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Answer Key 53
Chapter 8
Lesson 8.1.
8-1. a. ( x + 4)( y + x + 2) = xy + x^2 + 6 x + 4 y + 8 b. 18 x^2 +^9 x^!^2
8-2. a. (2 x + 3)( x + 2) b. (2 x + 1)(3 x + 2) c. no solution
d. (2 x + y )( y + 3) ; Conclusion. Not every expression can be factored.
8-3. a. (3 x + 1)(2 x + 5) = 6 x^2 + 17 x + 5
b. (5 x! 2)( y + 3) = 5 xy + 15 x! 2 y! 6 c. (4 x! 3)(3 x + 4) = 12 x^2 + 7 x! 12
8-4. The product of each diagonal is equal. 6 x^2! 5 = 30 x^2 and 2 x! 15 x = 30 x^2.
8-5. Diagonals: part (a) are both 30 x^2 , part (b) are both! 30 xy , part (c) are both!^144 x^2. Typical response: “The product of one diagonal always equals the product of the other diagonal.”
8-6. (2 x! 3)( x + 2 y! 4) = 2 x^2 + 4 xy! 11 x! 6 y + 12
8-7. a. 12 x^2 + 17 x! 5
b. 4 x^2! 28 x + 49
8-8.
8-9. a. m = 2 , (0,!! 12 )
b. m =! 3 , (0,!!7) c. m =! 23 , (0,!8) d. m = 0 , (0,!!2)
8-10. a. (0, –8); It is the constant in the equation. b. (–2, 0) and (4, 0); The product of the x- intercepts equals the constant term. c. (1, –9); Its x -coordinate is midway between the x- intercepts.
8-11. a.! 1 b.! 7.24 c.! " 4.
2
(^10) – 4
12 –3 0 7
0 7 – 0
9 3 x 5 x
6 x^2 2 x x –6 x
! 7 x^2 –7 x
54 Algebra Connections
Lesson 8.1.
8-12. a. (5 x! 2)(2 x! 7) = 10 x^2! 39 x + 14 b.! 35 x "! 4 x = 10 x^2 " 14 = 140 x^2
8-13. a. (2 x + 3)( x + 1)
b. One corner should contain 4 x , while the other should contain 6 x ; (3 x + 4)( x + 2). c. Their sum is 7 x , and their product is 12 x^2. d. The product 12 x^2 should be placed at the top of the diamond problem, 7 x at the bottom, and terms 3 x and 4 x should be in the middle. e. (2 x + 3)( x + 2)
8-14. a. One corner contains 6 x^2 , and the opposite corner contains 12.
b. The product of the x^2 and units terms (in this case, 72 x^2 ) goes on top, while the x -term (^17 x^ ) goes on bottom. d. (2 x + 3)(3 x + 4)
8-15. a. ( x + 3)( x + 6) b. (4 x! 3)( x + 5) c. (2 x! 3)(2 x! 1)
d. not factorable because there are no integers that multiply to get! 9 x^2 (the diagonal of the generic rectangle) and add to get 5 x.
8-16. a. ( x! 6)( x + 2) b. (2 x + 1)^2 c. ( x! 5)(2 x + 1)
d. ( x + 4)(3 x! 2)
8-17. a. x -intercepts (–1, 0) and (3, 0), y -intercept. (0, –3) b. x -intercept (2, 0), no y -intercept c. x -intercepts (–3, 0), (–1, 0), and (1, 0), y -intercept (0, 2) d. x -intercept (8, 0), y -intercept (0, –20)
8-18. a. (0, –9); It is the constant in the equation. b. (3, 0) and (–3, 0)
8-19. a. (6, 9) b. (0, 2)
8-20. a. x =! 1023 b. all numbers c. c = 0
8-21. y = 14 x + 400
56 Algebra Connections
Lesson 8.2.
8-45. a. 2 b. –3 c. " –6.
8-46. y =! 3 x + 25
8-47. y = 3 x! 5 ; m = 3 and b = 5
8-48. There is only one line of symmetry. horizontal through the middle.
8-49. a. x -intercepts (–2, 0) and (0, 0), y -intercept (0, 0) b. x -intercepts (–3, 0) and (5, 0), y -intercept (0, 3) c. x -intercepts (–1, 0) and (1, 0), y -intercept (0, –1) d. x -intercept (9, 0), y -intercept (0, 6)
8-50. a. 6 x^2 + x! 12 b. 25 x^2! 20 x + 4
Lesson 8.2.
8-51. a. Longest : Maggie, Highest : Jen b. Jen. (0, 0) and (8, 0), Maggie. (3, 0) and (14, 0), Imp. (2, 0) and (12, 0), Al. (10, 0) and (16, 0); the x- intercepts tell where the balloon was launched and where it landed. c. Jen. (4, 32), Maggie. (8.5, 30.25), Imp. (7, 25), and Al. (13, 27); maximum height.
8-52. You should be able to connect rule! table, table " graph, graph " situation, and table " situation.
8-53. a. One way to write the rule is y = ( x + 1)( x + 2) + 2. b. Yes
8-54. vertex. (4, –9), x- intercepts. (1, 0) and (7, 0), y -intercept. (0, 7)
8-55. a. 3 –7 6 –2 b. …it does not change the value of the number. c. It tells us that a = 0. d. 0 for all e. …the result is always 0.
8-56. a. x -intercepts (2, 0), (– 4, 0), and (3, 0), y -intercept. (0, 18) b. x -intercepts (3, 0) and (8, 0), y -intercept. (0, –3) c. x -intercept (1, 0) and y -intercept (0, – 4)
8-57. a. (–3, 0) b.! (^12)
8-58. a. no solution b. (7, 2)
2 4 6 8 10 12 14 16
5
10
15
20
25
30
Jen (^) Maggie Al Imp
Solution to part (a)
Answer Key 57
Lesson 8.2.
8-59. a. No; the y -intercept is not enough information. b. No; the parabola could vary in width and direction. c. Yes; solution shown at right.
8-60. a. y = 0 for all x- intercepts and x = 0 for all y -intercepts. b. (0, –12) c. 0 =^2 x^2 +^5 x^!^12 d. Not yet, because it has an x^2 term.
8-61. a. At least one of the two numbers must be zero. b. At least one of the three numbers must be zero. c. Typical response: “If the product of two or more numbers is zero, then you know that one of the numbers must be zero.”
8-62. a. 0 = (2 x! 3)( x + 4)
b. 2 x! 3 = 0 or x + 4 = 0 , so x = 23 or x =! 4. c. The roots are at ( 23 , !0) and (– 4, 0). d. The solution graph is shown at right.
8-63. This parabola should have roots (–3, 0) and (2, 0) and y -intercept (0, –6).
8-64. roots: (–1, 0) and (–2, 0), y -intercept: (0, 4)
8-65. a. One is a product and the other is a sum.
b. first: x =! 2 or x = 1 ; second: x =! (^12)
8-66. a. x^ =^2 or x^ =^!^8 b. x^ =^3 or x^ =^1 c. x =! 10 or x = 2.5 d. x = 7
8-67. a. The line x = 0 is the y -axis, so this system is actually finding where the line 5 x! 2 y = 4 crosses the y- axis. b. (0, –2)
8-68. a. 4; Since the vertex lies on the line of symmetry, it must lie halfway between the x -intercepts. b. (4, –2)
8-69. a. 2( x! 2)( x + 1) b. (^) 4( x! 3)^2
8-70. a. The symbol “#” represents “greater than or equal to” and the symbol “>” represents “greater than.” b. 5 > 3 c. x! 9 d. –2 is less than 7.
Answer Key 59
Lesson 8.2.
8-82. (1) " b, (2) " e, (3) " a, (4) " g, (5) " d, (6) " i
8-83. Letter A. The client should order the parabola y = ( x! 1)( x + 6).
Letter B. The parabola y = ( x! 5)^2 should be recommended. Letter C. The parabola y = !( x + 3)( x! 2) should be recommended.
8-84. a. y =! 2 x ( x! 8) =! 2 x^2 + 16 x b. y = !3( x! 10)( x! 16)
8-85. a. y = x^2 + 2 x! 8 b. y = x^2! 6 x + 9 c. y = x^2! 7 x
d. y =! x^2! 4 x + 5
8-86. m^ =^12 , (0, 4) 8-87. a. " –1.4 and " 0.3 b. The quadratic is not factorable.
8-88. a. x = 4 or x = –10 b. x = –8 or x = 1.
8-89. a. 4 b. –10 c. –8 d. 1.5; They are the same.
8-90. a. (1, –1) b. (–2, 12 )
Lesson 8.3.
8-91. a. The quadratic is not factorable. b. There are two roots ( x -intercepts). c. The intercepts are " –1.5 and 4.5.
8-92. a. a = 1 , b =! 3 , c =! 7 b. 3 ±^2 37 " 4.5 and –1.5; yes
8-93. a. Graphing and factoring with the Zero Product Property
8-94. a. x = –2 or! 13 b. x = 7 or –2.5 c. x = –0.5 or –0.75 d. no solution
8-96. a. x = 6 or 7 b. x =^23 or – 4 c. x = 0 or 5 d. x = 3 or –
8-97. x = 6 or 7; yes
8-98. no a. The parabola should be tangent to the x -axis. b. Answers vary, but the parabola should not cross the x -axis.
8-99. y = 12 x + 9 8-100. line. (a) and (c); parabola. (b) and (d)
8-101. A and D
8-102. a. false b. true c. true d. true e. true f. false g. true h. false
60 Algebra Connections
Lesson 8.3.
8-103. a. (3 x! 2)(2 x + 5) = 0 ,
x = (^23) or! (^52) b. a = 6 , b = 11 , c =! 10 , x = (^23)
or!^52 c. Yes
8-104. a. x = 5.5 or x = !5.
b. x = 2 or x =! (^12) c. x =! 3 or x = 14 d. x = (^56)
8-105. a. " 315 and –315 feet; The bases of the arch are 315 feet from the center. b. " 630 feet c. 630 feet; y -intercept
8-106. a. x = 5 b. x = 13 or –
c. x = –1 or 53 d. x = ± 43
8-107. x = 13 or –6; yes
8-108. a. y = ( x + 3)( x! 1) = x^2 + 2 x! 3
b. y = ( x! 2)( x + 2) = x^2! 4
8-109. If x = width, x (2 x + 5) = 403 ;
width = 13 cm.
8-110. (b) and (c) are solutions.
8-111. a. She solved for x when y = 0.
b. The y -intercept is (0, –5), and a shortcut is to solve for y when x = 0; y = 35 x! 5. c. x. (8, 0), y. (0, –12)
Lesson 8.3.
8-112. a. x =! 3 or x =! 9 b. x = !17.6 or x = !0. c. x =! 43 or x = 12 d. x = 4 e. no solution f. x = 2.5 or x = !0.
8-114. a. The Zero Product Property only works when a product equals zero. b. x^2! 3 x! 4 = 0 c. x = 4 or x = –1; no
8-115. a. x = 0 seconds and x = 2. seconds, so it is in the air for 2.4 seconds. b. at x^ =^ 1.2^ secs c. 2.88 feet d. The sketch should have roots (0, 0) and (2.4, 0) and vertex (1.2, –2.88).
8-116. If n = # nickels and q = # of quarters, 0.05 n + 0.25 q = 1.90 , n = 2 q + 3 , and n = 13, so Daria has 13 nickels.
8-117. a. x = ± 0.08 b. x = 29 or – 4 c. no solution d. x " 1.4 or –17.
8-118. While the expressions may vary, each should be equivalent to y = x^2 + 4 x + 3.
8-119. a. x = 2 b. x = 15 c. x = –2 d. all numbers
8-120. Line L has slope 4, while line M has slope 3. Therefore, line L is steeper.
8-121. D
