Download Chap. 9.1 Answers - Aspen High School - Algebra I

GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
3. Sample answer: Subtract
c from both sides, then
divide both sides by a.
c
If º}} is positive, find both
a
c
c
square roots of º}}. If º}}
a
a
is zero, the only solution
c
is 0. If º}} is negative,
a
there are no real solutions.
Skill Check
✓
3 APPLY
1. State the meanings of the symbols 兹苶, º兹苶, and ± 兹苶. the principal (or positive)
square root; the negative square root; both the positive and the negative square root
2. Give an example of a perfect square and an example of an irrational number.
苶
Sample answers: 121; 兹11
3. Explain how to find solutions of an equation of the form ax 2 + c = 0.
See margin.
If the statement is true, give an example. If false, give a counterexample.
4. Some numbers have no real square root. True. Sample answer: 兹º
苶1苶
5. No number has only one square root. False. Sample answer: 兹0苶 = 0
AVERAGE
6. All positive numbers have two different square roots.
True. Sample answer: º5 and 5 are both square roots of 25.
7. The square root of the sum of two numbers is equal to the sum of the square
roots of the numbers. False. Sample answer: 兹6苶+
苶苶
3 = 3; 兹6苶 + 兹3苶 = 4.8
12.
9. 兹0
苶.8
苶1苶
6
0.9
10. º兹0
苶.0
苶4苶
º0.2 11. ±兹9
苶
Day 1: pp. 507–510 Exs. 23, 26, 29,
and every 3rd Ex. through
Ex. 77, 78–81, 87,
95–100, 104, 107, 110, 111
±3
兹b苶2苶+
苶苶10苶a苶
10 ± 2兹b苶
13. ᎏᎏ
a
6
3, 7
14.
兹b苶2苶º
苶苶8a苶
BLOCK SCHEDULE
0
Solve the equation. If there is no solution, state the reason.
15. 2x 2 º 8 = 0
16. x 2 + 25 = 0
17. x 2 º 1.44 = 0 18. 5x 2 = º15
See margin.
See margin.
º2, 2
º1.2, 1.2
FALLING OBJECTS If an object is dropped from an initial height s, how
long will it take to reach the ground? Assume there is no air resistance.
19. s = 48
about 1.7 sec
20. s = 96 about 2.4 sec
21. s = 192
about 3.5 sec
22. If you double the height from which the object falls, do you double the
falling time? If not, why? No. Sample answer: The height is a function of the
square of the falling time rather than of the falling time.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 805.
FINDING SQUARE ROOTS Find all square roots of the number or write no
square roots. Check the results by squaring each root.
23. 49 º7, 7
27. 0
0
24. 144
º12, 12
28. 100
º10, 10
25. º4
no square roots
29. 0.09 º0.3, 0.3
26. º25
no square roots
30. 0.16 º0.4, 0.4
EVALUATING SQUARE ROOTS Evaluate the expression. Give the exact
value if possible. Otherwise, approximate to the nearest hundredth.
STUDENT HELP
31. º兹1
苶6苶9苶 º13
32. 兹6
苶4苶
HOMEWORK HELP
35. 兹0
苶.0
苶4苶 0.2
36. ±兹0
苶.7
苶5苶
Example 1: Exs. 23–30
Example 2: Exs. 31–38
Example 3: Exs. 39–44
continued on p. 508
Day 1: pp. 507–510 Exs. 23, 26, 29,
and every 3rd Ex. through
Ex. 77, 79–81, 87, 95,
96, 104, 107, 110, 111
ADVANCED
Evaluate the expression.
Evaluate the radical expression when a = 2 and b = 4.
18. There is no solution;
negative numbers have
no real square roots.
BASIC
Day 1: pp. 507–510 Exs. 23, 26, 29,
and every 3rd Ex. through
Ex. 77, 79–81, 95, 96,
104, 107, 110, 111
8. 兹3
苶6苶
16. There is no solution;
negative numbers have
no real square roots.
ASSIGNMENT GUIDE
8
33. 兹1
苶3苶
3.61
±0.87 37. º兹0
苶.1
苶
34. º兹1
苶2苶5苶
º11.18
º0.32 38. ±兹6
苶.2
苶5苶 ±2.5
EVALUATING EXPRESSIONS Evaluate 兹b
苶2苶º
苶苶4a苶c苶 for the given values.
39. a = 4, b = 5, c = 1
3
42. a = 2, b = 4, c = 0.5
2兹3苶
40. a = º2, b = 8, c = º8
0
43. a = º3, b = 7, c = 5
兹10
苶9苶
41. a = 3, b = º7, c = 6
undefined
44. a = 6, b = º8, c = 4
undefined
9.1 Solving Quadratic Equations by Finding Square Roots
507
pp. 507–510 Exs. 23, 26, 29, and
every 3rd Ex. through Ex. 77,
79–81, 87, 95, 96, 104, 107, 110,
111 (with Ch. 8 Assess.)
EXERCISE LEVELS
Level A: Easier
23–38, 101–104
Level B: More Difficult
39–77, 79–88, 95, 96, 105–112
Level C: Most Difficult
78, 89–94, 97–100
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 26, 38, 44, 50, 62, 74, 79. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 9
Resource Book, p. 24)
•
Transparency (p. 68)
COMMON ERROR
EXERCISE 29 Some students
may quickly write that the square
roots of 0.09 are ±0.03. Remind
them that when finding square
roots of decimals less than one, the
positive square root is larger than
the original decimal.
507
EVALUATING EXPRESSIONS Use a calculator to evaluate the
expression. Round the results to the nearest hundredth.
STUDENT HELP
COMMON ERROR
EXERCISE 73 After subtracting
12 from each side, students will
5
likely multiply both sides by ᎏᎏ
4
before concluding that there is no
solution. Point out that by being
observant, they can save a step.
After subtracting, the right side is
negative and the coefficient of x is
positive, so they know that the right
side will remain negative without
actually having to multiply.
HOMEWORK HELP
continued from p. 507
Example 4:
Example 5:
Example 6:
Example 7:
Exs. 45–53
Exs. 54–59
Exs. 60–77
Exs. 79–86
3 ± 4兹6苶
45. ᎏ
3
º2.27, 4.27
6 ± 4兹2苶
2 ± 5兹3苶
46. ᎏ º11.66, º0.34 47. ᎏ
º1
5
º1.33, 2.13
1 ± 6兹8苶
48. ᎏ
6
º2.66, 3.00
7 ± 3兹2苶 º11.24, º2.76 50. 2 ± 5兹6苶
49. ᎏ
ᎏ
º1
2
º5.12, 7.12
5 ± 6兹3苶
51. ᎏ
3
º1.80, 5.13
3 ± 4兹5苶
52. ᎏ
4
º1.49, 2.99
7 ± 0.3兹1
苶2
苶 º1.34, º0.99
53. ᎏᎏ
º6
QUADRATIC EQUATIONS Solve the equation or write no solution. Write
the solutions as integers if possible. Otherwise write them as radical
expressions.
54. x 2 = 36 º6, 6
55. b2 = 64 º8, 8
56. 5x 2 = 500 º10, 10
57. x 2 = 16 º4, 4
58. x 2 = 0 0
59. x 2 = º9 no solution
60. 3x 2 = 6 º兹2苶, 兹2苶
61. a 2 + 3 = 12 º3, 3
62. x 2 º 7 = 57 º8, 8
63. 2s 2 º 5 = 27 º4, 4
64. 5x 2 + 5 = 20 º兹3苶, 兹3苶 65. 7x 2 + 30 = 9 no solution
66. x 2 + 4.0 = 0 no solution 67. 6x 2 º 54 = 0 º3, 3
68. 7x 2 º 63 = 0 º3, 3
SOLVING EQUATIONS Use a calculator to solve the equation or write
no solution. Round the results to the nearest hundredth.
69. 4x 2 º 3 = 57
70. 6y2 + 22 = 34
71. 2x 2 º 4 = 10
º3.87, 3.87
º1.41, 1.41
º2.65, 2.65
2 2
1 2
4 2
72. ᎏᎏn º 6 = 2
73. ᎏᎏx + 12 = 5
74. ᎏᎏx + 3 = 8
5
3
2
º3.46, 3.46
no solution
º3.16, 3.16
2
2
75. 3x + 7 = 31
76. 6s º 12 = 0
77. 5a 2 + 10 = 20
º1.41, 1.41
º1.41, 1.41
º2.83, 2.83
78. CRITICAL THINKING Write your own example of a quadratic equation in the
form x 2 = d for each of the following.
2
a. An equation that has no real solution. x = d, for any number d < 0
b. An equation that has one solution. x 2 = 0
80. Sample answer:
t
0
1
1.5
1.75
1.8
1.9
1.95
c. An equation that has two solutions. x 2 = d, for any number d > 0
h
60
44
24
11
8.16
2.24
º0.84
about 1.95 sec
ROAD SAFETY In Exercises 79–82, a boulder falls off the top of a cliff
during a storm. The cliff is 60 feet high. Find how long it will take for the
boulder to hit the road below.
79. Write the falling object model for s = 60. h = º16t 2 + 60
80. Try to determine the time when h = 0 from an input-output table for this model.
See margin.
81. Solve the falling object model for h = 0. 兹3.
苶75
苶 ≈ 1.94 sec
82.
Writing Which problem solving method do you prefer? Why?
See margin.
FALLING OBJECT MODEL In Exercises 83–86, an object is dropped from
82. Sample answer: I prefer
solving the equation; it is a height s. How long does it take to reach the ground? Assume there is no
more straightforward and air resistance.
provides an exact
83. s = 144 feet
84. s = 256 feet
85. s = 400 feet
86. s = 600 feet
answer.
3 sec
4 sec
5 sec
about 6.1 sec
508
508
Chapter 9 Quadratic Equations and Functions
INT
STUDENT HELP
NE
ER T
87.
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 87–88.
88.
FOCUS ON
CAREERS
HOME COMPUTER SALES The sales S (in millions of dollars) of
home computers in the United States from 1988 to 1995 can be modeled
by S = 145.63t 2 + 3327.56, where t is the number of years since 1988. Use
this model to estimate the year in which sales of home computers will be
$36,000 million. 䉴 Source: Electronic Industries Association 2003
STUDENT HELP NOTES
COMPUTER SOFTWARE SALES The sales S (in millions of dollars) of
computer software in the United States from 1990 to 1995 can be modeled by
S = 61.98t 2 + 1001.15, where t is the number of years since 1990. Use this
model to estimate the year in which sales of computer software will be
$7200 million. 䉴 Source: Electronic Industries Association 2000
MINERALS In Exercises 89–94, use the following information.
Mineralogists use the Vickers scale to measure the hardness of minerals. The
hardness H of a mineral can be determined by hitting the mineral with a pyramidshaped diamond and measuring the depth d of the indentation. The harder the
mineral, the smaller the depth of the indentation. A model that relates mineral
hardness with the indentation depth (in millimeters) is Hd 2 = 1.89.
Use a calculator to find the depth of the indentation for the mineral with the given
value of H. Round to the nearest hundredth of a millimeter.
RE
FE
L
AL I
90. Gold: H = 50
0.19 mm
91. Galena: H = 80
0.15 mm
92. Platinum: H = 125
0.12 mm
93. Copper: H = 140
0.12 mm
94. Hematite: H = 755
0.05 mm
MINERALOGISTS
The properties and
distribution of minerals are
studied by mineralogists.
The Vickers scale applies to
thin slices of minerals that
can be examined with a
microscope.
INT
89. Graphite: H = 12
0.40 mm
NE
ER T
CAREER LINK
www.mcdougallittell.com
Test
Preparation
95. .
APPLICATION NOTE
EXERCISES 89–94 Although
graphite has the lowest Vickers
scale hardness number, 12, industrial diamonds may be synthesized
by subjecting graphite to very high
pressures and temperatures. The
mineral chromite is at the high end
of the scale, with a Vickers number
of 1206.
ENGLISH LEARNERS
EXERCISES 87–94
Understanding the language of
word problems can be extremely
difficult for English language learners. In Exercises 87 and 88, you may
need to clarify the usage of the
word where, common in mathematical language but different from
normal usage. For Exercises 89–94,
make sure students understand
both the vocabulary and the causeeffect relationship in the sentence
The harder the mineral, the smaller
the depth of the indentation.
CAREER NOTE
EXERCISES 89–94
Additional information about
mineralogists is available at
www.mcdougallittell.com.
QUANTITATIVE COMPARISON In Exercises 95–96, choose the statement
below that is true about the given numbers.
A
¡
B
¡
C
¡
D
¡
Homework Help Students can
find help for Exs. 87 and 88 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
The number in column A is greater.
The number in column B is greater.
The two numbers are equal.
The relationship cannot be determined from the given information.
For Lesson 9.1:
Column A
Column B
º兹1苶2苶1苶
º2兹1苶2苶1苶
A
The positive value of
x in 2x 2 + 14 = 46
C
96. . The positive value of
x in 2x 2 º 14 = 18
ADDITIONAL PRACTICE
AND RETEACHING
9.1 Solving Quadratic Equations by Finding Square Roots
509
• Practice Levels A, B, and C
(Chapter 9 Resource Book, p. 14)
• Reteaching with Practice
(Chapter 9 Resource Book, p. 17)
•
See Lesson 9.1 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
509
4 ASSESS
DAILY HOMEWORK QUIZ
Transparency Available
1. Find all square roots of 0.25.
±0.5
2. Evaluate –兹1.
苶44
苶. –1.2
2
3. Evaluate 兹b苶苶–苶苶4a苶c苶 for
a = 3, b = 3, and c = –6. 9
4. Using a calculator, evaluate
–3 ± 2兹5苶
ᎏ to the nearest
4
hundredth. –1.87, 0.37
SCIENCE
±1.53
7. A cliff diver jumps from an
84 ft cliff. Write a falling object
model for the time t it will take
the diver to hit the water.
Assume there is no air
resistance. h = –16t 2 + 84
1
2
that is, the exponential
term in both equations
is the same.
97. The NASA Lewis Research Center has two microgravity facilities. One
provides a 132-meter drop into a hole and the other provides a 24-meter
drop inside a tower. How long will each free-fall period be?
about 5.19 sec, about 2.21 sec
98. In Japan a 490-meter-deep mine shaft has been converted into a
microgravity facility. This creates the longest period of free fall currently
available on Earth. How long will a period of free-fall be? 10 sec
99. If you want to double the free-fall time, how much do you have to increase
the height from which the object was dropped? See margin.
100. CRITICAL THINKING How are these formulas similar?
1
2
h = º16t 2 + s when h is height, s is initial height, and t is time
www.mcdougallittell.com
MIXED REVIEW
FACTORING INTEGERS Write the prime factorization. (Skills Review, p. 777)
102. 24 23 ª 3
103. 72
2 3 ª 32
104. 108 22 ª 33
GRAPH AND CHECK Use a graph to solve the linear system. Check your
solution algebraically. (Review 7.1)
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 9.1 are available in
blackline format in the Chapter 9
Resource Book, p. 21 and at
www.mcdougallittell.com.
105. º3x + 4y = º5
4x + 2y = º8
(º1, º2)
106. 4x + 5y = 20
5
ᎏᎏx + y = 4
4
(0, 4)
1
107. ᎏᎏx + 3y = 18
2
2x + 6y = º12
(º48, 14)
SOLVING SYSTEMS In Exercises 109–111, use linear combinations to solve
the system. (Review 7.3)
ADDITIONAL TEST
PREPARATION
1. WRITING Describe how to
solve the quadratic equation
ax 2 + c = 0. Sample answer:
108. 12x º 4y = º32
Add –c to each side. Then divide
c
each side by a to get x 2 = – }}.
冪莦莦
c
c
the solution ± – }} . If – }} is
a
a
negative, there is no real solution.
510
º7x + y = 9
(º2, 2)
(º4, º19)
110. 8x º 5y = 100
1
2x + ᎏᎏy = 4
2
(5, º12)
BASKETBALL TICKETS You are selling tickets at a high school
basketball game. Student tickets cost $2 and general admission tickets cost
$3. You sell 2342 tickets and collect $5801. How many of each type of ticket
did you sell? (Review 7.2) 1225 student tickets, 1117 general admission tickets
112.
FLOWERS You are buying a combination of irises and white tulips for a
flower arrangement. The irises are $1 each and the white tulips are $.50. You
spend $20 total to purchase an arrangement of 25 flowers. How many of
each kind did you purchase? (Review 7.2) 15 irises and 10 white tulips
a
square root of both sides to get
109. 10x º 3y = 17
x + 3y = 4
111.
If – }} is nonnegative, take the
510
See margin.
d = ᎏᎏg(t 2) when d is distance, g is gravity, and t is time
EXTRA CHALLENGE
101. 11 11
c
a
In Exercises 97–100, use the following information.
the equation d = ᎏᎏg(t 2), where g is the acceleration due to gravity
2
6. Solve 3y + 4 = 11. Write the
result to the nearest hundredth.
CONNECTION
Scientists simulate a gravity-free environment called microgravity in free-fall
situations. A similar microgravity environment can be felt on free-fall rides at
amusement parks or when stepping off a high diving platform. The distance d
(in meters) that an object that is dropped falls in t seconds can be modeled by
(9.8 meters per second per second).
100. Sample answer: The
first equation represents
the distance a falling
object falls in a given
time t, and the second
represents the height of
the same object at time
t. Then h = s º d and
1
so º16t 2 = º}}g(t 2);
2
5. Solve x 2 – 7 = 13. Write the
result as a radical expression.
±兹20
苶
★ Challenge
99. 4 times; the distance
that an object is
dropped is a function of
the time squared. So
(2t)2 = 22 • t 2 = 4t 2
Chapter 9 Quadratic Equations and Functions