example 4
15. q 2  121
16. r 4  k 4
17. 6n 4  6
18. w 4  625
19. r 2  9t 2
20. 2c 2  32d 2
21. h 3  100h
22. h 4  256
23. 2x 3  x 2  162x + 81
24. x 2  4y 2
25. 7h 4  7p 4
26. 3c 3 + 2c 2  147c  98
27. 6k 2h 4  54k 4
28. 5a 3  20a
29. f 3 + 2f 2  64f  128
30. 3r 3  192r
31. 10q 3  1210q
32. 3xn 4  27x 3
33. p 3r 5  p 3r
34. 8c 3  8c
35. r 3  5r 2  100r + 500
36. 3t 3  7t 2  3t + 7
37. a 2  49
38. 4m 3 + 9m 2  36m  81
39. 3m 4 + 243
40. 3x 3 + x 2  75x  25
41. 12a 3 + 2a 2  192a  32
42. x 4 + 6x 3  36x 2  216x
43. 15m 3 + 12m 2  375m  300
44. GEOMETRY The drawing at the right is a square with a square cut out of it.
(4n + 1) cm
a. Write an expression that represents the area of the shaded region. b. Find the dimensions of a rectangle with the same area as the shaded region in the drawing. Assume that the dimensions of the rectangle must be represented by binomials with integral coefficients.
B
5
(4n + 1) cm
5
45. DECORATIONS An arch decorated with balloons was used to decorate the gym for the spring dance. The shape of the arch can be modeled by the equation y = 0.5x 2 + 4.5x, where x and y are measured in feet and the xaxis represents the floor. a. Write the expression that represents the height of the arch in factored form. b. How far apart are the two points where the arch touches the floor? c. Graph this equation on your calculator. What is the highest point of the arch? 46. DECKS Zelda is building a deck in her backyard. The plans for the deck show that it is to be 24 feet by 24 feet. Zelda wants to reduce one dimension by a number of feet and increase the other dimension by the same number of feet. If the area of the reduced deck is 512 square feet, what are the dimensions of the deck? 47 SALES The sales of a particular CD can be modeled by the equation S =  25m 2 + 125m, where S is the number of CDs sold in thousands, and m is the number of months that it is on the market. a. In what month should the music store expect the CD to stop selling? b. In what month will CD sales peak? c. How many copies will the CD sell at its peak?
502  Lesson 85  Quadratic Equations: Differences of Squares
Solve each equation by factoring. Confirm your answers using a graphing calculator.
C
48. 36w 2 = 121
49 100 = 25x 2
50. 64x 2  1 = 0
9 51. 4y 2  _ =0
1 2 52. _ b = 16 4
1 2 53. 81  _ x =0 25
54. 9d 2  81 = 0
9 55. 4a 2 = _
56.
16
64
MULTIPLE REPRESENTATIONS In this problem, you will investigate perfect square trinomials. a. Tabular Copy and complete the table below by factoring each polynomial. Then write the first and last terms of the given polynomials as perfect squares. Polynomial 2
4x + 12x + 9
Factored Polynomial (2x + 3)(2x + 3)
First Term 2
4x = (2x)
2
Last Term
Middle Term
9 = 32
9x 2  24x + 16 4x 2  20x + 25 16x 2 + 24x + 9 25x 2 + 20x + 4
b. Analytical Write the middle term of each polynomial using the square roots of the perfect squares of the first and last terms. c. Algebraic Write the pattern for a perfect square trinomial. d. Verbal What conditions must be met for a trinomial to be classified as a perfect square trinomial?
H.O.T. Problems
Use HigherOrder Thinking Skills
57. ERROR ANALYSIS Elizabeth and Lorenzo are factoring an expression. Is either of them correct? Explain your reasoning.
Lorenzo
Elizabeth 16x 4 – 25y 2 = (4x  5y)(4x + 5y)
4
16x – 25y 2 = (4x 2  5y)(4x 2 + 5y)
58. CHALLENGE Factor and simplify 9  (k + 3)2, a difference of squares. 59. CHALLENGE Factor x 16  81. 60. REASONING Write and factor a binomial that is the difference of two perfect squares and that has a greatest common factor of 5mk. 61. REASONING Determine whether the following statement is true or false. Give an example or counterexample to justify your answer. All binomials that have a perfect square in each of the two terms can be factored. 62. OPEN ENDED Write a binomial in which the difference of squares pattern must be repeated to factor it completely. Then factor the binomial. 63. WRITING IN MATH Describe why the difference of squares pattern has no middle term with a variable. connectED.mcgrawhill.com
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SPI 3102.3.10, SPI 3102.3.6
Standardized Test Practice 64. One of the roots of 2x 2 + 13x = 24 is 8. What is the other root? 3 A _
2 C _
3 B _ 2
2 D _
2
3
3
65. Which of the following is the sum of both solutions of the equation x 2 + 3x = 54? F 21 G 3
67. EXTENDED RESPONSE Two cars leave Cleveland at the same time from different parts of the city and both drive to Cincinnati. The distance in miles of the cars from the center of Cleveland can be represented by the two equations below, where t represents the time in hours.
H 3 J 21
66. What are the xintercepts of the graph of y = 3x 2 + 7x + 20? 5 A _ , 4 3
5 B _ , 4 3
Car A: 65t + 15
Car B: 60t + 25
a. Which car is faster? Explain. b. Find an expression that models the distance between the two cars. 1 c. How far apart are the cars after 2_ hours? 2
5 C _ ,4
3 5 D _, 4 3
Spiral Review Factor each trinomial, if possible. If the trinomial cannot be factored using integers, write prime. (Lesson 84) 68. 5x 2  17x + 14
69. 5a 2  3a + 15
70. 10x 2  20xy + 10y 2
Solve each equation. Check your solutions. (Lesson 83) 71. n 2  9n = 18
72. 10 + a 2 = 7a
74. SAVINGS Victoria and Trey each want to buy a scooter. In how many weeks will Victoria and Trey have saved the same amount of money, and how much will each of them have saved? (Lesson 61)
73. 22x  x 2 = 96 Trey
$8 per week $18 so far
Solve each inequality. Graph the solution set on a number line. (Lesson 51)
75. t + 14 ≥ 18
76. d + 5 ≤ 7
77. 5 + k > 1
78. 5 < 3 + g
79. 2 ≤ 1 + m
80. 2y > 8 + y
81. FITNESS Silvia is beginning an exercise program that calls for 20 minutes of walking each day for the first week. Each week thereafter, she has to increase her daily walking for a week by 7 minutes. In which week will she first walk over an hour a day? (Lesson 35)
Skills Review Find each product. (Lesson 78) 82. (x  6)2
83. (x  2)(x  2)
84. (x + 3)(x + 3)
85. (2x  5)2
86. (6x  1)2
87. (4x + 5)(4x + 5)
504  Lesson 85  Quadratic Equations: Differences of Squares
Victoria $5 per week $25 so far
Quadratic Equations: Perfect Squares Then
Now
Why?
You found the product of a sum and difference.
1 2
In a vacuum, a feather and a piano would fall at the same speed, or velocity. To find about how long it takes an object to hit the ground if it is dropped from an initial height of h 0 feet above ground, you would need to solve the equation 0 = 16t 2 + h 0, where t is time in seconds after the object is dropped.
(Lesson 78)
NewVocabulary perfect square trinomial
Factor perfect square trinomials. Solve equations involving perfect squares.
1 Factor Perfect Square Trinomials
In Lesson 78, you learned the patterns for the products of the binomials (a + b) 2 and (a  b) 2. Recall that these are special products that follow specific patterns. (a + b)2 = (a + b)(a + b)
Tennessee Curriculum Standards CLE 3102.3.8 Solve and understand solutions of quadratic equations with real roots. SPI 3102.3.3 Factor polynomials. SPI 3102.3.10 Find the solution of a quadratic equation and/or zeros of a quadratic function. Also addresses CLE 3102.3.2, ✓3103.3.9, and ✓3103.3.30.
(a  b)2 = (a  b)(a  b)
= a 2 + ab + ab + b 2
= a 2  ab  ab + b 2
= a 2 + 2ab + b 2
= a 2  2ab + b 2
These products are called perfect square trinomials, because they are the squares of binomials. The above patterns can help you factor perfect square trinomials. For a trinomial to be factorable as a perfect square, the first and last terms must be perfect squares and the middle term must be two times the square roots of the first and last terms. The trinomial 16x 2 + 24x + 9 is a perfect square trinomial, as illustrated below. 16x 2 + 24x + 9 Is the first term a perfect square? Yes, because 16x 2 = (4x)2.
Is the middle term twice the product of the square roots of the first and last terms? Yes, because 24x = 2(4x)(3).
Is the last term a perfect square? Yes, because 9 = 3 2.
KeyConcept Factoring Perfect Square Trinomials Symbols
a 2 + 2ab + b 2 = (a + b)(a + b) = (a + b) 2 a 2  2ab + b 2 = (a  b)(a  b) = (a  b) 2
Examples
x 2 + 8x + 16 = (x + 4)(x + 4) or (x + 4) 2 x 2  6x + 9 = (x  3)(x  3) or (x  3) 2
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StudyTip Recognizing Perfect Square Trinomials If the constant term of the trinomial is negative, the trinomial is not a perfect square trinomial, so it is not necessary to check the other conditions.
Example 1 Recognize and Factor Perfect Square Trinomials Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. a. 4y 2 + 12y + 9 1 Is the first term a perfect square?
Yes, 4y 2 = (2y) 2.
2 Is the last term a perfect square?
Yes, 9 = 3 2.
3 Is the middle term equal to 2(2y)(3)?
Yes, 12y = 2(2y)(3)
Since all three conditions are satisfied, 4y 2 + 12y + 9 is a perfect square trinomial. Write as a 2 + 2ab + b 2. 4y 2 + 12y + 9 = (2y) 2 + 2(2y)(3) + 3 2 = (2y + 3) 2 Factor using the pattern. b. 9x 2  6x + 4 1 Is the first term a perfect square?
Yes, 9x 2 = (3x)2.
2 Is the last term a perfect square?
Yes, 4 = 2 2.
3 Is the middle term equal to 2(3x)(2)? No, 6x ≠ 2(3x)(2). Since the middle term does not satisfy the required condition, 9x 2  6x + 4 is not a perfect square trinomial.
GuidedPractice 1A. 9y 2 + 24y + 16
1B. 2a 2 + 10a + 25
A polynomial is completely factored when it is written as a product of prime polynomials. More than one method might be needed to factor a polynomial completely. When completely factoring a polynomial, the Concept Summary can help you decide where to start. Remember, if the polynomial does not fit any pattern or cannot be factored, the polynomial is prime.
ConceptSummary Factoring Methods Steps
Step 1 Factor out the GCF.
Number of Terms any
Step 2 Check for a difference of squares or a perfect square trinomial.
2 or 3
Step 3 Apply the factoring patterns for x 2 + bx + c or ax 2 + bx + c (general trinomials), or factor by grouping.
3 or 4
Examples 4x 3 + 2x 2  6x = 2x(2x 2 + x  3) 9x 2  16 = (3x + 4)(3x  4) 16x 2 + 24x + 9 = (4x + 3)2 x 2  8x + 12 = (x  2)(x  6) 2x 2 + 13x + 6 = (2x + 1)(x + 6) 12y 2 + 9y + 8y + 6 = (12y 2 + 9y ) + (8y + 6) = 3y (4y + 3) + 2(4y + 3) = (4y + 3)(3y + 2)
506  Lesson 86  Quadratic Equations: Perfect Squares
Example 2 Factor Completely Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. a. 5x 2  80 Step 1 The GCF of 5x 2 and 80 is 5, so factor it out. Step 2 Since there are two terms, check for a difference of squares. 5x 2  80 = 5(x 2  16) = 5(x 2  4 2) = 5(x  4)(x + 4)
5 is the GCF of the terms. x 2 = x · x and 16 = 4 · 4 Factor the difference of squares.
b. 9x 2  6x  35 Step 1 The GCF of 9x 2, 6x, and 35 is 1. Step 2 Since 35 is not a perfect square, this is not a perfect square trinomial. Step 3 Factor using the pattern ax 2 + bx + c. Are there two numbers with a product of 9(35) or 315 and a sum of 6? Yes, the product of 15 and 21 is 315, and the sum is 6.
StudyTip Check Your Answer You can check your answer by: • Using the FOIL method. • Using the Distributive Property. • Graphing the original expression and factored expression and comparing the graphs. If the product of the factors does not match the original expression exactly, the answer is incorrect.
9x 2  6x  35 = 9x 2 + mx + nx  35 = 9x 2 + 15x  21x  35 = (9x 2 + 15x) + (21x  35) = 3x(3x + 5)  7(3x + 5) = (3x + 5)(3x  7)
Write the pattern. m = 15 and n = 21 Group terms with common factors. Factor out the GCF from each grouping. 3x + 5 is the common factor.
GuidedPractice 2A. 2x 2  32
2B. 12x 2 + 5x  25
2 Solve Equations with Perfect Squares
When solving equations involving repeated factors, it is only necessary to set one of the repeated factors equal to zero.
Example 3 Solve Equations with Repeated Factors Solve 9x 2  48x = 64. 9x 2  48x = 64 9x 2  48x + 64 = 0 (3x) 2  2(3x)(8) + (8) 2 = 0 (3x  8) 2 = 0 (3x  8)(3x  8) = 0 3x  8 = 0 3x = 8 8 x=_ 3
Original equation Add 64 to each side. Recognize 9x 2  48x + 64 as a perfect square trinomial. Factor the perfect square trinomial. Write (3x  8)2 as two factors. Set the repeated factor equal to zero. Add 8 to each side. Divide each side by 3.
GuidedPractice Solve each equation. Check your solutions. 3A. a 2 + 12a + 36 = 0
4 4 3B. y 2  _ y+_ =0 3
9
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You have solved equations like x 2  16 = 0 by factoring. You can also use the definition of a square root to solve the equation. x 2  16 = 0 x 2 = 16
ReadingMath Square Root Solutions ± √ 16 is read as plus or minus the square root of 16.
x = ± √ 16
Original equation Add 16 to each side. Take the square root of each side.
Remember that there are two square roots of 16, namely 4 and 4. Therefore, the solution set is {4, 4}. You can express this as {±4}.
KeyConcept Square Root Property Words
To solve a quadratic equation in the form x 2 = n, take the square root of each side.
Symbols
For any number n ≥ 0, if x 2 = n, then x = ± √ n.
Example
x 2 = 25 x = ± √ 25 or ±5
In the equation x 2 = n, if n is not a perfect square, you need to approximate the square root. Use a calculator to find an approximation. If n is a perfect square, you will have an exact answer.
Example 4 Use the Square Root Property Solve each equation. Check your solutions. a. (y  6)2 = 81 (y  6)2 = 81
Original equation
y  6 = ± √ 81
Square Root Property
y  6 = ±9
81 = 9 · 9
y=6±9 y=6+9
Add 6 to each side.
or y = 6  9
= 15
= 3
The roots are 15 and 3.
Separate into two equations. Simplify. Check in the original equation.
b. (x + 6)2 = 12 (x + 6)2 = 12 x + 6 = ± √ 12 x = 6 ± √ 12
Original equation Square Root Property Subtract 6 from each side.
The roots are 6 ± √ 12 or 6 + √ 12 and 6  √ 12 . 12 ≈ 2.54 and 6  √ 12 ≈ 9.46. Using a calculator, 6 + √
GuidedPractice 4A. (a  10)2 = 121
508  Lesson 86  Quadratic Equations: Perfect Squares
4B. (z + 3)2 = 26
RealWorld Example 5 Solve an Equation PHYSICAL SCIENCE During an experiment, a ball is dropped from a height of 205 feet. The formula h = 16t 2 + h 0 can be used to approximate the number of seconds t it takes for the ball to reach height h from an initial height of h 0 in feet. Find the time it takes the ball to reach the ground. At ground level, h = 0 and the initial height is 205, so h 0 = 205. Original Formula
0 = 16t 2 + 205
Replace h with 0 and h 0 with 205.
205 = 16t
Math HistoryLink Galileo Galilei (1564–1642) Galileo was the first person to prove that objects of different weights fall at the same velocity by dropping two objects of different weights from the top of the Leaning Tower of Pisa in 1589.
h = 16t 2 + h 0 2
12.8125 = t 2 ±3.6 ≈ t
Subtract 205 from each side. Divide each side by 16. Use the Square Root Property.
Since a negative number does not make sense in this situation, the solution is 3.6. It takes about 3.6 seconds for the ball to reach the ground.
GuidedPractice 5. Find the time it takes a ball to reach the ground if it is dropped from a bridge that is half as high as the one described above.
Check Your Understanding Example 1
Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 1. 25x 2 + 60x + 36
Example 2
= StepbyStep Solutions begin on page R12.
2. 6x 2 + 30x + 36
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 3. 2x 2  x  28
4. 6x 2  34x + 48
5. 4x 2 + 64
6. 4x 2 + 9x  16
Examples 3–4 Solve each equation. Confirm your answers using a graphing calculator. 7. 4x 2 = 36 9. 64y 2  48y + 18 = 9 Example 5
8. 25a 2  40a = 16 10. (z + 5) 2 = 47
11. PAINT While painting his bedroom, Nick drops his paintbrush off his ladder from a height of 6 feet. Use the formula h = 16t 2 + h 0 to approximate the number of seconds it takes for the paintbrush to hit the floor.
Practice and Problem Solving Example 1
Extra Practice begins on page 815.
Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it. 12. 4x 2  42x + 110
13. 16x 2  56x + 49
14. 81x 2  90x + 25
15 x 2 + 26x + 168 connectED.mcgrawhill.com
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Example 2
Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. 16. 24d 2 + 39d  18
17. 8x 2 + 10x  21
18. 2b 2 + 12b  24
19. 8y 2  200z 2
20. 16a 2  121b 2
21. 12m 3  22m 2  70m
22. 8c 2  88c + 242
23. 12x 2  84x + 147
24. w 4  w 2
25. 12p 3  3p
26. 16q 3  48q 2 + 36q
27. 4t 3 + 10t 2  84t
28. x 3 + 2x 2y  4x  8y
29. 2a 2b 2  2a 2  2ab 3 + 2ab
30. 2r 3  r 2  72r + 36
31. 3k 3  24k 2 + 48k
32. 4c 4d  10c 3d + 4c 2d 3  10cd 3
33. g 2 + 2g  3h 2 + 4h
Examples 3–4 Solve each equation. Confirm your answers using a graphing calculator. 34. 4m 2  24m + 36 = 0
35 (y  4)2 = 7
10 25 36. a 2 + _ a+_ =0
3 9 37. x 2  _ x+_ =0
7
49
2
38. x + 8x + 16 = 25 2
2
16
2
39. 5x  60x = 180
40. 4x = 80x  400
41. 9  54x = 81x 2
42. 4c 2 + 4c + 1 = 15
43. x 2  16x + 64 = 6
44. PHYSICAL SCIENCE For an experiment in physics class, a water balloon is dropped from the window of the school building. The window is 40 feet high. How long does it take until the balloon hits the ground? Round to the nearest hundredth. 45. SCREENS The area A in square feet of a projected picture on a movie screen can be modeled by the equation A = 0.25d 2, where d represents the distance from a projector to a movie screen. At what distance will the projected picture have an area of 100 square feet? Example 5
46. GEOMETRY The area of a square is represented by 9x 2  42x + 49. Find the length of each side. 47. GEOMETRY The area of a square is represented by 16x 2 + 40x + 25. Find the length of each side.
B
48. ELECTION For the student council elections, Franco is building the voting box shown with a volume of 672 cubic inches. a. Write a polynomial that represents the volume of the box. b. What are the dimensions of the voting box?
Vote V ote t for f Studdentt Council Student St Counci unciiill h+6
49. AQUARIUM Dexter has a fish tank shaped like a rectangular prism. It has a volume of 480 cubic inches. The height of the tank is 8 inches taller than the width, and the length of the tank is 6 inches longer than the width. a. Write a polynomial that represents the volume of the fish tank. b. What are the dimensions of the fish tank?
510  Lesson 86  Quadratic Equations: Perfect Squares
h
h–2
C
50. GEOMETRY The volume of a rectangular prism is represented by the expression 8y 3 + 40y 2 + 50y. Find the possible dimensions of the prism if the dimensions are represented by polynomials with integer coefficients. 51 POOLS Ichiro wants to buy an aboveground swimming pool for his yard. Model A is 42 inches deep and holds 1750 cubic feet of water. The length of the rectangular pool is 5 feet more than the width. a. What is the surface area of the water? b. What are the dimensions of the pool? c. Model B pool holds twice as much water as Model A. What are some possible dimensions for this pool? d. Model C has length and width that are both twice as long as Model A, but the height is the same. What is the ratio of the volume of Model A to Model C? 52. GEOMETRY Use the rectangular prism at the right. a. Write an expression for the height and width of the prism in terms of the length, . b. Write a polynomial for the volume of the prism in terms of the length.
H.O.T. Problems
8
= 14
4
Use HigherOrder Thinking Skills
53. ERROR ANALYSIS Debbie and Adriano are factoring the expression x 8  x 4 completely. Is either of them correct? Explain your reasoning.
Adriano
Debbie 8
4
4
2
2
x – x = x (x + 1)(x – 1)
8
4
4
x – x = x (x 2 + 1)(x – 1)(x + 1)
54. CHALLENGE Factor x n + 6 + x n + 2 + x n completely. 55. OPEN ENDED Write a perfect square trinomial equation in which the coefficient of the middle term is negative and the last term is a fraction. Solve the equation. 56. REASONING Find a counterexample to the following statement. A polynomial equation of degree three always has three real solutions. 57. WRITING IN MATH Explain how to factor a polynomial completely. 58. WHICH ONE DOESN’T BELONG? Identify the trinomial that does not belong. Explain. 4x 2  36x + 81
25x 2 + 10x + 1
4x 2 + 10x + 4
9x 2  24x + 16
59. OPEN ENDED Write a binomial that can be factored using the difference of two squares twice. Set your binomial equal to zero and solve the equation. 60.
E
WRITING IN MATH Explain how to determine whether a trinomial is a perfect square trinomial. connectED.mcgrawhill.com
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SPI 3102.3.10, SPI 3102.3.8, SPI 3102.3.9, SPI 3102.4.1
Standardized Test Practice 61. What is the solution set for the equation (x  3)2 = 25? A {8, 2} B {2, 8}
C {4, 14} D {4, 14}
63. At an amphitheater, the price of 2 lawn seats and 2 pavilion seats is $120. The price of 3 lawn seats and 4 pavilion seats is $225. How much do lawn and pavilion seats cost? F G H J
62. SHORT RESPONSE Write an equation in slopeintercept form for the graph shown below. y
0
x
$20 and $41.25 $10 and $50 $15 and $45 $30 and $30
64. GEOMETRY The circumference of a circle is 6π _ units. What is the area of the circle? 5
9π C _ units 2
3π A _ units 2
25
5 12π _ units 2 B 5
30π D _ units 2 25
Spiral Review Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. (Lesson 85) 65. x 2  16
66. 4x 2  81y 2
67. 1  100p 2
68. 3a 2  20
69. 25n 2  1
70. 36 – 9c 2
Solve each equation. Check your solutions. (Lesson 84) 71. 4x 2  8x  32 = 0
72. 6x 2  48x + 90 = 0
73. 14x 2 + 14x = 28
74. 2x 2  10x = 48
75. 5x 2  25x = 30
76. 8x 2  16x = 192
SOUND The intensity of sound can be measured in watts per square meter. The table gives the watts per square meter for some common sounds. (Lesson 72) 77. How many times more intense is the sound from busy street traffic than sound from normal conversation? 78. Which sound is 10,000 times as loud as a busy street traffic? 79. How does the intensity of a whisper compare to that of normal conversation?
Watts Per Square Meter
Common Sounds
10
11
rustling leaves
10
10
whisper
10
6
normal conversation
10
5
busy street trafﬁc
10
4
vacuum cleaner
10 1
front rows of rock concert
10
1
threshold of pain
10
2
military jet takeoff
Skills Review Find the slope of the line that passes through each pair of points. (Lesson 33) 80. (5, 7), (2, 3)
81. (2, 1), (5, 3)
82. (4, 1), (3, 3)
83. (3, 4), (5, 1)
84. (2, 3), (8, 3)
85. (5, 4), (5, 1)
512  Lesson 86  Quadratic Equations: Perfect Squares
Study Guide and Review Study Guide KeyConcepts Monomials and Factoring (Lesson 81) • The greatest common factor (GCF) of two or more monomials is the product of their common prime factors. Factoring Using the Distributive Property (Lesson 82) • Using the Distributive Property to factor polynomials with four or more terms is called factoring by grouping. ax + bx + ay + by = x(a + b) + y (a + b) = (a + b)(x + y ) • Factoring can be used to solve some equations. According to the Zero Product Property, for any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b equal zero.
Factoring Trinomials and Differences of Squares
KeyVocabulary difference of two squares (p. 499) factored form (p. 471) factoring (p. 476) factoring by grouping (p. 477) greatest common factor (GCF) (p. 471) perfect square trinomial (p. 505) prime polynomial (p. 495) quadratic equation (p. 488) Square Root Property (p. 508) Zero Product Property (p. 478)
(Lessons 83 through 85)
• To factor x 2 + bx + c, find m and p with a sum of b and a product of c. Then write x 2 + bx + c as (x + m)(x + p). • To factor ax 2 + bx + c, find m and p with a sum of b and a product of ac. Then write as ax 2 + mx + px + c and factor by grouping. 2
2
• a  b = (a – b)(a + b)
Perfect Squares and Factoring (Lesson 86) • For a trinomial to be a perfect square, the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. • For any number n ≥ 0, if x 2 = n, then x = ± √n.
VocabularyCheck State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence. x 2 + 5x + 6 is an example of a prime polynomial. 1. __________ 2. (x + 5)(x  5) is the factorization of a difference of squares. 3. 5x is the greatest common factor of 10x and 15xy 2. 4. (x + 5)(x  2) is the factored form of x 2  3x  10.
StudyOrganizer Be sure the Key Concepts are noted in your Foldable.
5. Expressions with four or more unlike terms can sometimes be factored by grouping. 6. The Zero Product Property states that if ab = 1, then a or b is 1. 7. x 2  12x + 36 is an example of a perfect square trinomial. 8. x  2 = 0 is a quadratic equation. 9. x 2  16 is an example of a perfect square trinomial. 10. The greatest common factor of 8x and 4x 2 is 4x. connectED.mcgrawhill.com
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Study Guide and Review Continued LessonbyLesson Review
811 Monomials and Factoring
CLE 3102.3.2, SPI 3102.3.3
(pp. 471–474)
Factor each monomial completely.
Example 1
11. 28x 3
12. 33x 2y 3
Factor 24a 2b 3 completely.
13. 68cd 3
14. 120mq
24a 2b 3 = 4 · 6 · a · a · b · b · b
Find the greatest common factor of each set of monomials. 15. 22b, 33c
16. 21xy, 28x 2y, 42xy 2
17. 6ab, 24ab 4
18. 10ab, 30a, 40a 2b
=2·2·2·3·a·a·b·b·b
Example 2 Find the greatest common factor of 12xy and 8xy 2. 12xy = 2 · 2 · 3 · x · y
19. HOME IMPROVEMENT A landscape architect is designing a stone path 36 inches wide and 120 inches long. What is the maximum size square stone that can be used so that none of the stones have to be cut?
8xy 2 = 2 · 2 · 2 · x · y · y
Factor each monomial. Circle the common prime factors.
The greatest common factor is 2 · 2 · x · y or 4xy.
CLE 3102.3.2, ✔3102.3.8, SPI 3102.3.3
822 Using the Distributive Property
(pp. 476–482)
Use the Distributive Property to factor each polynomial.
Example 3
20. 12x + 24y
Factor 12y 2 + 9y + 8y + 6.
21. 14x 2y  21xy + 35xy 2
12y 2 + 9y + 8y + 6
22. 8xy  16x 3y + 10y 23. a 2  4ac + ab  4bc
= (12y 2 + 9y ) + (8y + 6)
Group terms with common factors.
= 3y (4y + 3) + 2(4y + 3)
Factor the GCF from each group.
= (4y + 3)(3y + 2)
Distributive Property
2
24. 2x  3xz  2xy + 3yz 25. 24am  9an + 40bm 15bn
Example 4
Solve each equation. Check your solutions. 26. x (3x  6) = 0
27. 6x 2 = 12x
28. x 2 = 3x
29. 3x 2 = 5x
30. GEOMETRY The area of the rectangle shown is x 3  2x 2 + 5x square units. What is the length?
514  Chapter 8  Study Guide and Review
Solve x 2  6x = 0. Check your solutions. Write the equation so that it is of the form ab = 0.
x
x 2  6x = 0 Original equation x (x  6) = 0 Factor by using the GCF. x = 0 or x  6 = 0 Zero Product Property x=6 Solve. The roots are 0 and 6. Check by substituting 0 and 6 for x in the original equation.
CLE 3102.3.8, SPI 3102.3.3, SPI 3102.3.10
833 Quadratic Equations: x
2
+ bx + c = 0 (pp. 485–491)
Factor each trinomial. Confirm your answers using a graphing calculator. 31. x 2  8x + 15 2
33. x  5x  6
32. x 2 + 9x + 20 2
34. x + 3x  18
Solve each equation. Check your solutions.
Example 5 Factor x 2 + 10x + 21 b = 10 and c = 21, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the positive factors of 21, and look for the pair of factors with a sum of 10.
35. x 2 + 5x  50 = 0
Factors of 21
Sum of 10
36. x  6x + 8 = 0
1, 21
22
37. x 2 + 12x + 32 = 0
3, 7
10
2
38. x 2  2x  48 = 0
The correct factors are 3 and 7.
39. x 2 + 11x + 10 = 0
x 2 + 10x + 21 = (x + m)(x + p)
40. ART An artist is working on a painting that is 3 inches longer than it is wide. The area of the painting is 154 square inches. What is the length of the painting?
= (x + 3)(x + 7)
Write the pattern. m = 3 and p = 7
CLE 3102.3.8, SPI 3102.3.3, SPI 3102.3.10
844 Quadratic Equations: ax
2
+ bx + c = 0 (pp. 493–498)
Factor each trinomial, if possible. If the trinomial cannot be factored, write prime. 41. 12x 2 + 22x  14
Example 6 Factor 12a 2 + 17a + 6 a = 12, b = 17, and c = 6. Since b is positive, m + p is positive. Since c is positive, mp is positive. So, m and p are both positive. List the factors of 12(6) or 72, where both factors are positive.
2
42. 2y  9y + 3 43. 3x 2  6x  45 44. 2a 2 + 13a  24
Factors of 72
Sum of 17
1, 72
73
2, 36
38
3, 24
27
46. 2x  3x  20 = 0
4, 18
22
47. 16t 2 + 36t  8 = 0
6, 12
18
48. 6x 2  7x  5 = 0
8, 9
17
Solve each equation. Confirm your answers using a graphing calculator. 2
45. 40x + 2x = 24 2
49. GEOMETRY The area of the rectangle shown is 6x 2 + 11x  7 square units. What is the width of the rectangle?
2x  1
The correct factors are 8 and 9. 12a 2 + 17a + 6 = 12a 2 + ma + pa + 6 = 12a 2 + 8a + 9a + 6 = (12a 2 + 8a) + (9a + 6) = 4a(3a + 2) + 3(3a + 2) = (3a + 2)(4a + 3) So, 12a 2 + 17a + 6 = (3a + 2)(4a + 3).
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Study Guide and Review Continued CLE 3102.3.8, SPI 3102.3.3, SPI 3102.3.10
855 Quadratic Equations: Differences of Squares
(pp. 499–504)
Factor each polynomial.
Example 7
50. y 2  81
Solve x 2  4 = 12 by factoring. x 2  4 = 12
51. 64  25x 2
Original equation
x 2  16 = 0
52. 16a 2  21b 2
Subtract 12 from each side.
x 2  (4)2 = 0
53. 3x 2  3 Solve each equation by factoring. Confirm your answers using a graphing calculator. 54. a 2  25 = 0
55. 9x 2  25 = 0
56. 81  y 2 = 0
57. x 2  5 = 20
(x + 4)(x  4) = 0 x+4=0
Factor the difference of squares.
or x  4 = 0 Zero Product Property
x = 4
58. EROSION A boulder falls down a mountain into water 64 feet below. The distance d that the boulder falls in t seconds is given by the equation d = 16t 2. How long does it take the boulder to hit the water?
16 = 4 2
x = 4 Solve each equation.
The solutions are 4 and 4.
CLE 3102.3.8, SPI 3102.3.3, SPI 3102.3.10
866 Quadratic Equations: Perfect Squares
(pp. 505–512)
Factor each polynomial, if possible. If the polynomial cannot be factored write prime. 59. x 2 + 12x + 36
Example 8 Solve (x  9)2 = 144. (x  9)2 = 144 x  9 = ± √ 144
2
60. x + 5x + 25 2
x  9 = ± 12
61. 9y  12y + 4
x = 9 ± 12
62. 4  28a + 49a 2
Square Root Property 144 = 12 · 12 Add 9 to each side.
x = 9 + 12 or x = 9  12 Zero Product Property
63. x 4  1
x = 21
64. x 4  16x 2 Solve each equation. Confirm your answers using a graphing calculator. 2
Original equation
2
65. (x  5) = 121
66. 4c + 4c + 1 = 9
67. 4y 2 = 64
68. 16d 2 + 40d + 25 = 9
69. LANDSCAPING A sidewalk of equal width is being built around a square yard. What is the width of the sidewalk?
5PUBMBSFB GU
516  Chapter 8  Study Guide and Review
GU
x = 3
Solve.
CHECK (x  9)2 = 144
(x  9)2 = 144
(21  9)2 144
(3  9)2 144
(12)2 144
(12)2 144
144 = 144
144 = 144
Tennessee Curriculum Standards
Practice Test Factor each monomial completely. 1. 25x 2y 4
SPI 3102.3.10
Solve each equation. Confirm your answers using a graphing calculator. 14. y(y  14) = 0
2. 17ab 2
15. 3x(x + 6) = 0
3. 18c 5d 3
16. a 2 = 12a
4. GARDENING Conrado is planting 140 pumpkins in a rectangular arrangement in his garden. In what ways can he arrange them so that he has at least 4 rows of pumpkins, the same number of pumpkins in each row, and at least 6 pumpkins in each row?
17. MULTIPLE CHOICE Chantel is carpeting a room that has an area of x 2  100 square feet. If the width of the room is x  10 feet, what is the length of the room? F x  10 ft
Find the greatest common factor of each set of monomials. 2
5. 2a, 8a , 16a
3
2
6. 7c, 24d
7. 50g h, 120gh
2
G x + 10 ft H x  100 ft J 10 ft
2 2
8. 8q r , 36qr
9. MULTIPLE CHOICE The area of the rectangle shown below is 2x 2  x  15 square units. What is the width of the rectangle?
Factor each trinomial. 18. x 2 + 7x + 6
19. x 2  3x  28
20. 10x 2  x  3
21. 15x 2 + 7x  2
22. x 2  25
23. 4x 2  81
24. 9x 2  12x + 4
25. 16x 2 + 40x + 25
2x + 5
A x5
Solve each equation. Confirm your answers using a graphing calculator.
B x+3
26. x 2  4x = 21
C x3
27. x 2  2x  24 = 0
D 2x  3
28. 6x 2  5x  6 = 0
Use the Distributive Property to factor each polynomial. 10. 5xy  10x 11. 7ab +
14ab 2
2
+ 21a b
29. 2x 2  13x + 20 = 0
30. MULTIPLE CHOICE Which choice is a factor of x 4  1 when it is factored completely? A x2  1
Factor each polynomial. 2
B x1
12. 4x + 8x + x + 2
C x
13. 10a 2  50a  a + 5
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Preparing for Standardized Tests Solve MultiStep Problems Some problems that you will encounter on standardized tests require you to solve multiple parts in order to come up with the final solution. Use this lesson to practice these types of problems.
Strategies for Solving MultiStep Problems Step 1
Read the problem statement carefully. Ask yourself: • What am I being asked to solve? What information is given? • Are there any intermediate steps that need to be completed before I can solve the problem?
Step 2
Organize your approach. • List the steps you will need to complete in order to solve the problem. • Remember that there may be more than one possible way to solve the problem.
Step 3
Solve and check. • Work as efficiently as possible to complete each step and solve. • If time permits, check your answer.
SPI 3102.1.1
Test Practice Example Read the problem. Identify what you need to know. Then use the information in the problem to solve. A florist has 80 roses, 50 tulips, and 20 lilies that he wants to use to create bouquets. He wants to create the maximum number of bouquets possible and use all of the flowers. Each bouquet should have the same number of each type of flower. How many roses will be in each bouquet? A 4 roses
C 10 roses
B 8 roses
D 15 roses
518  Chapter 8  Preparing for Standardized Tests
Read the problem carefully. You are given the number of roses, tulips, and lilies and told that bouquets will be made using the same number of flowers in each. You need to find the number of roses that will be in each bouquet. Step 1
Find the GCF of the number of roses, tulips, and lilies.
Step 2
Use the GCF to determine how many bouquets will be made.
Step 3
Divide the total number of roses by the number of bouquets.
Step 1 Write the prime factorization of each number of flowers to find the GCF. 80 = 2 · 2 · 2 · 2 · 5 50 = 2 · 5 · 5 20 = 2 · 2 · 5 GCF = 2 · 5 = 10 Step 2
The GCF of the number of roses, tulips, and lilies tells you how many bouquets can be made because each bouquet will contain the same number of flowers. So, the florist can make a total of 10 bouquets.
Step 3
Divide the number of roses by the number of bouquets to find the number of roses in each bouquet. 80 _ =8 10
So, there will be 8 roses in each bouquet. The answer is B.
Exercises Read each problem. Identify what you need to know. Then use the information in the problem to solve.
3. What is the area of the square? x4
1. Which of the following values is not a solution to x 3  3x 2  25x + 75 = 0? A x=5
C x = 3
A x 2 + 16
B x=3
D x = 5
B 4x  16
2. There are 12 teachers, 90 students, and 36 parent volunteers going on a field trip. Mrs. Bartholomew wants to divide everyone into equal groups with the same number of teachers, students, and parents in each group. If she makes as many groups as possible, how many students will be in each group?
C x 2  8x  16 D x 2  8x + 16 4. Students are selling magazines to raise money for a field trip. They make $2.75 for each magazine they sell. If they want to raise $600, what is the least amount of magazines they need to sell?
F 6
H 12
F 121
H 202
G 9
J 15
G 177
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Standardized Test Practice Cumulative, Chapters 1 through 8 6. Which of the following is not a factor of the polynomial 45a 2  80b 2?
Multiple Choice Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.
F 5 G 3a  4b H 2a  5b
1. Karla makes gift baskets of cookies. Each basket contains an equal number of four different types of cookies. The chart shows Karla’s inventory of cookies. If she puts the greatest possible number of cookies in each basket, how many baskets can she make? Type of Cookie
Number
Chocolate Chip
54
Peanut Butter
45
Oatmeal Raisin
36
Sugar
J 3a + 4b 7. A rectangular gift box has dimensions that can be represented as shown in the figure. The volume of the box is 56w cubic inches. Which of the following is not a dimension of the box? w4
w
60
A 18
C 12
B 16
D 10
w5
A 6 in. B 7 in. C 8 in.
2. Refer to the information given in Exercise 1. How many of each type of cookie will be in each of the baker’s gift baskets? F 2
H 4
G 3
J 6
3. Factor the mn + 5m  3n  15. A (mn  3)(5)
C (m  5)(n + 3)
B (n  3)(m + 5)
D (m  3)(n + 5)
4. Which of the following is a solution to x 2 + 6x  112 = 0? F 14
H 6
G 8
J 12
D 12 in. 8. Factor the polynomial y 2  9y + 20. F ( y  2)( y  10) G ( y  4)( y  5) H ( y  2)( y  7) J ( y  5)( y + 2) 9. Which of the following numbers is less than zero? A 1.03 × 10 21 B 7.5 × 10 2 C 8.21543 × 10 10 D none of the above
5. Which of the following polynomials is prime? A 5x 2 + 34x + 24 B 4x 2 + 22x + 10 2
C 4x + 38x + 70 D 5x 2 + 3x + 4
520  Chapter 8  Standardized Test Practice
TestTakingTip Question 4 If time permits, be sure to check your answer. Substitute it into the equation to see if you get a true number sentence.
16. GRIDDED RESPONSE The amount of money that Humberto earns varies directly as the number of hours that he works as shown in the graph. How much money will he earn for working 40 hours next week? Express your answer in dollars.
Short Response/Gridded Response Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 10. GRIDDED RESPONSE Mr. Branson bought a total of 9 tickets to the zoo. He bought children tickets at the rate of $6.50 and adult tickets for $9.25 each. If he spent $69.50 altogether, how many adult tickets did Mr. Branson purchase?
Wages Earned
70
11. What is the domain of the following relation? {(2, 1), (4, 3), (7, 6)}
Cashews c
$6.90
Walnuts w
$5.60
20
1
2 3
4
5
6
7
Extended Response Record your answers on a sheet of paper. Show your work.
15. Marcel spent $24.50 on peanuts and walnuts for a party. He bought 1.5 pounds more peanuts than walnuts. How many pounds of peanuts and walnuts did he buy?
$3.80
30
Hours Worked
14. The equation h = –16t 2 + 40t + 3 models the height h in feet of a soccer ball after t seconds. What is the height of the ball after 2 seconds?
Peanuts p
40
0
13. GRIDDED RESPONSE Carlos bought a rare painting in 1995 for $14,200. By 2003, the painting was worth $17,120. Assuming that a linear relationship exists, write an equation in slopeintercept form that represents the value V of the painting after t years.
Price per pound
50
10
12. Ken just added 15 more songs to his MP3 player, making the total number of songs more than 84. Draw a number line that represents the original number of songs he had on his MP3 player.
Product
60
17. The height in feet of a model rocket t seconds after being launched into the air is given by the function h(t) = 16t 2 + 200t. a. Write the expression that shows the height of the rocket in factored form. b. At what time(s) is the height of the rocket equal to zero feet above the ground? Describe the real world meaning of your answer. c. What is the greatest height reached by the model rocket? When does this occur?
Need ExtraHelp? If you missed Question... Go to Lesson... For help with TN SPI...
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3102. 3.6
3102. 3.6
3102. 3.3
3102. 3.10
3102. 3.3
3102. 3.3
3102. 3.10
3102. 3.3
3102. 2.3
3102. 3.10
3102. 3.7
3102. 3.5
3102. 3.8
3102. 3.2
3102. 3.9
3102. 3.1
3102. 3.10
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