Download Chap. 9.5 Answers - Aspen High School - Algebra I

3 APPLY
ASSIGNMENT GUIDE
BASIC
Day 1: p. 536 Exs. 24–52 even
Day 2: pp. 537–538 Exs. 54–80
even, 84, 87–90, 92–102
even
AVERAGE
Day 1: p. 536 Exs. 24–52 even
Day 2: pp. 537–538 Exs. 54–80
even, 82–84, 87–90,
92–102 even
ADVANCED
Day 1: p. 536 Exs. 24–52 even
Day 2: pp. 537–538 Exs. 54–80
even, 82–90, 92–102 even
BLOCK SCHEDULE
pp. 536–538 Exs. 24–80 even,
82–84, 87–90, 92–102 even
EXERCISE LEVELS
Level A: Easier
87–91
Level B: More Difficult
23–84, 92–103
Level C: Most Difficult
85, 86
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 24, 32, 40, 48, 54, 64, 70, 72, 78.
See also the Daily Homework Quiz:
• Blackline Master (Chapter 2
Resource Book, p. 83)
•
Transparency (p. 72)
COMMON ERROR
EXERCISE 44 Students may look
at the equation and quickly assume
that c is 30 instead of –30. Remind
them to write a quadratic equation
in standard form before using the
quadratic formula. It may help students also to note that they can
multiply through by –1 to remove
the negative signs.
36–43. See Additional Answers
beginning on page AA1.
536
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
1. What formula can you use to solve any quadratic equation? the quadratic formula
2. Explain how to use the quadratic formula to solve º2x 2 + 5x = º7.
See margin.
3. What is the difference between the two models for vertical motion?
See margin.
Use the quadratic formula to solve the equation.
2. Sample answer: Rewrite
4. x 2 + 6x º 7 = 0 1, º7 5. x 2 º 2x º 15 = 0 5, º3 6. x 2 + 12x + 36 = 0 º6
the equation in standard
form. Then substitute
7. 4x 2 º 8x + 3 = 0
8. 3x 2 + x º 1 = 0
9. x 2 + 6x º 3 = 0
a = º2, b = 5, and c = 7
0.5, 1.5
See margin.
º3 + 2兹3苶, º3 º 2兹3苶
into the quadratic formula. Write in standard form. Use the
quadratic formula to solve the equation.
The solutions are
10. 2x 2 = –x + 6
11. 6x = –8x 2 + 2
12. 3 = 3x 2 + 8x
º5 + 兹5苶2苶
º苶4(º
苶2)
苶(7
苶)苶
1
3
1
}}} ,
2
3x 2 + 8x º 3 = 0; º3, }}
2(º2)
}
}
}
}
2x 2 + x º 6 =
0;
º2,
8x
+
6x
º
2
=
0;
º1,
3
2
2 14. ºx 2 + 4x = 3
4 15. 4x 2 + 4x = º1
13.
º14x
=
–2x
+
36
or x = º1, and
1
2
2
2
}
}
2x
º
14x
º
36
=
0;
º2,
9
ºx
+
4x
º
3
=
0;
1,
3
4x
+
4x
+
1
=
0;
º
2
º5 º 兹5苶2苶
º苶4(º
苶2)
苶(7
苶)苶
Find the x-intercepts of the graph of the equation.
}}} , or
2(º2)
16. y = x 2 º 11x + 24
17. y = x 2 + 10x + 16
18. y = 2x 2 + 4x º 30
3, 8
º2, º8
º5, 3
3. In one model, the object is 19. y = 2x 2 + 6x º 9
20. y = 5x 2 + 8x º 8
21. y = 4x 2 + 8x º 1
dropped, that is, its initial
See margin.
See margin.
See margin.
FIELD TARGET EVENT From a hot-air balloon directly over a target, you
velocity is 0. In the other, 22.
the object is thrown, and,
throw a marker with an initial downward velocity of º40 feet per second from
therefore, has a nonzero
a height of 180 feet. How long does it take the marker to reach the target? 2.33 sec
initial velocity.
º3 º 3兹3苶
º4 º 2兹14
苶
º2 º 兹5苶
21. }}
≈ º4.10, 20. }}
≈ º2.30,
≈ º2.12,
19. }}
2
5
2
º1 + 兹13
苶 º1 º 兹13
苶
8. }}, }}
º3 + 3兹3苶
º4 + 2兹14
苶
º2 + 兹5苶
6
6
}} ≈ 1.10
}} ≈ 0.70
}} ≈ 0.12
x = 3.5.
2
5
2
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 805.
FINDING VALUES Find the value of b 2 º 4ac for the equation.
23. x 2 º 3x º 4 = 0 25
24. 4x 2 + 5x + 1 = 0 9
26. 5w 2 º 3w º 2 = 0 49 27. s 2 º 13s + 42 = 0 1
1
29. 5x 2 + 5x + ᎏᎏ = 0 21
5
25. x 2 º 11x + 30 = 0 1
28. 3x 2 º 5x º 12 = 0 169
1
1
30. ᎏᎏa 2 + 5a º 8 = 0 41 31. ᎏᎏv 2 º 6v º 3 = 0 39
2
4
SOLVING EQUATIONS Use the quadratic formula to solve the equation.
1
46. x 2 º 11x + 16 = 0;
11 º 兹57
苶
}} ≈ 1.73,
2
11 + 兹57
苶
}} ≈ 9.27
2
STUDENT HELP
HOMEWORK HELP
Example 1:
Example 2:
Example 3:
Example 4:
Exs. 23–43
Exs. 44–52
Exs. 53–61
Exs. 71–80
32. 4x 2 º 13x + 3 = 0 }4}, 3 33. y 2 + 11y + 10 = 0
34. 7x 2 + 8x + 1 = 0 1
º10, º1
º1, ºᎏᎏ
35. º3y 2 + 2y + 8 = 0 4 36. 6n2 º 10n + 3 = 0
37. 9x 2 + 14x + 3 = 0 7
See margin.
º}}, 2 See margin.
3
38. 8m2 + 6m º 1 = 0
39. 7x 2 + 2x º 1 = 0
40. º6x 2 º 3x + 2 = 0
See margin.
See margin.
See margin.
1
41. ºᎏᎏx 2 + 6x + 13 = 0 42. º10a 2 + 3a + 2 = 0 43. º2d 2 º 5d + 19 = 0
2
See margin.
See margin.
See margin.
STANDARD FORM Write the quadratic equation in standard form. Solve
using the quadratic formula.
44. 2x 2 = 4x + 30
45.
2x 2 º 4x º 30 = 0; º3, 5
2
47. 5x º 1 = º6x
1 48.
6x 2 + 5x º 1 = 0; º1, }}
6 51.
2
50. º1 + 3x = 2x
1
2
3x º 2x º 1 = 0; º}}, 1
3
536
2 º 3x + x 2 = 0
46. 16 = ºx 2 + 11x
See margin.
2q 2 º 6 = –4q
49. 5z º 2z2 + 15 = 8 7
2q + 4q º 6 = 0; 1, º3
2z 2 º 5z º 7 = 0; º1, }}
º5c2 + 9c = 4
52. º16b = º8b 2 º 8 2
4
8b 2 º16b + 8 = 0; 1
5c 2 º 9c + 4 = 0; }}, 1
x 2 2º 3x + 2 = 0; 1, 2
Chapter 9 Quadratic Equations and Functions
5
3 º 3 3苶
2
兹
56. }}
≈ º1.10,
FINDING INTERCEPTS Find the x-intercepts of the graph of the equation.
3 + 3兹3苶
}} ≈ 4.10
2
º1 º 兹17
苶
≈ º2.56,
57. }}
2
º7 º 57
苶
2
53. y = 3x 2 º 6x º 24
54. y = 2x 2 º 6x º 8
55. y = 2x 2 º 2x º 12
4, º2
4,
º1
º2, 3
56. y = º2x 2 + 6x + 9
57. y = x 2 + x º 4
58. y = x 2 + 7x º 2
See margin.
See margin.
See margin.
59. y = º3x 2 º 2x + 1 1 60. y = º4x 2 + 8x º 2
61. y = º5x 2 + 5x + 5
º1, }}
See margin.
See margin.
3
CHOOSING A METHOD Solve the quadratic equation by finding square
roots or by using the quadratic formula. Explain why you chose the method.
62–70. See margin.
62. 6x 2 + 20x + 5 = 0
63. m2 = 32
64. x 2 º 625 = 0
º7 + 兹57
苶
}} ≈ 0.27
2
65. 4y 2 º 49 = 0
66. º2x 2 + 6x + 1 = 0
67. h2 + 18h + 81 = 0
2 º 2苶
2
68. 5x 2 = 25
69. 9y 2 º 3y = 1
70. v 2 = 8v + 2
2 + 兹2苶
}} ≈ 1.71
2
FIELD TARGET EVENT In Exercises 71–76, six balloonists compete in the
Field Target Event at a hot-air balloon festival. Calculate the amount of time
it takes for the marker to hit the target when thrown down from the given
initial height (in feet) with the given initial velocity (in feet per second).
º1 + 兹17
苶
}} ≈ 1.56
2
兹
58. }}
≈ º7.27,
兹
60. }}
≈ 0.29,
1 º 兹5苶
≈ º0.62,
61. }}
2
1 + 兹5苶
}} ≈ 1.62
2
71. s = 200; v = º50
72. s = 150; v = º25
73. s = 100; v = º10
2.30 sec
2.38 sec
2.21 sec
74. s = 150; v = º33
75. s = 80; v = º40
76. s = 50; v = 0
2.20 sec
1.31 sec
1.77 sec
VERTICAL MOTION In Exercises 77–80, use a vertical motion model to
find how long it will take for the object to reach the ground.
77. You drop keys from a window 30 feet above ground to your friend below.
Your friend does not catch them. 1.37 sec
78. An acorn falls 45 feet from the top of a tree. 1.68 sec
FOCUS ON
80. You throw a ball downward with an initial speed of 10 feet per second out of
a window to a friend 20 feet below. Your friend does not catch the ball. 0.85 sec
81.
82.
RE
FE
L
AL I
URBAN BIRDS
Cities provide a
habitat for many species of
wildlife including birds of
prey such as Peregrine
falcons and red-tailed
hawks.
83.
PEREGRINE FALCON A falcon
dives toward a pigeon on the ground.
When the falcon is at a height of
100 feet the pigeon sees the falcon,
which is diving at 220 feet per
second. Estimate the time the pigeon
has to escape. 0.44 sec
Falcon
100 ft
RED-TAILED HAWK A hawk
dives toward a snake. When the hawk
is at a height of 200 feet the snake
sees the hawk, which is diving at
105 feet per second. Estimate the
time the snake has to escape.
1.54 sec
BALD EAGLE An eagle dives
62–70. Sample explanations are
given.
–10 + 兹70
苶
–10 – 兹70
苶
62. }
≈ –0.27, }
≈
6
6
–3.06; quadratic formula; the
expression on the left cannot be
written as a perfect square.
63. 4兹2苶 ≈ 5.66, –4兹2苶 ≈ –5.66; finding
square roots; simplest method
64. –25, 25; finding square roots;
simplest method
7 7
65. – }}, }}; finding square roots;
2 2
simplest method
–3 + 兹11
苶
79. A lacrosse player throws a ball upward from her playing stick with an initial
height of 7 feet, at an initial speed of 90 feet per second. 5.70 sec
APPLICATIONS
APPLICATION NOTE
EXERCISE 81 It is important to
specify that the wings are folded so
that air resistance is reduced and
the dive is more like free fall.
Students may have seen films in
which sky divers fold in their arms
to make a faster descent. Ski
jumpers widen their arms and skis
so that they fall less quickly.
–3 – 兹11
苶
66. }
≈ –0.16, }
≈ 3.16;
–2
–2
quadratic formula; the expression
on the left cannot be written as a
perfect square.
67. –9; finding square roots; the
expression on the left can be
written as (h + 9) 2.
68. 兹5苶 ≈ 2.24, –兹5苶 ≈ –2.24; finding
square roots; simplest method
69 and 70. See Additional Answers
beginning on page AA1.
ADDITIONAL PRACTICE
AND RETEACHING
For Lesson 9.5:
Pigeon
toward a mouse. When the eagle is at a height of 125 feet the mouse sees the
eagle, which is diving at 145 feet per second. Estimate the time the mouse
has to escape. 0.79 sec
9.5 Solving Quadratic Equations by the Quadratic Formula
537
• Practice Levels A, B, and C
(Chapter 9 Resource Book, p. 73)
• Reteaching with Practice
(Chapter 9 Resource Book, p. 76)
•
See Lesson 9.5 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
537
Test
Preparation
4 ASSESS
DAILY HOMEWORK QUIZ
Transparency Available
1. Find the value of b 2 – 4ac for
–2x 2 + 7x – 6. 1
2. Use the quadratic formula to
solve 6x 2 – x – 15. – }32}, }53}
3. Write –17x = –6x 2 – 12 in
standard form. Solve using
the quadratic formula.
84. MULTI-STEP PROBLEM In parts (a)–(d), a batter hits a pitched baseball
when it is 3 feet off the ground. After it is hit, the height h (in feet) of the ball
at time t (in seconds) is modeled by
84. d. Sample answer: The speed
of the ball approaching the
h = º16t 2 + 80t + 3
batter, the speed of the
swinging bat, the angle of where t is the time (in seconds).
the bat, and the wind and
a. Find the time when the ball hits the ground in the outfield. 5.04 sec
weather conditions can
change the path of the
b. Write a quadratic equation that you can use to find the time when the
baseball. The dimensions of
baseball is at its maximum height of 103 feet. Solve the quadratic
the ball park also determine
equation. º16t 2 + 80t + 3 = 103, or º16t 2 + 80t º 100 = 0; 2.5 sec
whether a hit is a home run.
c.
4 3
3 2
6x 2 – 17x + 12 = 0; }}, }}
4. Find the x-intercepts of the
graph of y = 2x 2 – 5x + 1.
★ Challenge
5 – 兹17
苶
5 + 兹17
苶
} ≈ 0.22, } ≈ 2.28
4
4
5. Solve 9x 2 – 36 = 0 by finding
square roots or by using the
quadratic formula. –2, 2
6. The initial upward velocity of a
golf ball is 60 ft/sec. How long
will it take the ball to return to
the ground? 3.75 sec
1
2
85. x = 1}}; the x-intercepts are
冉 12 冊
EXTRA CHALLENGE
www.mcdougallittell.com
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 9.5 are available in
blackline format in the Chapter 9
Resource Book, p. 80 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Describe how to use
the x-intercepts of the graph of
a quadratic function to find the
vertex of the graph.
Sample answer: Find the
x-coordinate of the point midway
between the x-intercepts. This
is the x-coordinate of the vertex.
Substitute this value into the
equation of the function to find
the y-coordinate of the vertex.
92.
93.
4
5
6
⫺2 ⫺1
0
1
2
3
4
538
0
2
4
6
94–102. See Additional Answers
beginning on page AA1.
538
x
Writing Explain how you can use this two-part
form of the quadratic formula
兹b苶苶º
苶苶4a苶c苶
x = ᎏᎏ ± ᎏᎏ
ºb
2a
2
2a
to find the distance between the axis of symmetry of
a parabola and either of its x-intercepts. See margin.
⫺21
Ex. 85
EVALUATING EXPRESSIONS Evaluate the expression. (Review 1.3 for 9.6)
the axis of symmetry of the
2
88. ºy 2 when y = º1 º1
graph of y = ax 2 + bx + c. 87. x when x = º5 25
The given values of x are
89. º4xy when x = º2 and y = º6º48 90. y 2 º y when y = º2 6
the x-intercepts of the
equation, which are
NEQUALITIES Solve the inequality and graph the solution. (Review 6.3, 6.4)
equidistant from the axis I91–96.
See margin for graphs.
of symmetry. The distance 91. 2 ≤ x < 5
92. 8 > 2x > º4
93. º12 < 2x º 6 < 4
of each point from the axis
4 > x > º2
º3 < x < 5
of symmetry is
94. º3 < ºx < 1
95. |x + 5| ≥ 10
96. |2x + 9| ≤ 15
º1 < x < 3
x ≤ º15 or x ≥ 5
º12 ≤ x ≤ 3
苶2苶苶
苶c苶
兹b
º苶
4a
SKETCHING GRAPHS Sketch the graph of the function. Label the vertex.
}}.
2a
(Review 9.3) 97–102. See margin for graphs.
1
5
97. }}, º}}
3
3
97. y = 6x 2 º 4x º 1
98. y = º3x 2 º 5x + 3
99. y = º2x 2 º 3x + 2
5 61
98. º}}, }}
6 12
1
1
100. y = ᎏᎏx 2 + 2x º 1
101. y = 4x 2 º ᎏᎏx + 4
102. y = º5x 2 º 0.5x + 0.5
3 25
2º3)
4
99. º}}, }}
(º2,
4 8
103.
RECREATION There were 1.4 ª 107 people who visited Golden Gate
1 255
101. }}, 3}}
Recreation Area in California in 1996. On average, how many people visited
3
⫺2
86.
3
ºb
2a
2
⫺4
y
3
symmetry of the graph and show that it lies halfway
between the two x-intercepts. See margin.
86. x = }} is the equation of
1
0
85. VISUAL THINKING Write an equation of the axis of
MIXED REVIEW
冉 冊
冉 冊
冉 冊
冉 32 256 冊
1 41
102. 冉º}}, }}冊
20 80
91.
2.5 sec; this is the same as the value found in part (b).
d. CRITICAL THINKING What factors change the path of a baseball? What
factors would contribute to hitting a home run? See margin.
º3 and 6, and the midpoint of
(º3, 0) and (6, 0) is 1}}, 0 .
Use a graphing calculator to graph the function. Use the zoom feature
to approximate the time when the baseball is at its maximum height.
Compare your results with those you obtained in part (b).
per day? per month? 䉴 Source: National Park Service (Review 8.4)
about 3.84 ª 10 4 people; about 1.17 ª 10 6 people
Chapter 9 Quadratic Equations and Functions