Lehmann IA SSM Ch9

8/20/2019 Lehmann IA SSM Ch9

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275

Chapter 9

Radical Functions

Homework 9.1

1.

5 2 2/ 5 x x=

3. 4 3 3/ 4

x x=

5. 1/ 2

x x=

7. ( ) ( )3/ 7 372 9 2 9 x x+ = +

9. ( ) ( )4 4/77

3 2 3 2 x x+ = +

11. ( )1/ 2

7 4 7 4 x x+ = +

13. 49 7=

15. 50 25 2 25 2 5 2= ⋅ = =

17. 2 2/ 2 1

x x x x= = =

19. 8 8/ 2 4

x x x= =

21.6 6 3

36 36 6 x x x= ⋅ =

23.2 2

5 5 5 x x x= =

25.9 8 8 4

x x x x x x x= ⋅ = ⋅ =

27.5 4

4

2

24 4 6

4 6

2 6

x x x

x x

x x

= ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅

=

29. 3 8 2 8

2 8

4

80 16 5

16 5

4 5

x y x x y

x x y

xy x

= ⋅ ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅

=

31. 3 5 2 4

2 4

2

200 100 2

100 2

10 2

x y x x y y

x x y y

xy xy

= ⋅ ⋅ ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅ ⋅

=

33. ( ) ( ) ( )8 8/ 2 4

2 5 2 5 2 5 x x x+ = + = +

35. ( ) ( ) ( )

( )

( )

5 4

4

2

6 3 6 3 6 3

6 3 6 3

6 3 6 3

x x x

x x

x x

+ = + +

= + ⋅ +

= + +

37. 327 3=

39. 6 6 6/6 1

x x x x= = =

41. 3 33 338 8 2 x x x= ⋅ =

43. 5 55 5532 32 2 x x x= ⋅ =

45. 4 412 12 3481 81 3 x x x= ⋅ =

47. 6 6 6 6 617 12 5 12 5 2 5

x x x x x x x= ⋅ = ⋅ =

49. 3 317 15 2

3 315 23

35 2

125 125

125

5

x x x

x x

x x

= ⋅ ⋅

= ⋅ ⋅

=

51. 40 7 40 5 25 5

5 40 5 255 5 5

8 25

64 32 2

32 2

2 2

x y x y y

x y y

x y y

= ⋅ ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅

=

53. ( ) ( ) ( )5 5/5 15 6 6 6 6 xy xy xy xy= = =

55. ( ) ( ) ( )4 4/ 4 14 3 6 3 6 3 6 3 6 x x x x+ = + = + = +

57. ( ) ( ) ( )20 20/5 45 4 7 4 7 4 7 x x x+ = + = +

59. ( ) ( ) ( )24 24/ 4 64 7 7 7 x x x+ = + = +


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Homework 9.1 SSM : Intermediate Algebra

276

61. ( ) ( ) ( )

( )

( )

31 306 6

30 66

5 6

2 9 2 9 2 9

2 9 2 9

2 9 2 9

x x x

x x

x x

+ = + +

= + ⋅ +

= + +

63. 8 46 6/8 3/ 4 3

x x x x= = =

65. 6 34 4/ 6 2/3 2

x x x x= = =

67. ( ) ( )

( )

( )

10 10/1212

5/ 6

56

2 7 2 7

2 7

2 7

x x

x

x

+ = +

= +

= +

69.

6 14 14/ 6

7 /3

3 7

3 6

3 6 3

2 3

x x

x

x

x x

x x

x x

=

=

=

= ⋅

= ⋅

=

71. 6 36

3/ 6

1/ 2

27 3

3

3

3

=

=

=

=

73. 10 108 810

10 102 8

2 /10 8/10

1/5 4 /5

5 45

5 4

16 16

4

4

4

4

4

x x

x

x

x

x

x

= ⋅

= ⋅

= ⋅

= ⋅

= ⋅

=

75. a. 3 3 2 3 2 6

2 22 2 2 2

h h h hd

⋅= = ⋅ = =

⋅

b.

( )

6

2

6 1450

2

8700

2

10 87

2

5 87 46.64 miles

hd =

=

=

=

= ≈

The distance to the horizon from the top of

the Sears Tower is roughly 46.64 miles.

c.

( )

6

2

6 30,000

2

180,000

2

300 2

2

150 2 212.13 miles

hd =

=

=

=

= ≈

The distance to the horizon from the plane is

roughly 212.13 miles.

77. Answers will vary.

79. n n

n n

n n

n n n

abc a bc

a bc

a b c

a b c

= ⋅

=

= ⋅

=


3/36

SSM: Intermediate Algebra Homework 9.2

277

81. a.

i.

ii.

iii.

b.

The graphs in (a) coincide with the graph of

y x= .

c.

i.

ii.

iii.

d.

The graphs in (c) coincide with the graph of

y x= .

83. Answers will vary. See Key Points.

Homework 9.2

1. ( )4 5 4 5 9 x x x x+ = + =

3.

( )2.3 4.8 2.3 4.8 2.5 x x x x− = − = −

5.

( ) ( )

( ) ( )

3 5 2 3 6 3 7 5

3 5 7 5 2 3 6 3

3 7 5 2 6 3

10 5 4 3

x x x x

x x x x

x x

x x

+ − +

= + + −

= + + −

= −

7. ( )

( )

3 3

3

3

2 5 5 2 5 5

2 5 5

3 5

x x x x x x

x x

x x

+ − = − +

= − +

= − +

9.

( )

3 3

3

3

6 1 3 1 2 1

6 3 1 2 1

3 1 2 1

x x x

x x

x x

− − − − −

= − − − −

= − − −

11.

( ) ( )

3 3

3

3

5 3 4 2

5 3 4 2

8 6

x x x x

x x

x x

+ + +

= + + +

= +

13.

( ) ( )

6 64 4

64

4

3.7 1.1 4.2 4.2

3.7 1.1 4.2 4.2

2.6

x x x x

x x

x

− − +

= − + − +

=

15. ( ) ( )

( ) ( )

3 7 2 2

3 7 3 3 2 2

21 3 6 2

3 21 6 2

3 1 21 6 2

4 25

x x

x x

x x

x x

x

x

− + − +

= ⋅ − ⋅ + ⋅ − −

= − + − −

= − − + + −

= − − + + −

= − +

17. ( )5 2 4 5 2 5 4

10 20

x x

x

− + = − ⋅ − ⋅

= − −


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278

19. ( ) ( )

( )

3 3

3 3

3 3

3 3

7 1 7 1

7 7 1 7 7 1

7 7 7 7

7 7 7 7

14

x x

x x

x x

x x

+ − −

= ⋅ + ⋅ − ⋅ − −

= + − +

= − + +

=

21. 25 4 25 4

5 2

7

x x x x

x x

x

+ = +

= +

=

23. 3 20 2 45 3 4 5 2 9 5

3 4 5 2 9 5

3 2 5 2 3 5

6 5 6 5

12 5

x x x x

x x

x x

x x

x

+ = ⋅ + ⋅

= +

= ⋅ + ⋅

= +

=

25.

( )

3 2

2

5 49 5 49

5 49

5 7

5 7

2

x x x x x x x

x x x x

x x x x

x x

x x

− = ⋅ − ⋅

= −

= −

= −

= −

27.2 2

3 81 2 100 3 9 2 10

27 20

7

x x x x

x x

x

− = ⋅ − ⋅

= −

=

29.

( )

3 2

2

12 75 4 3 25 3

4 3 25 3

2 3 5 3

2 5 3

7 3

x x x x x x x

x x x x

x x x x

x x

x x

+ = ⋅ + ⋅

= +

= +

= +

=

31.

( )

3 3 3 35 2 3 2 2

3 3 33 2 23

3 32 2

3 2

3 2

27 8 27 8

27 8

3 2

3 2

x x x x x x x

x x x x

x x x x

x x

x x

− = ⋅ − ⋅

= −

= −

= −

=

33.

( )

4 4 4 411 7 8 3 4 3

4 4 4 48 3 4 3

4 42 3 2 3

42 3

42 3

16 3 16 3

16 3

2 3

2 3

x x x x x x x x

x x x x x

x x x x

x x

x x

− = ⋅ − ⋅

= −

= −

= −

= −

35.

2

3 2 3 2

6

6

6

x x x x

x x

x

x

⋅ = ⋅ ⋅ ⋅

= ⋅

=

=

37.

2

2

2 5 4 3 2 4 5 3

8 5 3

8 15

8 15

8 15

x x x x

x x

x

x

x

− ⋅ = − ⋅ ⋅ ⋅

= − ⋅

= −

= −

= −

39. ( )

2 2

2 2

2 7 7 2 2 7 7 2 7 2

2 7 7 2 7 2

2 49 2 14

2 49 2 14

2 7 2 14

14 2 14

x x x x x x x

x x x x

x x

x x

x x

x x

+ = ⋅ + ⋅

= ⋅ + ⋅

= +

= +

= ⋅ +

= +

41. ( )( )

( )

2

6 2 5 4

6 5 2 5 6 4 2 4

30 10 24 8

10 30 8 24

10 38 24

x x

x x x x

x x x

x x x

x x

− −

= ⋅ − ⋅ − ⋅ − −

= − − +

= − + + −

= − + −

43. ( )( )

( )

2

2 1 4

2 1 2 4 1 4

2 8 4

2 1 8 4

2 7 4

x x

x x x x

x x x

x x

x x

+ −

= ⋅ + ⋅ − ⋅ − ⋅

= + − −

= + − −

= − −


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280

73. a. ( ) ( )2 100 200

100 2.4932.2 32.2

f = = ≈

It takes roughly 2.49 seconds for an object

to fall 100 feet.

b. ( ) ( )2 1483 2966

1483 9.632.2 32.2

f = = ≈

The parachutist was in freefall for roughly

9.6 seconds. This is an underestimate

because the chute would slow down his

descent.

c. f is an increasing function. This makes sense

because the higher an object is, the longer it

should take to fall to the ground.

75.

1 1 3 2 11/ 2 62 3 6 6 6

1/ 33

x x

x x x x x x

− −

= = = = =

77. Begin by letting the two expressions be denoted

by W and Y . We know the following from the

problem statement:

9 7

3

W Y x

W Y x

+ = +

− = +

Adding the two equations together will eliminate

the Y terms and yield:

2 9 7 3

2 10 10

5 5

W x x

W x

W x

= + + +

= +

= +

Plug this into one of the original equations:

5 5 9 7

9 7 5 5

4 2

x Y x

Y x x

Y x

+ + = +

= + − −

= +

Thus, the two expressions are 5 5 x + and

4 2 x + .

79. a. 1/ 3 1/ 43 4

1 1

3 4

4 3

12 12

7/12

12 7

x x x x

x

x

x

x

+

+

= ⋅

=

=

=

=

b. 1/ 1/

1 1

k nk n

k n

n k

n k n k

n k

n k

n k n k

x x x x

x

x

x

x

+

+

⋅ ⋅

+

⋅

⋅ +

= ⋅

=

=

=

=

c.3 4 3, 4 x x k n→ = =

3 4 123 4 73 4 x x x x⋅ +

= =

The results are the same.

d. 5 7 355 7 125 7 x x x x

⋅ +

= =

81. Answers will vary. See Key Points.

Homework 9.3

1.

( )2

2 2 3

3 3 3

2 3

3

2 3

3

= ⋅

=

=

3.

( )2

8 8

8

8

x

x x x

x

x

x

x

= ⋅

=

=

5.

( )2

3 3 5

5 5 5

3 5

5

3 5

5

x

x x x

x

x

x

x

= ⋅

=

=


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281

7.

( )

2

5 5

2 2

5

2

5

2

x

x x x

x

x

x

x

= ⋅

⋅

=

=

9.

( )2

4 4 2

3 2 3 2 2

4 2

3 2

4 2

3 2

2 2

3

x

x x x

x

x

x

x

x

x

= ⋅

=

=

⋅

=

11.

( )2

4 4

2

2

2

x x

x

x x

x

x

x

x

=

= ⋅

=

=

13.

( )2

7 7

2 2

7 2

2 2

7 2

2

14

2

=

= ⋅

⋅

=

=

15.

( )2

2 2

2

2

2

x x

x

x x

x

x

x

x

=

= ⋅

⋅

=

=

17.

( )2

3 3

3

3 3

3

3

3

3

x x

x

x

x

=

= ⋅

⋅

=

=

19.

( )2

3 3 4

4 4 4

3 4

4

3 4

4

x

x x x

x

x

x

x

−

= ⋅

− − −

−

=

−

−

=

−

21.

3

3 3 3

3

3

3

3

3

2 2 25

5 5 25

2 25

5 25

2 25

125

2 25

5

= ⋅

=

⋅

=

=


8/36


282

23.

3

33 3

3

3

3

3

3

5 5 4

16 16 4

5 4

16 4

5 4

64

5 4

4

= ⋅

=

⋅

=

=

25.

3 2

3 3 3 2

3 2

3 2

3 2

3 3

3 2

4 4

5 5

4

5

4

5

4

5

x

x x x

x

x x

x

x

x

x

= ⋅

=

⋅

=

=

27.

3

33 32 2

3

3 2

6 6 4

42 2

6 4

2 4

x

x x x

x

x x

= ⋅

=

⋅

3

3 3

3

3 33

3

3

6 4

8

6 4

8

6 4

2

3 4

x

x

x

x

x

x

x

x

=

=

=

=

29.

4

44 43 3

4

4 3

7 7 4

44 4

7 4

4 4

x

x x x

x

x x

= ⋅

=

⋅

4

4 4

4

4 44

4

7 4

16

7 4

16

7 4

2

x

x

x

x

x

x

=

=

=

31.

( )

3 3

1/3 1/ 2

2

x x x

x x x

x x

x

= ⋅

⋅=

1 1

3 2

2 3

6 6

5/ 6

6 5

x

x

x

x

x

x

x

x

+

+

=

=

=

=

33.

5

53 5 3

5 25

5 53 2

2 2

2

x x

x

x x

=

= ⋅

5 2

5 3 2

5 2

5 5

5 2

2

2

2

x

x x

x

x

x

x

⋅=

⋅

=

=


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283

35.

4

42 4 2

4 24

4 42 2

4 4

9 9

4 9

9 9

x x

x

x x

=

= ⋅

4 2

4 2 2

4 2

4 4

4 2 2

2/ 4 2/ 4

1/ 2 1/ 2

4 9

9 9

36

81

6

3

6

3

6

3

6

3

x

x x

x

x

x

x

x

x

x

x

x

x

⋅=

⋅

=

=

=

=

=

37.

5

54 5 4

5 5

55 4

3 3

4 4

3 8

84

x x

x

x x

=

= ⋅

5

5 4

5

5 5

5

3 8

4 8

24

32

24

2

x

x x

x

x

x

x

⋅=

⋅

=

=

39.

( ) ( )22

1 1 5 3

5 3 5 3 5 3

1 5 1 3

5 3

5 325 3

5 3

22

−= ⋅

+ + −

⋅ − ⋅=

−

−=

−

−=

41.

( ) ( )

22

2 2 4 7

4 7 4 7 4 7

2 4 2 7

4 7

8 2 7

16 7

8 2 7

9

−= ⋅

+ + −

⋅ − ⋅=

−

−=

−

−=

43.

( ) ( )2

2

1 1 7

7 7 7

1 1 7

7

7

49

x

x x x

x

x

x

x

+= ⋅

− − +

⋅ + ⋅=

−

+=

−

45.

( ) ( )2

2

1

1 1 1

1

1

1

x x x

x x x

x x x

x

x x

x

+= ⋅

− − +

⋅ + ⋅=

−

+=

−

47.

( ) ( )

( )

( )

2 2

2

22

3 3 4 5

4 5 4 5 4 5

3 4 3 5

4 5

12 15

4 25

12 15

16 25

x x x

x x x

x x x

x

x x

x

x x

x

+= ⋅

− − +

⋅ + ⋅=

−

+

=

−

+=

−


10/36


284

49.

( ) ( )( )

( )

2 2

2

22

2 2 3 7

3 7 3 7 3 7

2 3 2 7

3 7

6 14

3 49

6 14

9 49

x x x

x x x

x x x

x

x x

x

x x

x

+= ⋅

− − +

⋅ + ⋅=

−

+

=

−

+=

−

51.

( ) ( )( ) ( )

( ) ( )

2 2

2 2

5 5 5

5 5 5

2 5 5

5

10 25

25

x x x

x x x

x x

x

x x

x

− − −= ⋅

+ + −

− +

=

−

− +=

−

53.

( ) ( )22

3 3 4

4 4 4

4 3 4 3

4

4 12 3

167 12

16

x x x

x x x

x x x x

x

x x x

x

x x

x

+ + += ⋅

− − +

⋅ + ⋅ + ⋅ + ⋅=

−

+ + +=

−

+ +=

−

55.

( ) ( )

( )

( )

2 2

2

22

2 5 2 5 3 1

3 1 3 1 3 1

2 3 2 1 5 3 5 1

3 1

6 2 15 5

3 1

6 17 5

9 1

x x x

x x x

x x x x

x

x x x

x

x x

x

+ + += ⋅

− − +

⋅ + ⋅ + ⋅ + ⋅=

−

+ + +

=

−

+ +=

−

57.

( ) ( )( )

( )

2 2

2

22

6 5 6 5 3 7

3 7 3 7 3 7

6 3 6 7 5 3 5 7

3 7

18 6 7 3 5 5 7

3 7

18 6 7 3 5 35

9 7

x x x

x x x

x x x x

x

x x x

x

x x x

x

+ + += ⋅

− − +

⋅ + ⋅ + ⋅ + ⋅=

−

+ ⋅ + ⋅ + ⋅

=

−

+ + +=

−

59. Student 1 did the work correctly. Student 2’s

error was to square the entire expression. This

changes the value of the expression.

61. Answers may vary. The student did not

rationalize the denominator correctly. For theradicand in the denominator to be a perfect cube,

x needs to be multiplied by2

x to yield

3 32 33 x x x ⋅ = .

3 2

3 3 3 2

3 2

3 2

3 2

3 3

3 2

5 5

5

5

5

x

x x x

x

x x

x

x

x

x

= ⋅

=

⋅

=

=

63.

2

3 3

3

3

x x x

x

x

x

x

x

= ⋅

=

=


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285

65.

( ) ( )

( )

2 2

2 2 2

2 2 2

2

2 2

x x x x x x

x x

x x

x x

+ − + − + += ⋅

+ +

+ −

=

+ +

( )

( )

2

2 2

2

2 2

1

2

x x

x x

x x

x x

+ −=

+ +

=

+ +

=

+ +

67.

3 31 1

2 1 2 1

3

2 1

x

x x x x x

x

x x x x x

x

x x

x

x x

− ⋅ −

=

+

⋅ +

−

=

+

3

2 1

3 2 1

3

2 1

x

x

x

x

x x

x x

x x

x x

−

+

=

− += ÷

−= ⋅

+

32 1

x

x

−=

+

( )

( ) ( )2 2

3 2 1

2 1 2 1

2 1 3 2 3 1

2 1

2 6 3

4 1

2 7 3

4 1

x x

x x

x x x x

x

x x x

x

x x

x

− −= ⋅

+ −

⋅ − ⋅ − ⋅ − −

=

−

− − +=

−

− +=

−

69. 2 3 5 9 5

2 9 5 3 5

2 6 5

x

x

x

+ =

= −

=

6 52

6 5 2

2 2

6 10

2

3 10

x =

= ⋅

=

=

71. Answers may vary.

1. Determine the conjugate of the denominator.

2. Multiply the original fraction by the fractionconjugate

conjugate

3. Find the product of the fractions and

simplify.

Homework 9.4

1. 2 y x=

0 0

1 2

4 4

9 6

16 8

x y

8

4

y

x4 8

3. y x= −

0 0

1 1

4 29 3

16 4

x y

−

−−

−

8

−8

4

−4

y

x


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5. 3 y x= +

0 3

1 4

4 5

9 6

16 7

x y

8

4

y

x4 8

7. 2 5 y x= −

0 5

1 3

4 1

9 116 3

x y

−

−

−

8

−4

y

x2

4

9. 3 4 y x= − +

0 4

1 1

4 2

9 5

16 8

x y

−

−

−

8

−4

y

x4

4

11. 2 y x= −

2 0

3 1

6 2

11 3

18 4

x y

8

4

y

x4 8

13. 2 y x= − +

2 0

1 1

2 2

7 3

14 4

x y

−

− −

−

−

−

8

−8

4

−4

y

x

15. 14

2 y x= −

4 0

15

28 1

313

2

20 2

x y

8

4

y

x4 8

17. 3 2 y x= + +

3 2

2 3

1 4

6 5

13 6

x y

−

−

8

4

4−4

y

x

19. 2 3 4 y x= − + −

3 4

2 6

1 8

6 10

13 12

x y

− −

− −

−

−

−

−8

−4

4−4

y

x


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287

21. 4 1 3 y x= − +

1 3

2 75 11

10 15

17 19

x y

16

8

y

x4 8

23. 4

4

x y

y x

+ =

= − +

0 4

1 3

4 2

9 1

16 0

x y

8

−4

y

x4

4

25. 2 6 8

2 6 8

3 4

y x

y x

y x

− =

= +

= +

0 4

1 7

4 10

9 13

16 16

x y

16

8

y

x4 8

27. 2 y x= −

0 0

1 2

4 49 6

16 8

x y

−

−−

−

8

−8

4

−4

y

x

Domain: 0 x ≥

Range: 0 y ≤

29. 2 y x= +

0 2

1 34 4

9 5

16 6

x y

8

4

y

x4 8

Domain: 0 x ≥

Range: 2 y ≥

31. 2 5 4 y x= − + +

5 4

4 21 0

4 2

11 4

x y

−

−

−

−

−

−4

4

4−4

y

x

Domain: 5 x ≥ −

Range: 4 y ≤

33. a. 0, 0, and 0a h k < = >

b. 0, 0, and 0a h k > < <

c. 0, 0, and 0a h k > < >

d. 0, 0, and 0a h k < > =

35. Answers may vary. One example: For the family

of curves y a x h k = − + , 3k = , and 6h = .

Let1 1

4, 3, 2, 1, , ,1, 2,3, and 42 2

a = − − − − − .

37. If 0a < , f has a maximum point at ( ),h k . If

0a > , f has a minimum point at ( ),h k .

39. ( )4 7 4 3

7 2 3

11

f = −

= ⋅ −

=


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288

41. ( )9 7 9 3

7 3 3

21 3

f c c

c

c

= −

= ⋅ −

= −

43. ( ) ( ) ( )3 5

3 5

2 2

h x x x

x x

x

= + + −

= + + −

= −

45. ( ) ( ) ( )5 3

5 3

8

h x x x

x x

= − − −

= − − −

= −

47. ( ) ( )

( )

3

5

3 5

5 5

xh x

x

x x

x x

+

=

−

+ += ⋅

− +

( ) ( )2 2

5 3 3 5

5

5 3 15

25

8 15

25

x x x x

x

x x x

x

x x

x

⋅ + ⋅ + ⋅ + ⋅=

−

+ + +=

−

+ +=

−

49. ( ) ( ) ( )5 9 4 1

5 9 4 1

9 8

h x x x

x x

x

= − + +

= − + +

= −

51. ( ) ( ) ( )4 1 5 9

4 1 5 9

10

h x x x

x x

x

= + − −

= + − +

= − +

53. ( )5 9

4 1

5 9 4 1

4 1 4 1

xh x

x

x x

x x

−=

+

− −= ⋅

+ −

( ) ( )2 2

5 4 5 1 9 4 9 1

4 1

20 5 36 9

16 1

20 41 9

16 1

x x x x

x

x x x

x

x x

x

⋅ − ⋅ − ⋅ + ⋅=

−

− − +=

−

− +=

−

55. ( ) ( ) ( )2 3 5 2 3 5

2 3 5 2 3 5

4

h x x x

x x

x

= − + +

= − + +

=

57. ( ) ( ) ( )2 3 5 2 3 5

2 3 5 2 3 5

6 5

h x x x

x x

= + − −

= + − +

=

59. ( )

( ) ( )( ) ( )

( ) ( )

2 2

2 2

2 3 5

2 3 5

2 3 5 2 3 5

2 3 5 2 3 5

2 2 2 3 5 3 5

2 3 5

4 12 5 45

4 45

xh x

x

x x

x x

x x

x

x x

x

−=

+

− −= ⋅

+ −

− +

=

−

− +=

−

61. ( ) ( ) ( )1 2 1 2

1 2 1 2

2 1

h x x x

x x

x

= + − + + +

= + − + + +

= +

63. ( ) ( ) ( )1 2 1 21 2 1 2

4

h x x x

x x

= + + − + −

= + + − + +

=


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289

65. ( )1 2

1 2

1 2 1 2

1 2 1 2

xh x

x

x x

x x

+ −=

+ +

+ − + −= ⋅

+ + + −

( ) ( )( ) ( )

( ) ( )

2 2

2 2

1 2 1 2 2

1 2

1 4 1 4

1 4

4 1 5

3

x x

x

x x

x

x x

x

+ − + +

=

+ −

+ − + +=

+ −

− + +=

−

67. a. The square root model would fit best

because it continues to increase slowly. The

quadratic function would reach a peak and then

start to decrease.

b. ( )20 3.9 20 280 297.44S = + ≈

The average test score in 2002 would be

about 297.

c. ( )23 3.9 23 280 298.70S = + ≈

The average test score in 2005 would be

about 299.

69. ( )6 0 f − =

71. ( )0 2.4 f =

73. 6 x = −

75. 3 x =

77. ( ) ( )2 3 2; 2 3 2 f x x g x x= + + = − + +

The graph looks like a parabola that opens to the

right. The relation is not a function because it

would fail the vertical line test.

79. Answers may vary. The graph of f can be found

by translating the graph of y a x= horizontally

by h units (left if 0h < and right if 0h > ), and

vertically by k units (up if 0k > and down if

0k < ).

Homework 9.5

1.

( )2

2

5

5

25

x

x

x

=

=

=

Check 25 x =

( )?

25 5

5 5 true

=

=

The solution is 25.

3.

( ) ( )2 2

2

2

4

x

x

x

= −

= −

=

Check 4 x =

( )?

?

4 2

2 2 false

=−

=−

There are no real solutions.

5.

( )2

2

3 1 5

3 6

2

2

4

x

x

x

x

x

− =

=

=

=

=

Check 4 x =

( )?

3 4 1 5

5 5 true

− =

=

The solution is 4.

7.

( )2

2

4 5 2 10

7 14

2

2

4

x x

x

x

x

x

− = −

− = −

=

=

=

Check 4 x =


16/36


290

( ) ( )?

4 5 4 2 4 10

6 6 true

− = −

− = −

The solution is 4.

9.

( )2

2

3 7 24 9 7

12 7 24

7 2

7 2

7 4

4

7

x x

x

x

x

x

x

− = −

=

=

=

=

=

Check4

7 x =

?

4 43 7 24 9 77 7

18 18 true

− =−

− = −

The solution is4

7.

11.

( )2

2

1 2

1 2

1 4

5

x

x

x

x

− =

− =

− ==

Check 5 x =

( )?

5 1 2

2 2 true

− =

=

The solution is 5.

13.

( ) ( )2

2

5 7 8

5 7 8

5 7 64

5 71

71

5

x

x

x

x

x

− = −

− = −

− ==

=

Check71

5 x =

?

?

715 7 8

5

8 8 false

− =−

=−


15.

( )2

2

10 6 3 100

6 3 10

6 3 10

6 3 100

6 97

97

6

x

x

x

x

x

x

+ =

+ =

+ =

+ ==

=

Check97

6 x =

?9710 6 3 100

6

100 100 true

+ =

=

The solution is97

6.

17.

( ) ( )2 2

3 1 2 6

3 1 2 6

3 1 2 6

5

x x

x x

x x

x

+ = +

+ = +

+ = +=

Check 5 x =

( ) ( )?

3 5 1 2 5 6

4 4 true

+ = +

=

The solution is 5.

19.

( ) ( )( )

2 2

2 1 5 0

2 1 5

2 1 5

4 1 5

4 4 5

4 1

1

4

x

x

x

x

x

x

x

− − =

− =

− =

− =

− =− =

= −

Check1

4 x = −

?12 1 5 0

4

0 0 true

− − − =

=

The solution is1

4− .


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291

21.

( )

22

4.91 3.18 7.14 2.19

2.193.18 7.14

4.91

2.19

3.18 7.14 4.91

x

x

x

− − = −

− =

− =

2

2

2

2.193.18 7.14

4.91

2.193.18 7.14

4.91

2.197.14

4.91

3.18

2.3078

x

x

x

x

− =

= +

+ =

=

Check 2.3078 x =

( )?

4.91 3.18 2.3078 7.14 2.19

2.19 21.19 true

− − =−

− = −

The solution is approximately 2.3078.

23.

( ) ( )

( )( )

2 2

2

2

3 3 5

3 3 5

3 3 10 25

13 22 0

11 2 0

x x

x x

x x x

x x

x x

+ = −

+ = −

+ = − +

− + =

− − =

11 0 or 2 011 or 2

x x

x x − = − == =

Check 11 x =

( ) ( )?

3 11 3 11 5

6 6 true

+ = −

=

Check 2 x =

( ) ( )?

?

3 2 3 2 5

3 3 false

+ = −

=−

The solution is 11.

25.

( ) ( )2 2

12 13 2 3

12 13 3 2

12 13 3 2

x x

x x

x x

+ + =+ = −

+ = −

( )

( )( )

2

2

2

12 13 9 12 4

9 24 9 0

3 3 8 3 0

3 3 1 3 0

x x x

x x

x x

x x

+ = − +

− − =

− − =

+ − =

3 1 0 or 3 0

1 or 3

3

x x

x x

+ = − =

= − =

Check1

3 x = −

?

?

1 112 13 2 3

3 3

5 1 false

− + + = −

=−

Check 3 x =

( ) ( )?

12 3 13 2 3 3

9 9 true

+ + ==

The solution is 3.

27.

( ) ( )2 2

2

2

3 4 3

3 4 3

3 4 3

3 4 6 9

3 5 0

x x

x x

x x

x x x

x x

+ − =

+ = +

+ = +

+ = + +

+ + =

( ) ( )( )( )

2

3 3 4 1 52 1

3 11

2

x − ± −=

− ± −=


29.

( ) ( )

2

222

2 2

5 1 3

5 1 3

5 1 6 9

8

x x x

x x x

x x x x

x

− + = −

− + = −

− + = − +=

Check 8 x =

( ) ( ) ( )?

28 5 8 1 8 3

5 5 true

− + = −

=

The solution is 8.


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292

31.

( ) ( )

( )

2 2

22

2 12

2 12

4 4 12

4 8

2

2

4

x x

x x

x x x

x

x

x

x

+ = +

+ = +

+ + = +

=

=

=

=

Check 4 x =

( ) ( )?

2 4 4 12

4 4 true

+ = +

=

The solution is 4.

33.

( ) ( )2 2

1 5

1 5

2 1 5

2 4 2

x x

x x

x x x

x x

− = −

− = −

− + = −

− =

( ) ( )

( )( )

22

2

2

2

2

4 4

5 4 0

4 1 0

x x

x x

x x x

x x

x x

− =

− =

− + =

− + =

− − =

4 0 or 1 0

4 or 1

x x

x x

− = − =

= =

Check 4 x =

( ) ( )?

4 1 5 4

1 1 true

− = −

=

Check 1 x =

( ) ( )?

?

1 1 5 1

0 2 false

− = −

=

The solution is 4.

35.

( ) ( )2 2

2 1

2 1

2 1

x x

x x

x x

− = −

= +

= +

( ) ( )22

2

2

2 2 1

1 2

1 2

2 1 4

6 1 0

x x x

x x

x x

x x x

x x

= + +

− =

− =

− + =

− + =

( ) ( ) ( )( )

( )

26 6 4 1 1

2 1

6 32

2

3 2 2

x

− − ± − −=

±=

= ±

Check 3 2 2 ( 5.83) x + ≈

( ) ( )?

?

5.83 2 5.83 1

1.00 1 true

− =−

− =−

Check ( )3 2 2 0.172 x − ≈

( ) ( )?

?

0.172 2 0.172 1

0.172 1 false

− =−

− =−

The solution is 3 2 2+ .

37.

( ) ( )2 2

1 7

1 7

1 14 49

x x

x x

x x x

+ = − −

+ = − −

+ = + +

( ) ( )2 2

14 48

7 24

7 24

49 576

576

49

x

x

x

x

x

= −

−

= −

=

=

Check ( )576

11.7649

x x = ≈

( ) ( )?

?

11.76 1 11.76 7

3.57 10.43 false

+ =− −

=−



19/36


293

39.

( ) ( )2 2

3 5 4

3 4 5

3 4 5

3 16 8 5 5

8 5 24

5 3

x x

x x

x x

x x x

x

x

− + + =

− = − +

− = − +

− = − + + ++ =

+ =

( )2

25 3

5 9

4

x

x

x

+ =

+ ==

Check 4 x =

( ) ( )?

4 3 4 5 4

4 4 true

− + + =

=

The solution is 4.

41.

( ) ( )2 2

4 6 2

4 6 2

4 6 4 6 4

x x

x x

x x x

− = + +

− = + +

− = + + + +

( )2

2

4 6 14

76

2

76

2

4964

25

4

x

x

x

x

x

+ = −

+ = −

+ = −

+ =

=

Check25

4 x =

?

?

25 254 6 2

4 4

3 11 false

2 2

− = + +

=


43.

( ) ( )2 2

2 1 3 2 2

2 1 2 3 2

2 1 2 3 2

x x

x x

x x

− + − =

− = − −

− = − −

( ) ( )

( )

( )( )

22

2

2

2

2 1 4 4 3 2 3 2

4 3 2 3

4 3 2 3

16 3 2 6 9

48 32 6 9

42 41 0

41 1 0

x x x

x x

x x

x x x

x x x

x x

x x

− = − − + −− = +

− = +

− = + +

− = + +

− + =

− − =

41 0 or 1 0

41 or 1

x x

x x

− = − == =

Check 41 x =

( ) ( )?

?

2 41 1 3 41 2 2

20 2 false

− + − =

=

( ) ( )?

2 1 1 3 1 2 2

2 2 true

− + − =

=

The solution is 1.

45.

( )

( )

22

22

2 3

2 3

2 9

11

11

121

x

x

x

x

x

x

− =

− =

− =

=

=

=

Check 121 x =

( )?

121 2 3

3 3 true

− =

=

The solution is 121.

47. 1 2

12

x

x

x x x

x

+ =

⋅ + = ⋅


20/36


294

( ) ( )

( )( )

22

2

2

1 2

1 2

1 2 4

2 1 0

1 1 0

x x

x x

x x x

x x

x x

+ =

+ =

+ + =

− + =

− − =

1 0

1

x

x

− =

=

Check 1 x =

( )( )

?11 2

1

2 2 true

+ =

=

The solution is 1.

49.

( )

13 2

2

12 2 3 2

2

x

x

x x x

x

= − +

+

+ ⋅ = + − +

+

( ) ( )

( )

22

2

2

2

1 3 2 2

3 2 3

3 2 3

9 2 6 9

9 18 6 9

3 9 0

x x

x x

x x

x x x

x x x

x x

= + − −

+ = +

+ = +

+ = + +

+ = + +

− − =

( ) ( ) ( )( )

( )

23 3 4 1 9

2 1

3 3 5

2

x

− − ± − − −=

±=

Check ( )3 3 5

4.852

x

+≈

( )( )

?13 4.85 2

4.85 2

0.382 0.382 true

= − +

+

=

Check ( )

3 3 5

1.85412 x

−

≈ −

( )( )

?13 1.8541 2

1.8541 2

2.62 2.62 true

= − − +

− +

=

The solutions are3 3 5

2

±.

51.

( ) ( )

3 4 7 1

3 7 4 1

3 7 4 1

4 5

x x

x x

x

x

+ − +

− + +

− + +

− +

53.

( )2

2

3 4 7 1 7

4 5 7

4 12

3

3

9

x x

x

x

x

x

x

+ − + = −

− + = −

− = −

=

=

=

Check 9 x =

( ) ( )?

?

3 9 4 7 9 1 7

7 7 true

+ − + =−

− =−

The solution is 9.

55. ( )( )3 1 3

3 1 3 1 3

3 3 3

x x

x x x x

x x x

+ + =

⋅ + ⋅ + ⋅ + ⋅ =

+ + + =

( ) ( )

( )

22

2

2

4 0

4

4

16

16 0

16 0

x x

x x

x x

x x

x x

x x

+ =

= −

= −

=

− =

− =

0 or 16 0

0 or 16

x x

x x

= − =

= =

Check 0 x =

( )( ) ( )( )?

0 3 0 1 3

3 3 true

+ + =

=

Check 16 x =

( )( ) ( )( )?

?

16 3 16 1 3

35 3 false

+ + =

=

The solution is 0.


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295

57. ( )( )3 1

3 1 3 1

3 3

4 3

x x

x x x x

x x x

x x

+ +

⋅ + ⋅ + ⋅ + ⋅

+ + +

+ +

59. ( ) 5 7 f x x= −

( )2

2

5 7 0

7 0

7 0

7 0

7

x

x

x

x

x

− =

− =

− =

− ==

Check 7 x =

( )

?

5 7 7 0 0 0 true

− ==

The x-intercept is ( )7,0 .

61. ( ) 3 3 4 15h x x= − + −

( )2

2

3 3 4 15 0

3 3 4 153 4 5

3 4 5

3 4 25

3 21

7

x

x x

x

x

x

x

− + − =

− + =− + =

− + =

− + =− =

= −

Check 7 x = −

( )?

3 3 7 4 15 0

0 0 true

− − + − =

=

The x-intercept is ( )7,0− .

63. ( ) 3 2 8 f x x x= − − +

( ) ( )2 2

3 2 8 0

3 2 8

3 2 8

3 2 8

2 10

5

x x

x x

x x

x x

x

x

− − + =

− = +

− = +− = +

==

Check 5 x =

( ) ( )?

3 5 2 5 8 0

0 0 true

− − + =

=


65. ( ) 2 4 3 5h x x x= + + −

( ) ( )( ) ( )

2 2

2 4 3 5 0

2 4 3 5

2 4 3 5

4 4 9 5

4 16 9 45

5 61

61

5

x x

x x

x x

x x

x x

x

x

+ + − =

+ = − −

+ = − −

+ = −

+ = −

− = −=

Check61

5 x =

?

?

61 612 4 3 5 0

5 5

36 0 false

5

+ + − =

=

No real number solution. There are no

x-intercepts.


22/36


296

67. ( ) 3 7 f x x= −

( )2

2

3 7 1

3 6

2

2

4

x

x

x

x

x

− = −

=

==

=

Check 4 x =

( )?

3 4 7 1

1 1 true

− =−

− = −

When 4 x = , ( ) 1 f x = − .

69. ( ) 2 4 5 f x x= − − +

( ) ( )2 2

2 4 5 7

2 4 2

4 1

4 1

4 1

5

x

x

x

x

x

x

− − + =

− − =

− = −

− = −

− ==

Check 5 x =

( )?

?

2 5 4 5 7

7 7 false

− − − =

− =

No real number solutions. There is no value of x that would make ( ) 7 f x = .

71. a. Somewhere during the 1970’s average test

scores declined. During the 1980’s and

1990’s test scores have been improving to

near the 1970 level.

b.

( )22

2

3.9 280 305

3.9 25

25

3.9

25

3.9

2541.09

3.9

t

t

t

t

t

+ =

=

=

=

= ≈

According to the model, average test scores

will return to 305 in 2023.

c.

( )2

2

2

3.9 280 500

3.9 220

220

3.9

220

3.9

2203182.12

3.9

t

t

t

t

t

+ =

=

=

=

= ≈

According to the model, the average test

score will reach the maximum score of 500

in 5164. Model breakdown has occurred.

The model assumes that the average score

will continue to rise.

73. No. When the student squared both sides of the

equation, the work on the left side was incorrect.

The student did not square the factor 5 on the left

side.

( ) ( )2

5 3 25 3 x x+ = +

75. 3 4

2 6

y x

y x

= −

= − +

Since the left hand sides are equal, set the right

hand sides equal to each other and solve the

resulting equation.

( )2

2

3 4 2 6

5 10

2

2

4

x x

x

x

x

x

− = − +

==

=

=

Substitute this value into either original equation

and solve for y.

( )3 4 4 2 y = − =

The solution is ( )4,2 .

77. The left hand side was not squared properly in

the second line.


23/36


297

Homework 9.6

1. ( )0,3 and ( )4,5

Substitute the point ( )0,3 into the equation

y a x b= + .

3 0

3

a b

b

= +

=


3 y a x= + and solve for a.

5 4 3

2 2

1

a

a

a

= +

=

=

The equation is 3 y x= + .

3. ( )0,2 and ( )9,6 Substitute the point ( )0,2 into the equation

y a x b= + .

2 0

2

a b

b

= +

=



6 9 2

3 4

4

3

a

a

a

= +

=

=

The equation is4

23

y x= + .

5. ( )0,2 and ( )3,5


y a x b= + .

2 0

2

a b

b

= +

=



5 3 2

3 3

3

3

3

a

a

a

= +

=

=

=

The equation is 3 2 y x= ⋅ + or 3 2 y x= + .

7. ( )1,2 and ( )4,3

Substitute the points into the equation

y a x b= + .

2 1

3 4

a b

a b

= +

= +

Rewrite as:

2

2 3

a b

a b

+ =

+ =

Solve the resulting system. Multiply the first

equation by 1− and add to the second equation.

2

2 3

1

a b

a b

a

− − = −

+ =

=


y x b= + and solve for b.

2 1

1

b

b

= +

=

The equation is 1 y x= + .

9. ( )2,4 and ( )3,5


y a x b= + .

4 2

5 3

a b

a b

= +

= +

Rewrite as:

1.4142 4

1.7321 5

a b

a b

+ =

+ =



1.4142 4

1.7321 5

0.3179 1

3.15

a b

a b

a

a

− − = −

+ =

=

≈


3.15 y x b= + and solve for b.

4 3.15 2

0.45

b

b

= +

≈ −

The equation is roughly 3.15 0.45 y x= − .


24/36


298

11. ( )2,6 and ( )5,4


y a x b= + .

6 2

4 5

a b

a b

= +

= +

Rewrite as:

1.4142 6

2.2361 4

a b

a b

+ =+ =


equation by 1− and add to the second equation.1.4142 6

2.2361 4

0.8129 2

2.43

a b

a b

a

a

− − = −+ =

= −≈ −


2.43 y x b= − + and solve for b.

6 2.43 2

9.44

b

b

= − +≈

The equation is roughly 2.43 9.44 y x= − + .

13. Increase the value of b to shift the graph up.

15. a. Start by plotting the data.

Answers may vary. A square root model

may be reasonable. Use the points ( )0,10

and ( )18,34 .


y a x b= + .

10 0

10

a b

b

= +=

Substitute the point ( )18,34 into the

equation 10 y a x= + and solve for a.

34 18 10

245.66

18

a

a

= +

= ≈

( ) 5.65 10 f t t = +

b. ( ) ( )26 5.65 26 10 38.81 f = + ≈

In 2008, 38.8% of single-parent fathers will

have never been married.

c.

( )

22

5.65 10 39

5.65 29

29

5.65

29

5.65

26.35

t

t

t

t

t

+ =

=

=

= ≈

In 2008, 39% of single-parent fathers will

have never been married.

d. ( )0,10 . In 1982, 10% of single-parent

fathers had never been married.



seems to be appropriate. Use the points

( )0,0 and ( )52.5,1.94 .


y a x b= + .

0 0

0

a b

b

= +=

Substitute the point ( )52.5,1.94 into the

equation y a x= and solve for a.

1.94 52.5

0.27

a

a

=≈

( ) 0.27S h h=


25/36


299

b. i. Graph all three functions.

The model ( )S h appears to fit the best.

ii. ( ) ( ) ( )0 0; 0 0.327; 0 0.165S L Q= = = S models the situation best near 0 since

it is the only model that passes through

the origin.

iii. Zoom out.

Q is not possible since it indicates that

the falling time will reach 0 for larger

drop heights.

iv. S models the situation the best and has

no problems with larger h.

v.2

32.2

2

32.2

0.249

hT

h

h

=

=

=

This is close to the model

( ) 0.27S h h= .

c.

( )2

2

0.27 3

3

0.27

3

0.27

123.46

h

h

h

h

=

=

=

≈

According to the model, the height of thecliff is roughly 123.46 feet.

d. ( )1250 0.27 1250 9.55S = ≈ It would take about 9.55 seconds for the

baseball to reach the ground if it were

dropped from the top of New York City’s

Empire State Building.



seems reasonable.

Use the points ( )0,0 and ( )3,68 .

Substitute the point ( )0,0 in the equation

y a x b= + .

0 0

0

a b

b

= +=


y a x= and solve for a.

68 3

39.26

a

a=≈

( ) 39.26S t t =

b. i. Both models seem to fit the data pretty

well but the square root model seems to

be a little better fit.

ii. As the length of release time increases,

the percent of criminals that were arrested

again cannot decrease since the

percentages are cumulative. The square

root function is an increasing function.

c. ( )4 39.26 4 78.52S = =

About 78.5% of convicts released from stateprison have been arrested for another crime

after being out of prison for 4 years or less.


26/36

Homework 9.6 SSM: Intermediate Algebra

300

d.

( )

22

100 39.26

100

39.26

100

39.26

6.5

t

t

t

t

=

=

= ≈

According to the model, after about 6.5

years, all convicts released from state prison

will have been arrested for a new crime.

This is not realistic since it is likely that

some released convicts will never be

arrested for another crime.


A square root model my fit reasonably well.

Use the points ( )2,54.3 and ( )4,73 .


y a x b= + .

54.3 2

73 4

a b

a b

= +

= +

Rewrite as:

1.4142 54.3

2 73

a b

a b

+ =+ =

Solve the system of equations. Multiply the

first equation by 1− and add to the secondequation.

1.4142 54.3

2 73

0.5858 18.7

31.92

a b

a b

a

a

− − = −+ =

=≈

Substitute the point ( )2,54.3 into the

equation 31.92 y x b= + and solve for b.

54.3 31.92 2

54.3 31.92 29.16

b

bb

= +

= −≈

( ) 31.92 9.16 f n n= +

b. ( ) ( )7 31.92 7 9.16 93.61 f = + ≈

93.6% of 7th

births occurred despite the use

of contraception.

c.

( )2

2

100 31.92 9.16

31.92 90.84

90.84

31.92

90.84

31.92

8.1

n

n

n

n

n

= +=

=

=

≈

All 8th

births occurred despite the use of

contraception.

d. The higher the birth order, the higher the

percent of births that happened despite the

use of contraception.


A square root function may fit reasonably

well. Use the points ( )2,85.3 and ( )6,160 .


y a x b= + .

85.3 2

160 6

a b

a b

= += +

Rewrite as:

1.41 85.3

2.45 160

a b

a b

+ =+ =

Multiply the first equation by –1 and add to

the second equation.

1.41 85.3

2.45 160

1.04 74.7

71.83

a b

a b

a

a

− − = −+ =

=≈

Substitute ( )2,85.3 into the equation


85.3 71.83 2

85.3 71.83 2

16.28

b

b

b

= +

= −≈ −


27/36

SSM: Intermediate Algebra Chapter 9 Review Exercises

301

( ) 71.83 16.28 f n n= − .

b. f is an increasing function. This suggests that

the greater the number of people living with

the child, the less likely the child is to try to

take control.

Chapter 9 Review Exercises

1.73/ 7 3

x x=

2.1/ 2

x x=

3. ( ) ( )2/9 292 1 2 1 x x+ = +

4. ( ) ( )7 7/55 3 4 3 4 x x+ = +

5. 16 16

4

x x

x

=

=

6.6 6

6

3

8 4 2

4 2

2 2

x x

x

x

= ⋅

=

=

7.7 6

6

3

3 3

3

3

x x x

x x

x x

= ⋅

=

=

8.8 6 6/8

3/ 4

4 3

x x

x

x

=

=

=

9.3 310 9

3 9 3

3 3

24 8 3

8 3

2 3

x x x

x x

x x

= ⋅

=

=

10. ( ) ( ) ( )

( ) ( )

( ) ( )

27 25 25 5

25 25 5

5 25

6 11 6 11 6 11

6 11 6 11

6 11 6 11

x x x

x x

x x

+ = + ⋅ +

= + +

= + +

11. ( )2 5 3 2 5 3

0

0

x x x x

x

− + = − +

= ⋅

=

12.

( ) ( )

3 3

3 3

3

3

5 2 7 4

5 7 2 4

5 7 2 4

12 2

x x x x

x x x x

x x

x x

− + +

= + − +

= + + − +

= +

13. ( )

( ) ( )

3 3

3 3

3 3

3

3

5 4 2 8

20 5 2 8

20 8 5 2

20 8 5 2

28 7

x x x x

x x x x

x x x x

x x

x x

− − +

= − − +

= + − −

= + + − −

= −

14. ( )

2

3 7 3 3 7

3 21

3 21

3 21

x x x x x

x x x

x x

x x

− = ⋅ − ⋅

= ⋅ −

= −

= −

15. ( )( )

2

4 3 2 1

4 2 3 2 4 1 3 1

8 6 4 3

8 2 3

8 2 3

x x

x x x x

x x x x

x x

x x

− +

= ⋅ − ⋅ + ⋅ − ⋅

= ⋅ − + −

= − −

= − −

16. ( )( )

( )

( )

( )2

2 3 8 4 2

2 3 4 8 4 3 2 8 2

2 12 32 6 16

2 12 38 16

24 76 32

x x

x x x x

x x x x

x x

x x

+ +

= ⋅ + ⋅ + ⋅ + ⋅

= ⋅ + + +

= + +

= + +


28/36

Chapter 9 Review Exercises SSM: Intermediate Algebra

302

17. ( )( ) ( ) ( )2

21 1 1

1

x x x

x

+ − = −

= −

18. ( )( ) ( ) ( )2 2

3 4 3 4 3 4

9 16

x x x

x

+ − = −

= −

19. ( ) ( ) ( )( ) ( )2 2 23 3 3

3 2 3

2 5 2 2 2 5 5

4 20 25

x x x

x x

+ = + +

= + +

20. ( ) ( ) ( )( ) ( )2 2

25 2 5 2 5 2 2

25 20 4

x x x

x x

− = − +

= − +

21.1/ 4 1/554

1 1

4 5

5 4

20 20

9/20

20 9

x x x x

x

x

x

x

+

+

= ⋅

=

=

=

=

22.

( )

33 1/ 66

1/ 31/ 6

1 1

6 3

1/18

18

x x

x

x

x

x

⋅

=

=

=

=

=

23.

1/ 44

1/ 66

1 1

4 6

3 2

12 12

1/12

12

x x

x x

x

x

x

x

−

−

=

=

=

=

=

24.2

2 2 2

2

2 2

2

2

x x

x

x

= ⋅

=

⋅

=

25.3 3

3

3

3

x x

x

x x

x

x x

x

x

=

= ⋅

⋅=

⋅

=

26.

3 2

3 3 3 2

3 2

3 2

3 2

3 3

3 2

5 5

5

5

5

x

x x x

x

x x

x

x

x

x

= ⋅

=

⋅

=

=

27.

5

52 5 2

5 35

5 52 3

7 7

27 27

7 9

27 9

x x

x

x x

=

= ⋅

5 3

5 2 3

5 3

5 5

5 3

7 9

27 9

63

243

63

3

x

x x

x

x

x

x

⋅=

⋅

=

=


29/36


303

28.

( )

( ) ( )2

2

5 5 3

3 3 3

5 3

3

15 5

9

x

x x x

x

x

x

x

−= ⋅

+ + −

−

=

−

−=

−

29.

( )

( ) ( )22

2 2 2 3

2 3 2 3 2 3

2 2 3

2 3

4 6

4 9

x

x x x

x

x

x

x

+= ⋅

− − +

+

=

−

+=

−

30.

( ) ( )2 2

5 4 5 4 2 3

2 3 2 3 2 3

5 2 4 2 5 3 4 3

2 3

10 8 15 12

4 9

10 23 12

4 9

x x x

x x x

x x x x

x

x x x x

x

x x

x

− − −= ⋅

+ + −

⋅ − ⋅ − ⋅ + ⋅=

−

⋅ − − +=

−

− +=

−

31. ( ) ( ) ( )3 5 2 4

3 5 2 4

7

h x x x

x x

x

= + + −

= + + −

= − +

32. ( ) ( ) ( )3 5 2 4

3 5 2 4

7 3

h x x x

x x

x

= + − −

= + − +

= +

33. ( ) ( )( )3 5 2 4

3 2 5 2 3 4 5 4

6 10 12 20

12 14 10

h x x x

x x x x

x x x

x x

= + −

= ⋅ + ⋅ − ⋅ − ⋅

= + − −

= − − +

34. ( )

( ) ( )22

3 5

2 4

3 5 2 4

2 4 2 4

3 2 5 2 3 4 5 4

2 4

xh x

x

x x

x x

x x x x

x

+=

−

+ += ⋅

− +

⋅ + ⋅ + ⋅ + ⋅=

−

( )( )

6 10 12 20

4 16

12 26 10

4 16

2 6 13 5

2 2 8

6 13 5

2 8

x x x

x

x x

x

x x

x

x x

x

+ + +=

−

+ +=

−

+ +

=

−

+ +=

−

35.

( )2

2

3 4 13

3 9

3

3

9

x

x

x

x

x

+ =

=

=

=

=

Check 9 x =

( )?

3 9 4 13

13 13 true

+ =

=

The solution is 9.

36.

( )2

2

2 1 4 7

2 1 3

2 1 3

2 1 9

2 8

4

x

x

x

x

x

x

+ + =

+ =

+ =

+ =

=

=

Check 4 x =

( )?

2 4 1 4 7

7 7 true

+ + =

=

The solution is 4.


30/36


304

37.

( )

( )( )

22

2

2

4 5

4 5

4 5

4 5 0

5 1 0

x x

x x

x x

x x

x x

+ =

+ =

+ =

− − =

− + =

5 0 or 1 0

5 or 1

x x

x x

− = + =

= = −

Check 5 x =

( ) ( )?

4 5 5 5

5 5 true

+ =

=

Check 1 x = −

( ) ( )?

?

4 1 5 1

1 1 false

− + = −

=−

The solution is 5.

38.

( ) ( )

( )( )

2 2

2

2

2 4 2

2 4 2

2 4 4 4

6 8 0

4 2 0

x x

x x

x x x

x x

x x

− = −

− = −

− = − +

− + =

− − =

4 0 or 2 0

4 or 2

x x

x x

− = − =

= =

Check 4 x =

( ) ( )?

2 4 4 4 2

2 2 true

− = −

=

Check 2 x =

( ) ( )?

2 2 4 2 2

0 0 true

− = −

=

The solutions are 4 and 2.

39.

( ) ( )

( )( )

2 2

2

2

6

6

6

12 36

13 36 0

9 4 0

x x

x x

x x

x x x

x x

x x

+ =

= −

= −

= − +

− + =

− − =

9 0 or 4 0

9 or 4

x x

x x

− = − =

= =

Check 9 x =

( ) ( )?

9 6 9

9 9 true

+ =

=

Check 4 x =

( ) ( )?

?

4 6 4

8 4 false

+ =

=

The solution is 9.

40.

( ) ( )2 2

13 4 5 20

13 4 5 20

13 4 5 20

8 24

3

x x

x x

x x

x

x

+ = −

+ = −

+ = −

= −

= −

Check 3 x = −

( ) ( )?

?

13 3 4 5 3 20

35 35

− + = − −

− = −

There is no real solution.

41.

( ) ( )

( ) ( )

2 2

2 2

2 1 1 3

2 1 1 3

2 1 1 2 3 3

2 3 5

2 3 5

x x

x x

x x x

x x

x x

− = + +

− = + +

− = + + + +

+ = −

+ = −

( )

( )( )

2

2

2

4 3 10 25

4 12 10 25

14 13 0

13 1 0

x x x

x x x

x x

x x

+ = − +

+ = − +

− + =

− − =

13 0 or 1 0

13 or 1

x x

x x

− = − =

= =

Check 13 x =

( ) ( )?

2 13 1 1 13 3

5 5 true

− = + +

=

Check 1 x =

( ) ( )?

?

2 1 1 1 1 3

1 3 false

− = + +

=

The solution is 13.


31/36


305

42.

( ) ( )2 2

2 9 7

2 7 9

2 7 9

2 49 14 9 9

14 9 56

x x

x x

x x

x x x

x

+ + + =

+ = − +

+ = − +

+ = − + + +

+ =

( )2

2

9 4

9 4

9 16

7

x

x

x

x

+ =

+ =

+ =

=

Check 7 x =

( ) ( )?

7 2 7 9 7

7 7 true

+ + + =

=

The solution is 7.

43. 2 y x= −

0 0

1 2

4 4

9 6

16 8

x y

−

−

−

−

8

−8

4

−4

y

x

44. 3 1 y x= +

0 1

1 4

4 7

9 10

16 13

x y

8

4

y

x4 8

45. 5 3 y x= − − +

5 3

6 2

9 114 0

21 1

x y

−

1 6

−4

y

x8

4

46. 2 4 1 y x= + −

4 1

3 1

0 3

5 5

12 7

x y

− −

−

−4

4

2−2

y

x

47. ( ) 4 7 2 1 f x x x= − − +

( ) ( )2 2

4 7 2 1 0

4 7 2 1

4 7 2 1

4 7 2 1

2 8

4

x x

x x

x x

x x

x

x

− − + =

− = +

− = +

− = +

=

=

Check 4 x =

( ) ( )?

4 4 7 2 4 1 0

0 0 true

− − + =

=


32/36


306


48. ( ) 3 2 9g x x= − + +

( )2

2

3 2 9 0

3 2 9

2 3

2 3

2 9

7

x

x

x

x

x

x

− + + =

− + = −

+ =

+ =

+ =

=

Check 7 x =

( )?

3 7 2 9 0

0 0 true

− + + =

=


49. Increase b to raise the y-intercept and decrease a

to lower the rate of increase. Sketches may vary.

50. The equation is of the form y a x b= + .

The y-intercept is ( )0,2 so 2b = .



9 8 2

8 7

72.475

8

a

a

a

= +

=

= ≈

The equation is roughly 2.475 2 y x= + .




3 y a x= + and solve for a.8 4 3

2 5

5

2

a

a

a

= +

=

=

The equation is5

32

y x= + .





3 9 7

3 4

4

3

a

a

a

= +

= −

= −

The equation is4

73

y x= − + .

53. ( )2,5 and ( )3,6


y a x b= + .

5 2

6 3

a b

a b

= +

= +

Rewrite as:

1.4142 5

1.7321 6

a b

a b

+ =

+ =

Solve the system of equations. Multiply the first


1.4142 5

1.7321 6

0.3179 1

3.15

a b

a b

a

a

− − = −

+ =

=

≈



5 3.15 2

5 3.15 2

0.55

b

b

b

= +

= −

≈

The equation is roughly 3.15 0.55 y x= + .


33/36

SSM: Intermediate Algebra Chapter 9 Test

307

54. ( )3,7 and ( )5,4


y a x b= + .

7 3

4 5

a b

a b

= +

= +

Rewrite as:

1.7321 7

2.2361 4

a b

a b

+ =

+ =



1.7321 7

2.2361 4

0.504 3

5.95

a b

a b

a

a

− − = −

+ =

= −

≈ −


5.95 y x b= − + and solve for b.

7 5.95 3

7 5.95 3

17.31

b

b

b

= − +

= +

≈

The equation is roughly 5.95 17.31 y x= − + .


A square root model, y a x b= + , seems

reasonable.

Use the points ( )0,16 and ( )4,28.8 .



equation 16 y a x= + and solve for b.

28.8 4 16

2 12.8

6.4

a

a

a

= +

=

=

( ) 6.4 16 f t t = +

b. ( )0,16 ; In 1998 there were 16 million U.S.

households with a Sony PlayStation.

c. 6.4 16 36

6.4 20

3.125

9.77

t

t

t

t

+ =

=

=

≈

In 2008, 36 million households in the United

States will have a Sony PlayStation.

d. ( )5 6.4 5 16 30.31 f = + ≈

The model predicts a slightly larger number

of households, but the values are fairly

close.

Chapter 9 Test

1. 9 8

8

4

32 16 2

16 2

4 2

x x x

x x

x x

= ⋅

=

=

2. 3 322 21

3 21 3

7 3

64 64

64

4

x x x

x x

x x

= ⋅

=

=

3. ( ) ( ) ( )

( ) ( )

( ) ( )

27 24 34 4

24 34 4

6 34

2 8 2 8 2 8

2 8 2 8

2 8 2 8

x x x

x x

x x

+ = + ⋅ +

= + +

= + +

4. 1/ 33

1/ 55

1 1

3 5

4 2

36

2

3

x x

x x

x−

=

=

5 3

15 15

2/15

15 2

2

32

3

2

3

x

x

x

−

=

=

=


34/36

Chapter 9 Test SSM: Intermediate Algebra

308

5.

( ) ( )2 2

1 1 2 3

2 3 2 3 2 3

2 1 2 3 1 3

2 3

2 2 3 3

4 9

2 5 3

4 9

x x x

x x x

x x x x

x

x x x

x

x x

x

+ + += ⋅

− − +

⋅ + ⋅ + ⋅ + ⋅=

−

+ + +=

−

+ +=

−

6. 4 4

4 4

4 4

4 4

4

3 12 2 8 27 5

3 4 3 8 9 3 2 5

3 2 3 8 3 3 2 5

6 3 24 3 2 5

18 3 3

x x x x

x x x x

x x x x

x x x x

x x

+ − −

= ⋅ − ⋅ + −

= ⋅ − ⋅ + −

= − + −

= − −

7. ( )2

3 6 5 3 6 3 5

18 15

18 15

x x x x x

x x

x x

− = ⋅ − ⋅

= −

= −

8. ( )( )

2

2 4 3 5

2 3 4 3 2 5 4 5

6 12 10 20

20 2 6

x x

x x x x

x x x

x x

+ −

= ⋅ + ⋅ − ⋅ − ⋅

= + − −

= − + +

9. ( )( ) ( ) ( )22

4 3 4 3 4 3

16 9

x x x

x

+ − = −

= −

10. ( ) ( ) ( )( ) ( )2 2 25 5 5

5 2 5

4 3 4 2 4 3 3

16 24 9

x x x

x x

− = − +

= − +

11.

1/

1/

1 1

nn

k k

n k

k n

kn kn

k n

kn

kn k n

x x

x x

x

x

x

x

−

−

−

−

=

=

=

=

=

12. ( ) ( ) ( )7 3 4 5

7 3 4 5

2 11

h x x x

x x

x

= − + +

= − + +

= +

13. ( ) ( ) ( )7 3 4 5

7 3 4 5

3 8

h x x x

x x

x

= − − +

= − − −

= −

14. ( ) ( )( )7 3 4 5

7 4 3 4 7 5 3 5

28 12 35 15

15 23 28

h x x x

x x x x

x x x

x x

= − +

= ⋅ − ⋅ + ⋅ − ⋅

= − + −

= − + +

15. ( )7 3

4 5

7 3 4 5

4 5 4 5

xh x

x

x x

x x

−=

+

− −= ⋅

+ −

( ) ( )22

7 4 3 4 7 5 3 5

4 5

28 12 35 15

16 25

15 47 28

16 25

x x x x

x

x x x

x

x x

x

⋅ − ⋅ − ⋅ + ⋅=

−

− − +=

−

− +=

−

16. ( ) ( )8 6 4 8 1

6 4 9

6 4 3

6

f = − +

= −

= − ⋅

= −

17.

( )

22

2 6 4 1

4 1 8

1 2

1 2

1 4

3

x

x

x

x

x

x

− = − +

+ =

+ =

+ =

+ =

=

Check 3 x =

( )?

?

2 6 4 3 1

2 2 true

− = − +

− =−

Therefore, 3 x = when ( ) 2 f x = − .


35/36

SSM: Intermediate Algebra Chapter 9 Test

309

18. 2 3 1 y x= − + +

3 1

2 1

1 36 5

13 7

x y

−− −

−−−

−4

4

4−4

y

x

19. a. We need 0a < and 0k ≥ , or we need

0a > and 0k ≤ . In either case, h can beany real number.

b. ( ) f x a x h k = − +

( )2

2

2

2

2

2

0a x h k

a x h k

k x h

a

k x h

a

k x ha

k x h

a

− + =

− = −

− = −

− = −

− =

= +

The x-intercept is2

2, 0

k h

a

+

.

20.

( )2

2

2 3 13

2 10

5

5

25

x

x

x

x

x

+ =

=

=

==

Check 25 x =

( )?

2 25 3 13

13 13 true

+ =

=

The solution is 25.

21.

( )2

2

3 5 4 27

5 4 9

5 4 9

5 4 81

5 85

17

x

x

x

x

x

x

− =

− =

− =

− ===

Check 17 x =

( )?

3 5 17 4 27

27 27 true

− =

=

The solution is 17.

22.

( ) ( )2 2

3 2 9 0

9 2 3

9 2 3

x x

x x

x x

− + − =

− = −

− = −

( ) ( )

( )

2 2

2

2

9 4 12 9

12 5

12 5

144 25

25 144 0

25 144 0

x x x

x x

x x

x x

x x

x x

− = − +

=

=

=

− =

− =

0 or 25 144 0

1440 or25

x x

x x

= − =

= =

Check 0 x =

( ) ( )?

?

3 2 0 9 0 0

6 0 false

− + − =

=

Check144

25 x =


36/36

Chapter 9 Test SSM: Intermediate Algebra

?144 1443 2 9 0

25 25

0 0 true

− + − =

=

The solution is

144

25 .

23. ( ) 3 2 4 2 2 1 f x x x= − − +

( ) ( )( ) ( )

2 2

3 2 4 2 2 1 0

3 2 4 2 2 1

3 2 4 2 2 1

9 2 4 4 2 1

18 36 8 4

10 40

4

x x

x x

x x

x x

x x

x

x

− − + =

− = +

− = +

− = +

− = +=

=

Check 4 x =

( ) ( )?

3 2 4 4 2 2 4 1 0

0 0 true

− − + =

=


24. Decrease b to lower the y-intercept and increase

a to increase the rate of increase. Graphs may

vary.

25. Substitute the points ( )2,4 and ( )5,6 into the

equation y a x b= + .4 2

6 5

a b

a b

= +

= +

Rewrite as:

1.4142 4

2.2361 6

a b

a b

+ =+ =


equation by 1− and add to the second equation.1.4142 4

2.2361 6

0.8219 2

2.43

a b

a b

a

a

− − = −+ =

=

≈



4 2.43 2

4 2.43 2

0.56

b

b

b

= +

= −≈

The equation is 2 43 0 56y x= +



seems reasonable.

Use the points ( )0,20.5 and ( )60,43.4 .

The y-intercept is ( )0,20.5 so 20.5b = .


equation 20.5 y a x= + and solve for a.

43.4 60 20.5

60 22.9

2.96

a

a

a

= +

=≈

( ) 2.96 20.5 f t t = +

b. ( )72 2.96 72 20.5 45.62 f = + ≈ According to the model, the median height

of 6-year-old boys is about 45.6 inches.

c.

( )2

2

36 2.96 20.5

2.96 15.5

15.5

2.96

15.5

2.96

27.42

t

t

t

t

t

= +

=

=

=

≈

The median height of 27-month-old boys is

3 feet.

d. The h-intercept is ( )0,20.5 . The median

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Lehmann IA SSM Ch9