chapter 9 radical expressions and negative so this is not defined using 4. 25 121 5 …€¦ ·...

Chapter 9 Radical Expressions and Equations

Section 9.1. Introduction to Radical Expressions

Practice 9.1.11. Since 42 = 16 16 = 4

2. Since 02 = 0 0 = 0

3. Since (3/8)2 = (9/64) 9 64

= 3 8

4. Since (13)2 = 132 132 = 13

5. Since (0.5)2 = 0.25 0 . 25 = 0 . 5

6. Since (0.07)2 = 0.0049 0 . 0049 = 0 . 07

Practice 9.1.21. 0 . 03 = 0 . 1732 ≈ 0 . 2

2. 2550 = 50. 4975 ≈ 50

3. 879. 985 = 29. 6645≈ 29. 66

4. 50. 01 = 7 . 07177≈ 7 . 1

Practice 9.1.31. − 49 = - 7

2. − 81 The radicand is negative sothis is not defined using realnumbers.

3. − 1 = - 1

4. − − 121 The radicand is negative so this is not defined using realnumbers.

Practice 9.1.41. 5 = 6 49 = 5 + 6•7 = 5 +42 = 47

2. 7 - 5 25 - 9 = 7 - 5 16 = 7-5•4 = 7- 20 = -13

3. 8 - 12 − 4 The radicand is

negative so this is not defined usingreal numbers.

4. − 25 + 121 = - 5 + 11 = 6

5. − 16 − 81 = - 4 - 9 = - 13

6. 8 - 4 • 16 − 25 • 36 = 8 - 4•4 - 5•6 = 8 - 16 - 30 = -38

7. 12 - 2• 9 + − 49 • 4 The radicand is negative so this is not defined using realnumbers.

8. (5 + 81 )(3 - 121 ) = (5 + 9)(3 - 11) = 14(-8) = -112

9. (2 - 16 )(4 - − 64 )The radicand is negative so this is not defined using realnumbers.

Practice 9.1.51. a = 9.1, b = 1.3

c = a 2 + b 2

c = ( 9 . 1 ) 2 + ( 1 . 3 ) 2

c = 82. 81 + 1 . 69c = 84. 5 c = 9.2

2. a = 408, b = 1700c = 4082 + 17002

c = 3 , 056, 464c = 1748.3 = 1750 rounded

3. s = 2 gdg = 32, d = 36s = 2 • 32• 36s = 2304s = 48

4. g = 32 and d = 49s = 2 • 32• 49s= 3136s = 56

5. g = 9.8 and d = 15s = 2 • 9 . 8 • 15s = 294

254 CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

s = 17

Exercise Set 9.11. 82 = 64 so 64 = 8

3. (9 11

)2 = (81121

) so 81121

= 9 11

5. (0.2)2 = 0.04 so 0 . 04 = 0 . 2

7. (1.1)2 = 1.21 so 1 . 21 = 1 . 1

9. 15. 4 = 3 . 924 ≈ 4

11. 254. 82 = 15. 963 ≈16. 0

13. 0 . 05 = 0 . 2236 ≈ 0 . 22

15. − 25 The radicand is negative sothis is not defined for real numbers.

17. − 169 = - 13

19. − − 4 The radicand is negative sothis is not defined for real numbers.

21. 15 + 8 64 = 15 + 8•8 = 15 + 64 = 79

23. 36 - 2 49 = 36 -2•7 = 36 - 14 = 22

25. − 36 + 64 =-6 +8 = 2

27. 3 4 - 5 16 = 3•2 - 5•4 = 6 - 20 = -14

29. 8 + 4 9 + 36 • 49 = 8+4•3 +6•7 = 8 + 12 + 42 = 62

31. (12 - 100 ) ( 4 + 25 ) = (12 - 10)(4 + 5) = 2• 9 = 18

33. ( 4 + 2 4 ) ( 5 - 3 25 ) =(4 + 2•2) (5 - 3•5) = (4 + 4) (5 - 15) = 8(-10) = -80

35. c = a 2 + b 2

a = 15, b = 18c = 152 + 182

c = 549

c =23

37. a = 0.0013, b = 0.056c = 0 . 00132 + 0 . 0562

c = 0 . 0031377c = 0.056

39. a = 735, b = 5010c = 7352 + 50102

c = 25, 640, 325c = 5060

41. a = 105.050, b = 60.002c = ( 105. 050) 2 + ( 60. 002) 2

c = 14635. 743c = 120.98

43. s = 2 5 d d = 405 feets = 2 5 • 405s = 2 2025s = 2•45 = 90 mph

45. d = 80 feets = 2 5 • 80s = 2 400s = 2•20 = 40 mph

47. 117 = 3•3•13

48.

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #7

49. a(a2 - ab + b2) +b(a2 - ab + b2) =a3 - a2b + ab2 + a2b - ab2 + b3 = a3 + b3

255CHAPTER 9 RADICAL EXPRESSIONS AND EQUATIONS

50.125h 11 − 75h 9

− 5 h 7 = 125h 11

− 5 h 7 − 75h 9

− 5 h 7 =

-25h4 +15h2

51.

3 x 2 + 3 x + 3 + 5

2 x - 2

2 x - 2 6 x 3 + 0 x 2 + 0 x - 1 6 x 3 − 6 x 2 Subtract

6 x 2 + 0 x 6 x 2 - 6 x Subtract

6 x - 1 6 x - 6 Subtract

5

52. ( ( 2 a ) ( 3 ab2 )

30a 3 ) 3 = ( 6 a 2 b 2

30a 3 ) 3 = ( b 2

5 a ) 3

=b 6

125a 3

53. a2b3ab-2 = a3b

54. ( t 3 v

w − 1 ) (

t − 4

w 3 ) = t

− 1 v

w 2 = v

tw2

55.

x 4

y 2

x 3

y 5

= x 4

y 2 •

y 5

x 3 = xy3

56.2 m − 3

( 2 m ) − 3 •

3 m − 2

( 3 m ) − 2 =

2 ( 2 m ) 3

m 3 •

3 ( 3 m ) 2

m 2 =

2 • 8 m 3

m 3 •

3 • 9 m 2

m 2 = 16• 27 = 432

57. (3s +2t)(3s - 2t) = (3s)2 - (2t)2 =

9s2 - 4t2

58.18x 3

9 y 2 •

y 5

2 x = 2 • 9 x 3

9 y 2 •

y 5

2 x = x 2 y 3

59. 1 - 1 5 x

+ 3 2 x

=

1 • 10x 10x

- 1 5 x

• 2 2

+ 3 2 x

• 5 5

=

10x - 2 + 1510x

= 10x + 1310x

60. 45 - 14x + x2 = x2 - 14x + 45 = (x - 9)(x - 5)

61. 12x2 - 30x + 12 = 6(2x2 - 5x + 2) =6(2x - 1)(x - 2)

62. x2 + 6x + 9 = (x+3)(x+3) = (x +3)2

63. 10x + 10x2 - 75x - 35 = 10x2 - 65x - 35 = 5(2x2 - 13x - 7)5(2x + 1)(x - 7)

64. 6(x -1) = 2 3

(x-9)

3•6(x-1) = 3•2 3

(x-9)

18x - 18 = 2(x - 9)18x - 18 = 2x - 1816x = 0x = 0

65. x2 - 64 = 0(x +8)(x - 8) = 0x + 8 = 0 or x - 8 = 0x = -8 or x = 8

66. x2 - 9 = (x+3)(x-3) = LCD2

x - 3 − 3

x + 3 = 12

x 2 − 9 2

x - 3 − 3

x + 3 = 12

( x + 3 ) ( x - 3 )

( x + 3 ) ( x - 3 ) 2

x - 3 − ( x + 3 ) ( x - 3 )

3 x + 3

=

( x + 3 ) ( x - 3 ) 12

( x + 3 ) ( x - 3 ) 2(x+3)-3(x-3) = 122x + 6 - 3x + 9 = 12-x + 15 = 12-x = -3


x = 32

3 - 3 − 3

3 + 3 = 12

3 2 − 9 2/0 - 3/6 = 12/0 Undefined3 is a restricted value. There is nosolution.

67. Eq 1: 3x - 11y = -5Eq 2: 7x + 2y = 162 times Eq 1: 2(3x - 11y) = 2(-5)11 times Eq 2: 11(7x + 2y) = 11•16Eq 1 transformed: 6x - 22y = -10 Eq 2 trAnsformed: 77x +22y=176Add together 83x = 166x = 2Substitute to solve for y:3•2 - 11y = -56 - 11 y = -5-11y = -11y = 1The solution is the ordered pair(2,1).

68. i = prti = 17000•0.065•3 = 3315Total = $17,000 + $3,315 =$20,315

69.48 right

65 questions= x right

38 questions65x = 38•4865x = 1824x = 28.06 or 28 questions

70. Let x = number of years since 1928y = current gold medal distance y = 39.4 + 0.51x


Section 9.2 Simplifying Radical Expressions

Practice 9.2.11. 121w = 121 w = 11 w

2. 81xy = 81 xy = 9 xy

3.16

t 2 = 16

t 2 = 4

t

4.25y 2 t

z 2 = 25y 2 t

z 2 = 5 y t

z

5. ( 3 a ) 2 = 3a if a ≥ 0

Practice 9.2.21. 2 • 2 d = 4 d = 4 d = 2 d

2. 18 • 8 b = 144b = 12 b

3.5 w h

w = 5 wh

w = 5 h

4.48w g

3 w = 48wg

3 w = 16g =

16 g = 4 g

Practice 9.2.31. 8 = 4 • 2 = 4 • 2 = 2 2

2. 75 = 25• 3 = 25 • 3 = 5 3

3. 128 = 64• 2 = 64 • 2 = 8 2

4.2449

= 24

49= 4 • 6

7 =

4 • 6 7

= 2 6 7

Practice 9.2.41. 5000 = 5 4 • 2 3 = 5 2 2 2 =

50 2

2. 432 = 2 • 2 • 2 • 2 • 3 • 3 • 3 = 2 4 • 3 3 = 2 2 • 3 3 = 12 3

3. 392 = 2 • 2 • 2 • 7 • 7 = 2 3 • 7 2 = 2•7 2 14 2

4. 968 = 2 • 2 • 2 • 11• 11 = 2 3 • 112 = 2 • 11 2 = 22 2

Practice 9.2.51. d 36 = d 18 since 36÷2 = 18

2. x 13 = x 6 x since 13÷2 = 6 with remainder 1.

3. y 81m 32 = y 40m 16 y since 81÷2=40 with remainder 1 and 32÷2 =16

4. x 5 y 12z 9 = x 2 y 6 z 4 xz since 5÷2= 2 with remainder 1, 12÷2 = 6and 9÷2 = 4 with remainder 1.

5. m 49n 4 r 25 = m 24n 2 r 12 mr since49÷2=24 with remainder 1, 4÷2 = 2and 25÷2 = 12 with remainder 1.


Practice 9.2.61. 48a 4 b 6 = 2 4 • 3 a 4 b 6 =

2 2 a 2 b 3 3 = 4 a 2 b 3 3

2. 75x 8 y 10 = 3 • 5 2 x 8 y 10 = 5 x 4 y 5 3

3. 128r 15s 11t = 2 7 r 15s 11t = 2 3 r 7 s 5 2 rst = 8 r 7 s 5 2 rst

4. 44x 17y 13z 8 = 2 2 • 11x 17y 13z 8 = 2 x 8 y 6 z 4 11xy

5. x 2 − 4 x + 4 = ( x - 2 ) 2 = x - 2

6. x 2 + 12x + 36 = ( x + 6 ) 2 = x + 6

Practice 9.2.71. 7 m 49m 7 = 7 m 7 2 m 7 =

7 m•7 m 3 m = 49m 4 m

2. 2 b 48b 25 = 2 b• 2 4 • 3 b 25 = 2 b•2 2 b 12 3 b = 8 b 13 3 b

3.6 x 4 72x 3 y 5

8 y = 3 x 4 2 • 6 2 x 3 y 5

4 y =

3 x 4 • 6 xy2 2 xy4 y

= 9 x 5 y 2 xy2

4.2 m 45m 7 n 4

3 n 4 = 2 m 3 2 • 5 m 7 n 4

3 n 4 =

2 m•3 m 3 n 2 5 m

3 n 4 = 2 m 4 5 m

n 2

Exercise Set 9.21. 49m = 49 m = 7 m

3.s 4

= s

4 = s

2

5.10

y 2 = 10

y 2 = 10

y

7. ( 5 y ) 2 = 5 y

9. 4 x 2 = 4 x 2 = 2 x

11. 169x 4 = 169 x 4 = 13x 2

13. 7 • 7 t = 7 2 t = 7 t

15.44 rw

11w = 44rw

11w = 4 r = 2 r

17. 27 = 9 3 = 3 3

19. 50 = 25 2 = 5 2

21. 200 = 100 2 = 10 2

23.409

= 40

9 = 4 10

9 = 2 10

3

25.1225

= 12

25= 4 3

25= 2 3

5

27. 150 = 2 • 3 • 5 2 = 5 6

29. 5832 = 2 3 • 3 6 = 2 • 3 3 2 = 54 2

31. x 15 = x 7 x

33. a 17 = a 8 a

35. x 7 y 9 z 3 = x 3 y 4 z xyz

37. r 42s 34t 16 = r 21s 17t 8

39. 12y 5 = 4 • 3 y 5 = 2 y 2 3 y

41. 50a 9 b 15 = 5 2 • 2 a 9 b 15 = 5 a 4 b 7 2 ab

43. 100w 25t 16 = 102 w 25t 16 = 10w 12t 8 w

45. 80r 2 s 5 t 3 = 2 4 • 5 r 2 s 5 t 3 = 2 2 rs2 t 5 st = 4 rs2 t 5 st


47. 60x 20y 12z 13 = 2 2 • 3 • 5 x 20y 12z 13 =

2 x 10y 6 z 6 15z

49. x 2 + 16x + 64 = ( x + 8 ) 2 = | x + 8 |

51. x 2 − 8 x + 16 = ( x - 4 ) 2 = | x - 4 |

53. 4 x 2 + 20x + 25 = ( 2 x + 5 ) 2 = |2 x + 5 |

55. 4 x 25x 10 = 4 x•5 x 5 = 20x 6

57. 8 s 36s 11 = 8 s•6 s 5 s = 48s 6 s

59. 9 wx 144w 9 x 4 = 9 wx•12w 4 x 2 w = 108w 5 x 3 w

61. 4 x 3 y 4 121x 11y 16 = 4 x 3 y 4 • 11x 5 y 8 x = 44x 8 y 12 x

63.10a 3 27a 36b 10

15b = 2 a 3 3 3 a 36b 10

3 b =

2 a 3 • 3 a 18b 5 3 3 b

= 2 a 21b 4 3

65.8 x 50x 6 y 7

5 y = 8 x 25• 2 x 6 y 7

5 y =

8 x•5 x 3 y 3 2 y 5 y

= 8 x 4 y 2 2 y

67. Let r = speed at long play2r = speed at standarddistance = rate• time1(r) + 1.5(2r) = 246r + 3r = 2464r = 246r = 61.5 meters per hour

68. Let t = time of slow traint - 1 = time of faster trainDistances are equal.distance = rate •time48t + 64(t -1) = 24448t + 64t - 64 = 244112t = 308t = 2.75 hours10am + 2.75 hrs = 12:45 pm

69. 2y = 6-4xy = 3 - 2xm = -2 and the y intercept is (0,3).

70. 81x2 - 1 = (9x+1)(9x-1)

71. (0,-3) and (3,0)

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #8

72.6 x - 30

30x 2 − 15x - 675= 6 ( x - 5 )

15( 2 x 2 − x - 45) =

6 ( x - 5 ) 15( 2 x + 9 ) ( x - 5 )

= 2 5 ( 2 x + 9 )

73.

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #9

74.


-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #10

75.5 6

x - 1 2

≥ 2 x - 4

6 • 5 6

x - 6 • 1 2

≥ 6 • 2 x - 6 • 4

5x - 3 ≥12x - 24-7x ≥ - 21x ≤ 3

76. 5x2 - 45 = 05(x2 - 9) = 05(x+3)(x-3) = 0x + 3 = 0 or x - 3 = 0x = -3 or x = 3

77. LCD = x(x + 2)2

x + 2 + 1

x = 5

x + 2

x ( x + 2 ) 2

x + 2 + x ( x + 2 )

1 x

= x ( x + 2 ) 5

x + 2 2x + x + 2 = 5x3x + 2 = 5x2 = 2xx = 1

78. Eq 1: x + y = 3Eq 2: 5x = 15 - 5yEq 2 divided by 5: x = 3 - ySubstitute Eq 2 in Eq 1:3-y + y = 33 = 3This is an identity. There are aninfinite number of solutions.Solve for y: y = 3-xThe solutions may be written (x,3-x).

79. Eq 1: 4x + 7y = -5

Eq 2: 6x + 8y = 103 times Eq 1: 3(4x + 7y) = 3(-5)-2 times Eq 2: -2(6x + 8y) = -2(10)Eq 1 transformed: 12x + 21y = -15Eq 2 transformed: -12x - 16y = -20Add together 5y = -35y = -7Substitute to solve for x: 4x+7(-7) = -54x - 49 = -54x = 44x = 11The solution is the ordered pair (11,-7).

80. -(3x+4) -(8-2x) = -3x - 4 - 8 + 2x =-x - 12

81.1812

÷ 4530

= 1812

• 3045

=

2 • 3 • 3 2 • 2 • 3

• 2 • 3 • 5 3 • 3 • 5

= 1

82. -m-1(-m3)(-2m-2) = 2

83. ( 2 a 3 b

) 2 ( a

2 b − 1 ) 3 (

a 2

b ) − 1 =

4 a 2

9 b 2 •

a 3

8 b − 3 •

a - 2

b − 1 = a 3

18b − 2 = a 3 b 2

18

84. 3.4 x 104 + 5.2 + 103 = 34000+5200 = 39200 = 3.92 x 104

85. (x - 15p)(x + 15p) = x2 - 225p2

86. ( 3 5

m 3 n + 2 ) + (

125

m 3 n - mn+ 2 ) +

( 6 mn- 2 ) =

= 155

m 3 n + 5 mn+ 2 = 3 m 3 n + 5 mn+ 2

87. 12x2 + 44x + 40 = 4(3x2 + 11x + 10) =4(3x + 5)(x + 2)

88. 48x3 - 104x2 + 48x = 8x(6x2 - 13x + 6) = 8x(3x-2)(2x-3)

89. a)Let x = number of years since


1932y = the winning timey = 79.4 - 0.33xb) 2008 - 1932 = 76 = xy = 79.4 -0.33(76) y = 54.32 secondsc)79.4 - 0.33x <6019.4 <0.33xx >58.8x > 59 years after 1932Occurred in 1991

Section 9.3 Multiplying and Dividing Radicals

Practice 9.3.11. 32 12 = 2 5 • 2 2 • 3 = 2 7 • 3 =

2 3 2 • 3 = 8 6

2. 75 27 = 3 • 5 2 • 3 3 = 3 4 • 5 2 = 32•5 = 9•5 = 45

3.15x 7 y

35x

y 3 = 3 • 5 x•5 • 7 x

7 y 4 =

3 • 5 2 • 7 x 2

7 y 4 = 5 x 3

y 2

4. 8 x 2 y 3 12x 4 y 5 = 2 3 • 2 2 • 3 x 2 y 3 x 4 y 5 = 2 5 • 3 x 6 y 8 = 2 2 x 3 y 4 3 = 4 x 3 y 4 3

5. 15m 11n 7 75m 5 n 12 = 3 • 5 m 11n 7 • 3 • 5 2 m 5 n 12 = 3 2 • 5 3 m 16n 19 = 3 • 5 m 8 n 9 5 n

15m 8 n 9 5 n

Practice 9.3.2

1.3

11= 3

11•

11

11= 3 11

( 11) 2 =

3 1111

2.7

12= 7

12•

3

3 = 7 3

36=

7 3 6

3.10

5 •

5

5 = 10 5

( 5 ) 2 •

10 5 5

=

2 5

4.11

7 a = 11

7 a •

7 a

7 a = 77a

( 7 a ) 2 =

77a 7 a

5.4 m 2

6 m = 4 m 2

6 m •

6 m

6 m =

4 m 2 6 m

( 6 m ) 2 = 4 m 2 6 m

6 m =

2 m 6 m 3


6.12y 2

8 y = 12y 2

8 y •

2 y

2 y =

12y 2 2 y

16y 2 = 12y 2 2 y

4 y = 3 y 2 y

Exercise Set 9.31. 48 36 = 2 4 • 3 2 2 • 3 2 =

2 2 • 2 • 3 3 = 24 3

3. 10 5 = 2 • 5 • 5 = 2 • 5 2 = 5 2

5. 20x 10x 9 = 2 • 10x•10x 9 = 2 • 102 x 10 = 10x 5 2

7.8 t

q 3

24t 4

q = 8 t•8 • 3 t 4

q 4 =

8 2 • 3 t 5

q 4 = 8 t 2 3 t

q 2

9. 12m 45m 11 = 2 2 • 3 • 3 2 • 5 m 12 = 2 • 3 m 6 3 • 5 = 6 m 6 15

11. 54x 2 y 3 27x 10y 12 = 2 • 3 3 • 3 3 x 12y 15 = 2 • 3 6 x 12y 15

= 3 3 x 6 y 7 2 y = 27x 6 y 7 2 y

13. 54x 3 y 5 12x 3 y 4 = 2 • 3 3 2 2 • 3 x 6 y 9 = 2 3 • 3 4 x 6 y 9 =

2 • 3 2 x 3 y 4 2 y = 18x 3 y 4 2 y

15.4

6 = 4

6 •

6

6 = 4 6

6 = 2 6

3

17.6

13= 6

13•

13

13= 6 13

13

19.8

2 = 8

2 •

2

2 = 8 2

( 2 ) 2 =

8 2 2

= 4 2

21.7

6 m = 7

6 m •

6 m

6 m =

42m

( 6 m ) 2 = 42m

6 m

23.3

5 t = 3

5 t •

5 t

5 t = 15t

( 5 t ) 2 = 15t

5 t

25.5 x 3

10x = 5 x 3

10x •

10x

10x =

5 x 3 10x

( 10x ) 2 = 5 x 3 10x

10x = x 2 10x

2

27.7 d 2

11d = 7 d 2

11d •

11d

11d =

7 d 2 11d

( 11d ) 2 = 7 d 2 11d

11d = 7 d 11d

11

29. 3(x-xy) -y2 = 3x - 3xy - y2

3(-2) - 3(-2)(-3) - (-3)2 = -6 - 18 - 9 = - 33

30. Eq 1: 9x + y = 4Eq 2: 3x - 13 = 2ySolve Eq 1 for y: y = 4 - 9xSubstitute in Eq 2:3x - 13 = 2(4 - 9x) 3x - 13 = 8 - 18x21x = 21x = 1Substitute to solve for y:9(1) + y = 49 + y = 4y = -5The solution is the ordered pair(1,-5).

31. Let t = time of freight traint - 3/4 = time of passenger traindistance = rate • timeThe distances are equal.40t = 45(t - 3/4)40t = 45t - 135/4-5t = -135/4-20t = -135t = 6.75 hours8:15 + 6 hr and 45 min = 3 pmNo, it will not have caught up.

32. amount = base • rateLet x = percent of copper in 35 gram


bar. 40(0.60) + 35x = (40+35)0.5324+35x = 75•0.5324+35x = 39.7535x = 15.75x = 0.45 or 45%

33. d = 1/2 n(n - 3)14 = 1/2 (n2 - 3n)28 = n2 - 3nn2 - 3n - 28 = 0(n - 7)(n + 4) = 0n - 7 = 0 or n + 4 = 0n = 7 sides or n = -4 disregard

We can’t have a -4 sides.

34. f(x) = 2x2 - 3x + 9f(-1) = 2(-1)2 - 3(-1) + 9f(-1) = 2 + 3 + 9 = 14

35. 3 w 3 − 4 w 2 + w + 4 - 3

w + 2

w + 2 3 w 4 + 2 w 3 − 7 w 2 + 6 w + 5 3 w 4 + 6 w 3 Subtract

− 4 w 3 − 7 w 2 − 4 w 3 − 8 w 2 Subtract

w 2 + 6 w w 2 + 2 w Subtract

4 w + 5 4 w + 8 Subtract

- 3

36. (10z-11y)(10z+11y) = 100z2 -121y2

37. -p( 5 t p

) 2 ( 2 p 5 t

) = - p ( 25t 2

p 2 ) (

2 p 5 t

) =

- 10t

38. (3xy)3(-2xy)-2=33x3y3(-2)-2x-2y-2 = 33(-2)-2xy =27xy

4

39. (4d-5)(3d +7) = 12d2 +28d - 15d - 3512d2 + 13d - 35

40. ( − 4 m 2

n − 2 ) (

− 4 m − 2

n 3 ) − 2 (

m − 4

n ) =

( − 4 m 2 n 2

1 ) (

( − 4 ) − 2 m 4

n − 6 ) (

m − 4

n ) =

( − 4 m 2 n 2

1 ) (

m 4 n 6

( - 4 ) 2 ) (

m − 4

n ) =

( − 4 m 2 n 2

1 ) (

m 4 n 6

16) (

m − 4

n ) =

− m 2 n 7

4

41. ( b − 2

− 2 b ) − 3 = b 6

( - 2 ) − 3 b − 3 =

( - 2 ) 3 b 6 b 3 = - 8 b 9

42.3

m − 1

5

4

m

= 5 m ( 3

m − 1

5 )

5 m ( 4

m )

=

5 • 3 - m 5 • 4

= 15 - m 20

43. 54m 11n 3 = 2 • 3 3 m 11n 3 = 3 m 5 n 2 • 3 mn = 3 m 5 n 6 mn

44. 8x2 - 20x - 12 = 4(2x2 - 5x - 3)=4(2x + 1)(x - 3)

45. 9x2 - 16 = (3x + 4)(3x - 4)

46. 2x3 + 32x2 + 6x3 - 40x =8x3 + 32x2 - 40x = 8x(x2 + 4x - 5)= 8x(x + 5)(x - 1)

47. x2 - 3x - 10 = 0(x - 5)(x + 2) = 0x - 5 = 0 or x + 2 = 0x = 5 or x = -2

48. 12x2 + 15x = -312x2 + 15x + 3 = 03(4x2 + 5x + 1) = 03(4x + 1)(x + 1) = 04x + 1 = 0 or x + 1 = 04x = -1 or x = -1x = -1/4 or x = -1

49. x2 - 5x - 14 = (x - 7)(x + 2) = LCD27

x 2 − 5 x - 14+ 2

x + 2 = 3

x - 7


27( x + 2 ) ( x - 7 )

+ 2 x + 2

= 3 x - 7

( x + 2 ) ( x - 7 ) 27

( x + 2 ) ( x - 7 ) + ( x + 2 ) ( x - 7 )

2 x + 2

=

( x + 2 ) ( x - 7 ) 3

x - 7 27 + 2(x -7) = 2(x + 2)27 + 2x - 14 = 2x + 413 + 2x = 2x + 413 = 4This is a contradiction. There is nosolution.

50. a) 80 years after 1869 (1949) therewere 30.7 million students enrolled.b)This represents the number ofstudents enrolled 100 years after

1869 (1969).c) t years after 1869 there were 40million students enrolled.d)The number of students enrolled tyears after 1869 is 4 times the

number of students enrolled in1869.

Section 9.4 Adding and Subtracting Radical Expressions

Practice 9.4.11. 12 3 + 8 3 = 20 3

2. 5 3 + 2 5 + 6 3 + - 5 = 11 3 + 5

3. 12 7 + 4 11 − 4 7 + 6 11 = 8 7 + 10 11

4. 22 5 y − 8 y 5 y This cannot besimplified any further.

5. 3 t − 8 3 t = - 7 3 t

6. 12 6 y + 8 6 y − 7 6 y = 13 6 y

7. 7 m + 15 7 m 5

= 7 m + 3 7 m =

4 7 m

8. 3 8 s − 8 3 s 6

= 3 8 s − 4 3 s 3

Practice 9.4.21. 7 75 + 12 = 7 3 • 5 2 + 2 2 • 3 =

7 • 5 3 + 2 3 = 35 3 + 2 3 =37 3

2. 8 50 − 8 = 8 2 • 5 2 − 2 3 = 8 • 5 2 − 2 2 = 40 2 - 2 2 = 38 2

3. 4 12 + 50 − 75 + 2 8 = 4 3 • 2 2 + 2 • 5 2 − 3 • 5 2 + 2 2 3 = 4 • 2 3 + 5 2 − 5 3 + 2 • 2 2 = 8 3 + 5 2 − 5 3 + 4 2 = 3 3 + 9 2

4. 3 27 + 32 − 12 + 4 50 =


3 3 3 + 2 5 − 2 2 • 3 + 4 2 • 5 2 = 3 • 3 3 + 2 2 2 − 2 3 + 4 • 5 2 = 9 3 + 4 2 − 2 3 + 20 2 = 7 3 + 24 2

5. 4 m 50m + 6 8 m 3 = 4 m 2 • 5 2 m + 6 2 3 m 3 = 4 m•5 2 m + 6 • 2 m 2 m = 20m 2 m + 12m 2 m = 32m 2 m

6. 2 49t + 64t − t = 2 • 7 t + 8 t − t = 14 t + 8 t − t = 21 t

Practice 9.4.31. 5 ( 20 + 2 ) + 45 =

100 + 2 5 + 9 • 5 = 10 + 2 5 + 3 5 = 10 + 5 5

2. 7 y ( y + 4 ) - 6 y = 7 y 2 + 28 y − 6 y = 7 y + 22 y

3. 3 t ( t + 3 ) - 4 t = 3 t 2 + 9 t − 4 t = 3 t + 5 t

4. ( 3 x + 3 ) 2 + 2 3 x = ( 3 x ) 2 + 2 • 3 • 3 x + ( 3 ) 2 + 2 3 x = 3 x + 6 3 x + 9 + 2 3 x =3 x + 8 3 x + 9

5. 5 3 + 1

3 = 5 3 + 1

3 •

3

3 =

5 3 + 3

( 3 ) 2 = 5 3 + 3

3 =

15 3 3

+ 3 3

= 16 3 3

6.2

5 − 5 = 2

5 •

5

5 − 5 =

2 5 5

− 5 = 2 5 5

− 5 5 5

=

− 3 5 5

7.1

3 + 3

7 − 7 =

1

3 •

3

3 + 3

7 •

7

7 − 7 =

3 3

+ 3 7 7

− 7 7 7

= 3 3

- 4 7 7

Exercise Set 9.4

1. 6 7 + 15 7 = 21 7 3. 5 3 + 2 3 − 8 3 = - 3

5. 8 5 s + 9 5 s − 3 5 s = 14 5 s

7. 3 5 + 2 2 x + 7 5 − 5 2 x = 10 5 − 3 2 x

9. 6 a − 4 6 a 2

= 6 a − 2 6 a = - 6 a

11. 11 5 c − 12 5 c 4

=

11 5 c − 3 5 c = 8 5 c

13. 3 5 n + 18 5 n 3

= 3 5 n + 6 5 n =

9 5 n

15. 2 t - 2 w This cannot besimplified.

17.5 2

2 s − 2 s = 5 2

2 s − 2 2 s 2

=

3 2 s 2

19. 7 8 + 32 = 7 2 3 + 2 5 = 7 • 2 2 + 2 2 2 = 14 2 + 4 2 = 18 2

21. 4 8 − 6 50 = 4 2 3 − 6 2 • 5 2 = 4 • 2 2 − 6 • 5 2 = 8 2 − 30 2 = - 22 2

23. 4 36y + 2 49y − 8 25y =


4 • 6 y + 2 • 7 y − 8 • 5 y = 24 y + 14 y − 40 y = - 2 y

25. 5 12 − 3 50 + 2 75 − 6 8 = 5 4 • 3 − 3 25• 2 + 2 25• 3 − 6 4 • 2 = 5 • 2 3 − 3 • 5 2 + 2 • 5 3 − 6 • 2 2 = 10 3 − 15 2 + 10 3 − 12 2 = 20 3 − 27 2

27. 5 x 12x + 4 75x 3 = 5 x 4 • 3 x + 4 25• 3 x 3 = 5 x•2 3 x + 4 • 5 x 3 x = 10x 3 x + 20x 3 x = 30x 3 x

29. 3 r 24r − 2 54r 3 − 2 r 6 r = 3 r 4 • 6 r − 2 9 • 6 r 3 − 2 r 6 r = = 3 r•2 6 r − 2 • 3 r 6 r − 2 r 6 r = 6 r 6 r − 6 r 6 r − 2 r 6 r = - 2 r 6 r

31. 2 ( 8 + 2 ) + 3 32 = 16 + 2 2 + 3 16• 2 =

4 + 2 2 + 3 • 4 2 = 4 + 2 2 + 12 2 = 4 + 14 2

33. 6 ( 6 + 3 ) + 28 = 36 + 3 6 + 4 • 7 =

6 + 3 6 + 2 7

35. 3 ( 12 − 6 ) + 27 = 36 − 6 3 + 9 • 3 =

6 - 6 3 + 3 3 = 6 - 3 3

37. 6 w ( w + 4 ) + 36w = 6 w 2 + 24 w + 6 w = 6 w + 30 w

39. ( 5 a − 3 ) 2 + 2 5 a = ( 5 a ) 2 − 2 • 3 5 a + 9 + 2 5 a = 5 a − 6 5 a + 9 + 2 5 a = 5 a - 4 5 a + 9

41. ( 2 w − 6 ) 2 − 3 w 2 = ( 2 w ) 2 − 2 • 6 2 w + ( - 6 ) 2 − 3 w 2 = 2 w - 12 2 w + 36 - 3 w 2

43. 2 5 + 1

5 = 2 5 + 1

5 •

5

5 =

2 5 + 5 5

= 2 5 ( 5 5

) + 5 5

=

10 5 + 5 5

= 11 5 5

45. 4 3 − 2

3 = 4 3 − 2

3 •

3

3 =

4 3 − 2 3 3

= 4 3 ( 3 3

) − 2 3 3

=

12 3 − 2 3 3

= 10 3 3

47.1

8 + 1

12=

1

8 •

2

2 + 1

12•

3

3 =

2

16+ 3

36= 2

4 + 3

6

49.2

5 + 5

6 − 5 =

2

5 •

5

5 + 5

6 •

6

6 − 5 =

2 5 5

+ 5 6 6

− 5 =

2 5 5

+ 5 6 6

− 5 ( 5 5

) =

2 5 − 5 5 5

+ 5 6 6

=

- 3 5 5

+ 5 6 6

51. The base is 5 and the exponent is 2.-52 = -25

52. Let x = the first integer; x +1 =second integer; x + 2 = thirdinteger

x + (x + 1) + (x + 2) = 2(x + 2) +9x + x+ 1 +x +2 = 2x + 4 + 93x + 3 = 2x + 13x = 10x + 1 = 10 + 1 = 11x + 2 = 10 + 2 = 12


53. y = 5x - 1 m = 5 Slope of a perpendicular line is -1/5.y = mx + b0 = -1/5(0) + bb = 0y = -1/5x

54. Percent for math and computersciences =241,000,000/4,192,000,000 x100%=5.7%Total in 1990 = 9,679,000,000Percent for math and computersciences =841,000,000/8,838,000,000x 100%=9.5%The percent for math increased.

55. -12x2 + 14x + 6 = -2(6x2 -7x - 3)-2(3x + 1)(2x - 3)

56.15x - 15

10x 2 − 10= 15( x - 1 )

10( x 2 − 1 ) =

3 ( x − 1 ) 2 ( x + 1 ) ( x - 1 )

= 3 2 ( x + 1 )

57. 8(25 -2•32) = 8(25 - 2•9) = 8(25-18)=

8(7) = 56

58. 1 - 1 6 x

+ 5 2 x

=

1 ( 6 x 6 x

) - 1 6 x

+ 5 2 x

( 3 3

) =

6 x - 1 + 156 x

= 6 x + 146 x

= 2 ( 3 x + 7 ) 2 • 3 x

=

3 x + 7 3 x

59.x 2 − 4 x - 3

• x 2 − x - 6

x 2 + 4 x + 4 =

( x + 2 ) ( x - 2 ) x - 3

• ( x - 3 ) ( x + 2 ) ( x + 2 ) ( x + 2 )

= x - 2

60.648x 2 y 2

9 x 3 = 2 3 • 3 4 x 2 y 2

9 x 3 =

2 • 3 2 xy 2

9 x 3 = 2 • 9 xy 2

9 x 3 = 2 y 2

x 2

61. ( 64 − 9 ) 2 = ( 8 - 3 ) 2 = ( 5 ) 2 = 25

62. 12a 9 b 2 = 4 • 3 a ( 3 b ) =( 3 b ) • 2 3 a = 6 b 3 a

63. -(f-2g-3h)-(6h-f)-(-2g+3h)=-f+2g+3h-6h+f+2g-3h =4g - 6h

64. 0 = x2 - 11x + 240 = (x - 8)(x - 3)x - 8 = 0 or x - 3 = 0x = 8 or x = 3

65. -30 -61x = 30x2

30x2 + 61x + 30 = 0ac = 30•30 = 900Product Sum20(45) 6525(36) 6130x2 + 25x + 36x + 30 = 05x(6x + 5) + 6(6x + 5) = 0(5x + 6)(6x + 5) = 05x + 6 = 0 or 6x + 5 = 05x = -6 or 6x = -5x = -6/5 or x = -5/6

66. 0.15(12-x)+0.25x=5+0.1x1.8 -0.15x + 0.25x = 5 +0.1x1.8 +0.10x = 5+0.1x1.8 = 5This is a contradiction. There is nosolution.

67.7 m

= 3 4

− m - 5 m

4 m•7 m

= 4 m•3 4

− 4 m•m - 5

m 28 = 3m -4(m-5)28 = 3m - 4m + 208 = -mm = -8

68. The domain is -7 ≤x ≤6. The rangeis

2 ≤ y≤ 7.

69. a)E = 70.3 +0.16tb)E(60) = 70.3 +0.16(60) =79.9By 2020 (1960 + 60) the average lifeexpectancy will be about 79.9 years.c)60 = 70.3 +0.16t-10.3 =0.16t


t = -64.375 years This means in about 1896 (1960 -

64) the life expectancy was 60 years.d)E(0) = 70.32(70.3) = 140.6140.6 =70.3 +0.16t70.3 = 0.16tt = 439.375 yearsIn 2399 the average life expectancywill be twice the average lifeexpectancy in 1960.

Section 9.5 Solving Equations that Involve Square Roots

Practice 9.5.11. 5 x = 10

( 5 x ) 2 = ( 10) 2

5x = 100x = 20Check: 5 • 20 = 10

100 = 1010=10

2. 6 = 3 4 x ( 6 ) 2 = ( 3 4 x ) 2

36 = 9(4x)36 = 36xx = 1Check: 6 = 3 4 ( 1 ) 6 = 3 4 6 = 3•26 = 6

3. x + 5 = 3 ( x + 5 ) 2 = ( 3 ) 2


x + 5 =9x = 4Check: 4 + 5 = 3

9 = 3 3=3

4. 5 = x - 10( 5 ) 2 = ( x - 10 ) 2

25 = x - 10x = 35Check: x - 10 = 35 - 10 = 25 = 5

Practice 9.5.21. 5 x − 15 = - 5

5 x = 10( 5 x ) 2 = ( 10) 2

5x = 100x = 20Check: 5 • 20 - 15 = 100 − 15

2. 11 - 6 x = - 1 − 6 x = - 12( − 6 x ) 2 = ( - 12) 2

6x = 144x = 24Check: 11 -

6 ( 24) = 11- 144 = 11- 12= - 1

3.6 - x 5

+ 1 = 2

6 - x 5

= 1

5 • 6 - x 5

= 1 • 5

6 - x = 5 ( 6 - x ) 2 = ( 5 ) 2

6 -x = 25-x = 19x = -19

Check: 6 - ( - 19)

5 + 1 =

6 + 195

+ 1 =

255

+ 1 = 5 5

+ 1 = 1 + 1 = 2

4. x + 6 = 25x + 6 3

( x + 6 ) 2 = ( 25x + 6 3

) 2

x + 6 = 25x + 6 9

9(x+6) = 25x + 69x + 54 = 25x + 648 = 16xx = 3

Check: 3 + 6 = 25• 3 + 6 3

9 = 813

3 = 9/33 = 3

5. 5 x - 7 = 3 x + 1 ( 5 x - 7 ) 2 = ( 3 x + 1 ) 25(x-7) = 3x + 125x - 175 = 3x + 122x = 176x = 85 8 - 7 = 3 • 8 + 1 5 1 = 255 = 5

Practice 9.5.31. 30 - x = x

30 - x = x2 Square both sidesx2 + x - 30 = 0(x + 6)(x -5) = 0x + 6 = 0 or x - 5 = 0x = -6 or x = 5Check x = -6:

30 - ( - 6 ) = 36 = 6 6 is not a solution.Check x = 5: 30 - 5 = 25 = 5 5 is a solution.

2. 2 x + 4 = - 6 2x+4 = 36 Square both sides.2x = 32x = 16Check: 2 ( 16) + 4 = 36 = 6 ≠ - 6 16 is not a solution. There is nosolution.

3. 4 = 4 x + 7 - 3 = 4 x 9 = 4x Square both sides.x = 2.25Check: 4 ( 2 . 25) + 7 = 9 + 7 =


3 + 7 = 10≠ 42.25 is not a solution. There is nosolution.

4. 2 x = 4 x + 8 4x2 =4x+8 Square both sides.4x2 - 4x - 8=04(x2 -x -2) = 04(x -2)(x+1) = 0x - 2 = 0 or x + 1 = 0x = 2 or x = -1Check x=2:2(2) = 4 ( 2 ) + 8 4 = 16 = 42 is a solution.Check x = -1:2(-1) = 4 ( - 1 ) + 8 -2 = 4 -2 ≠2-1 is not a solution.

Practice 9.5.41. d = 1 . 5 h

h = 6 + 12 = 18 feetd = 1 . 5 ( 18) d = 27d = 5.2 miles

2. d = 1 . 5 h 2.2 = 1 . 5 h 4.84= 1.5h Square both sides.h = 3.3 feet

3. d = 1 . 5 h 8 = 1 . 5 h 64 = 1.5h Square both sides.h = 42.7 feet42.7 - 7 = 35.7 feet for the platform.

Practice 9.5.51. v = µgR

v = 0 . 10( 9 . 8 ) 200v = 196v = 14 meters per second

2. v = µgR12 = 9 . 8 • 110µ 144 = 1078µµ = 0.13

Exercise Set 9.51. 2 x = 10

2x = 100 Square both sides.x = 50Check is left to student.

3. 4 = 2 w 16 = 2w Square both sides.w = 8

5. 5 2 x = 1525•2x = 225 Square both sides.50x = 225x = 4.5

7. 4 3 t = 2416•3t = 57648t = 576t = 12

9. 2 x + 3 = 5 2x + 3 = 25 Square both sides.2x = 22x = 11

11. 3 = 2 x - 5 9 = 2x - 5 Square both sides.2x = 14x = 7

13. y + 4 = 7 y + 4 = 49y = 45

15. x 2 − 5 = x x2 - 5 = x2

-5 = 0This is a contradiction. There is nosolution.

17. 7 x − 2 = 5 7 x = 7

7x = 49 Square both sides.x = 7

19. 8 = 3 x + 2 6 = 3 x 36 = 3x Square both sides.x = 12

21. 6 x − 7 = - 1


6 x = 6 6x = 36 Square both sides.x = 6

23. 6 x − 5 = 7 6 x = 12

6x = 144x = 24

25. 2 x − 10 = - 4 2 x = 6

2x = 36 Square both sidesx = 18

27. 7 - 2 x = 1 - 2 x = − 6 2x = 36 Square both sides.x = 18

29. 3 = - 2 + 5 x 5 = 5 x 25 = 5xx = 5

31. 2 = 4 + 6 x 2

− 3

5 = 4 + 6 x 2

10 = 4 + 6 x 100 = 4 + 6x Square both sides.96 = 6xx = 16

33.4 - 8 x 3

− 3 = - 1

4 - 8 x 3

= 2

4 - 8 x = 6 4 - 8x = 36 Square both sides.-8x = 32x = -4

35.6 - 5 x 2

− 5 = - 2

6 - 5 x 2

= 3

6 - 5 x = 6 6 - 5x = 36 Square both sides.-5x = 30

x = -6

37. 5 3 x - 20 = 11x + 1225(3x -20) = 11x + 12 Square bothsides.75x - 500 = 11x + 1264x = 512x = 8

39. 2 10x - 4 = 35x + 4 4(10x-4) = 35x + 440x - 16 = 35x + 45x = 20x = 4

41. y + 72 = y y + 72 = y2 Square both sides. y2 - y - 72 = 0(y -9)(y + 8) = 0y - 9 = 0 or y + 8 = 0y = 9 or y = -8Check: y = 9

9 + 72 = 81 = 9 = y 9 is a solution.Check y = -8:

− 8 + 72 = 64 = 8 ≠ - 8 -8 is not a solution.

43. 5 - 2 x = 7 - 2 x = 2 2x = 4 Square both sides.x = 2Check : 5 - 2 • 2 = 5 - 4 = 5 - 2 = 3 ≠ 7 2 is not a solution.There is nosolution.

45. x = 20 - x x2 = 20 -x Square both sides.x2 + x - 20 = 0(x + 5)(x - 4) = 0x + 5 = 0 or x - 4 = 0x = -5 or x = 4Check x = -5:-5 = 20 - ( - 5 ) -5 = 25-5 ≠5 -5 is not a solution.Check x = 4:4 = 20 - 4 4 = 16 4 = 4 4 is a solution.


47. 4 - 16x = - 104-16x = 100 Square both sides.-16x = 96x = 6Check:

4 - 16( − 6 ) = 100 = 10≠ - 10There is no solution.

49. 2 x = 20x + 244x2 = 20x + 24 Square both sides.4x2 - 20x - 24 = 04(x2 - 5x - 6) = 04(x - 6)(x + 1) = 0x - 6 = 0 or x + 1 = 0x = 6 or x = -1Check x = 6:2(6) = 20• 6 + 2412 = 14412 = 126 is a solution.Check x = -1:2(-1) = 20( - 1 ) + 24-2 = 4 -2≠2-1 is not a solution.

51. p = 3 p + 18p2 = 3p +18 Square both sides.p2 -3p -18 = 0(p- 6)(p+3) = 0p - 6 = 0 or p + 3 = 0p = 6 or p = -3Check p = 6: 6 = 3 ( 6 ) + 186 = 366 = 6; 6 is a solution.Check p = -3: -3 = 3 ( - 3 ) + 18-3 = 9 -3 ≠ 3; -3 is not a solution.

53. − 4 = 6 x + 2 − 6 = 6 x 36 = 6x Square both sides.x = 6Check:

6 • 6 + 2 = 36 + 2 = 6 + 2 = 8 ≠ - 4 6 is not a solution.

55. h = 6 + 16 = 22 feetd = 1 . 5 ( 22) = 33 = 5 . 7 miles

57. 15 = 1 . 5 h 225 = 1.5h Square both sides.150 feet = h150 - 5 = 145 feet taller

59. d = 1 . 5 • 5 = 7 . 5 = 2 . 7 milesd = 1 . 5 • 4 = 6 . 0 = 2 . 4 miles2.7 - 2.4 = 0.3 miles farther

61.

56 feet

ScottJoline

106 feet

x

(56)2 +x2 =(106)2

3136 +x2 =11236

x2 -8100 = 0

(x + 90)(x - 90) = 0x + 90 = 0 or x - 90 = 0x = -90 or x = 90-90 is not valid so x = 90 feet is theheight of the kite.

62. m = 1 - 0 1 - 1

= 1 0

undefined.

63. y = mx + by = 1x - 2

64. m=1 - 1

6 - ( - 8 ) = 0

14= 0

Since m= 0 , the equation is y = 1

65. 4x2 - 27x + 18 = (4x -3)(x - 6)

66. 5x2 - 9x - 18 = (5x + 6)(x - 3)


67. 36x3y2z = 22•32x3y2z24xy4z2 = 23•3xy4z2

40x2y3z6 = 23•5x2y3z6

GCF = 22xy2z = 4xy2z

68. 8x2 - 8x = 08x(x-1) = 08x = 0 or x - 1 = 0x = 0 or x = 1

69. Eq 1: 3x + y = 2Eq 2: 6x + 2y = -5-2 times Eq 1: -6x -2y = -4Add together 0 = -9This is a contradiction. There is nosolution.

70. 2 + 1 b + 1

= b + 2 b + 1

2 ( b + 1 ) + ( b + 1 ) 1

b + 1 = b + 2

b + 1 ( b + 1 )

2b + 2 +1 = b + 22b + 3 = b + 2b = -1

Check: 2 + 1 - 1 + 1

= - 1 + 2 - 1 + 1

2 + 1/0 = 1/0 This is undefined. There is no solution.

71. x2 - 2x = x(x - 2)x2 - 4 = (x +2)(x -2)LCD = x(x - 2)(x + 2)

3

x 2 − 2 x + 1

x 2 − 4 =

3 x ( x - 2 )

+ 1 ( x + 2 ) ( x - 2 )

=

3 x ( x - 2 )

• x + 2 x + 2

+ x x

• 1

( x + 2 ) ( x - 2 ) =

3 ( x + 2 ) + x x ( x + 2 ) ( x - 2 )

= 3 x + 6 + x x ( x + 2 ) ( x - 2 )

=

4 x + 6 x ( x + 2 ) ( x - 2 )

72.( 2 xy3 ) 4 ( x 2 y ) 2

( 4 xy2 ) 2 = 16x 4 y 12x 4 y 2

16x 2 y 4 = x 6 y 10

73. 8 - 3(x-2) - (5 - 5x) = 8 - 3x + 6 - 5 + 5x = 9 + 2x

74. 5 - 9 + 2 ( 3 - 25 ) =5 - 3 + 2(3 - 5) = 2 + 2(-2) = 2 - 4 = -2

75.18s 4 s 5

6 s 7 = 9 • 2 s 4 s 5

6 s 7 =

9 • 4 • 2 s 6

6 s 7 = 3 • 2 s 3 2

6 s 7 = 2

s 4

76. 24 + 5 54 − 3 216 = 4 • 6 + 5 9 • 6 − 3 36• 6 =

2 6 + 5 • 3 6 − 3 • 6 6 = 2 6 + 15 6 − 18 6 = - 6

77. The x intercept is (-8,0) and the y-intercept is (0,2).

78. Yes, it passes the vertical line test.


Section 9.6 Quadratic Equations

Practice 9.6.11. 3x2 + 7x - 6 = 0

a = 3, b = 7, c = -6

x = − 7 ± 7 2 − 4 ( 3 ) ( - 6 )

2 ( 3 )

x = − 7 ± 49 + 72

6

x = − 7 ± 121

6

x = − 7 ± 11

6 Solution 1 Solution 2x =(-7+11)/6 x = (-7 -11)/6x = 4/6 x = -18/6x = 2/3 x = -3

2. x2 - 5x + 1 = 0a = 1, b = -5, c = 1

x = − ( - 5 ) ± ( - 5 ) 2 − 4 ( 1 ) ( 1 )

2 ( 1 )

x =5 ± 25 − 4

2 = 5 ± 21

2 Solution 1 Solution 2

x = 5 + 21

2 x =

5 - 212

3. -x2 + 3x + 7 = 0a = -1, b = 3, c= 7

x = − 3 ± ( 3 ) 2 − 4 ( - 1 ) ( 7 )

2 ( - 1 )

x= − 3 ± 9 + 28

− 2 = − 3 ± 37

− 2 Solution 1 Solution 2

x = − 3 + 37

− 2 x =

− 3 - 37− 2

x = 3 - 37

2 x =

3 + 372

4. 2.1x2 - x - 5.4 =0a = 2.1, b = -1, c = -5.4

x = − ( - 1 ) ± ( - 1 ) 2 − 4 ( 2 . 1 ) ( - 5 . 4 )

2 ( 2 . 1 )

x = 1 ± 1 + 45. 36

4 . 2 = 1 ± 46. 36

4 . 2

Solution 1 Solution 2

x = 1 + 46. 36

4 . 2 x =

1 - 46. 364 . 2

x =1 + 6 . 80882

4 . 2 x=

1 - 6 . 808824 . 2

x =1.85924 x = -1.38305

Practice 9.6.21. 4x = 2x2 -1

2x2 - 4x - 1 = 0a = 2, b = -4, c = -1

x =− ( - 4 ) ± ( - 4 ) 2 − 4 ( 2 ) ( - 1 )

2 • 2

x=4 ± 16 + 8

4 = 4 ± 24

4 = 4 ± 2 6

4

x =4 4

± 2 6

4 = 1 ±

6 2


x = 1 + 6 2

x =1 - 6 2

2.1 4

= 1 2

x 2 − 3 x

0 = 1 2

x 2 − 3 x - 1 4

a = 1/2, b = -3 , c = -1/4

x=− ( - 3 ) ± ( - 3 ) 2 − 4 ( 1

2 ) ( - 1

4 )

2 ( 1

2 )

x=3 ± 9 + 1

2

1 = 3 ±

192

• 2 2

=

x = 3 ± 382


x =3 + 382

x=3 - 382

3. -2 = 4x2 - x4x2 - x + 2 = 0a = 4, b = -1, c = 2

x = − ( - 1 ) ± ( - 1 ) 2 − 4 ( 4 ) ( 2 )

2 ( 4 )


x= 1 ± 1 − 32

8 = 1 ± − 31

8 There is no real solution. Theradicand is negative.

4. 20+ 2x =0.1x2 0.1x2 - 2x - 20 = 0a = 0.1, b = -2, c = -20

x =− ( - 2 ) ± ( - 2 ) 2 − 4 ( 0 . 1 ) ( - 20)

2 ( 0 . 1 )

x= 2 ± 4 + 8

0 . 2 = 2 ± 12

0 . 2 =

2 ± 2 3 0 . 2

= 10± 10 3

Solution 1 Solution 2 x= 10 + 10 3 x = 10 - 10 3

5.1 9

x 2 − 4 3

x = - 4

1 9

x 2 − 4 3

x + 4 = 0

a = 1/9, b = -4/3, c = 4

x =− ( − 4

3 ) ± ( − 4

3 ) 2 − 4 ( 1

9 ) ( 4 )

2 ( 1

9 )

x=

4

3 ± 16

9 − 16

9

2

9

= 4

3 ± 0

2

9

x = 4 3

• 9 2

= 6

Practice 9.6.41. a)y = -16x2 +60x + 7

y = -16(4)2 + 60(4) + 7y = -256 +240 +7 = -9 or 9 feet below ground levelb)40 = -16x2 + 60x + 7-16x2 + 60x - 33 = 0a = -16, b = 60, c=-33

x =− 60± ( 60) 2 − 4 ( - 16) ( - 33)

2 ( - 16)

x = − 60± 3600 − 2112

− 32=

x= − 60± 1488

− 32= − 60± 38. 57

− 32Solution 1 Solution 2

x =− 60 + 38. 57

− 32 x=

− 60 - 38. 57− 32

x=0.67 seconds x =3.08

secondsc) When the ball hits the ground theheight is 0.0 = -16x2 + 60x + 7a = -16, b = 60, c = 7

x=− 60± ( 60) 2 − 4 ( - 16) ( 7 )

2 ( - 16)

x=− 60± 3600 + 448

− 32=

x=− 60± 63. 62

− 32Solution 1 Solution 2

x=− 60 + 63. 62

− 32 x=

− 60 - 63. 62− 32

x=-0.11 (Invalid) x = 3.86 sec.

Exercise Set 9.61. 3x2 - 2x - 5 = 0

a = 3, b = -2, c = -5

x= − ( - 2 ) ± ( - 2 ) 2 − 4 ( 3 ) ( - 5 )

2 ( 3 )

x=2 ± 4 + 60

6 = 2 ± 64

6 = 2 ± 8

6 Solution 1 Solution 2x = -6/6 = -1 x = 10/6 =5/3

3. 0 = 2x2 + 5x + 2a = 2, b = 5, c = 2

x =− 5 ± 5 2 − 4 • 2 • 2

2 • 2

x=− 5 ± 25 − 16

4 = − 5 ± 9

4

x =− 5 ± 3

4 Solution 1 Solution 2x = -8/4 = -2 x = -2/4 = -1/2

5. x2 - 9x + 4 = 0a = 1, b = -9, c = 4

x =− ( - 9 ) ± ( - 9 ) 2 − 4 ( 1 ) ( 4 )

2 ( 1 )

x = 9 ± 81 − 16

2 = 9 ± 65

2 Solution 1 Solution 2

x = 9 + 65

2 x=

9 - 652


7. -4x2 +x+2=0a = -4, b = 1, c = 2

x =− 1 ± 1 2 − 4 ( - 4 ) ( 2 )

2 ( - 4 )

x =− 1 ± 1 + 32

− 8 = − 1 ± 33

− 8


x =− 1 + 33

− 8 x =

− 1 − 33− 8

x = 1 - 33

8 x =

1 + 338

9. 9x2 = 12x -49x2 - 12x + 4 =0a = 9, b = -12 , c = 4

x =− ( - 12) ± ( - 12) 2 − 4 ( 9 ) ( 4 )

2 ( 9 )

x = 12± 144 − 144

18= 12± 0

18

x = = 12

18=

2

3

11. 3x2 + 3x - 214

= 0

a = 3, b = 3, c = -21/4

x =− 3 ± 3 2 − 4 ( 3 ) ( − 21

4 )

2 ( 3 )

x=− 3 ± 9 + 63

6

x = − 3 ± 726

= − 3 ± 6 2 6

x= - 1 2

± 2


x = - 1 2

+ 2 x= - 1 2

- 2

13. -2.07x +0.05x2 -1.15 =00.05x2 - 2.07x - 1.15= 0a = 0.05, b = -2.07, c = -1.15

x = − ( - 2 . 07) ± ( - 2 . 07) 2 − 4 ( 0 . 05) ( - 1 . 15)

2 ( 0 . 05)

x= 2 . 07± 4 . 2849+ 0 . 23

0 . 1

x = 2 . 07± 4 . 51490 . 1

= 2 . 07± 2 . 12480 . 1


x = 2 . 07- 2 . 1248

0 . 1 x= 2 . 07+ 2 . 1248

0 . 1

x = - 0.548 x = 41.948

15. 4x2 + 2x = -24x2 +2x +2 =0a = 4, b = 2, c = 2

x=− 2 ± ( 2 ) 2 − 4 ( 4 ) ( 2 )

2 ( 4 )

x =− 2 ± 4 − 32

8 = − 2 ± − 24

8

The radicand is negative. There is no real solution.

17. -x +0.5x2 +0.5 =00.5x2 - x + 0.5 = 0a = 0.5, b = -1, c = 0.5

x =− ( - 1 ) ± ( - 1 ) 2 − 4 ( 0 . 5 ) ( 0 . 5 )

2 ( 0 . 5 )

x =1 ± 1 − 1

1 = 1 ± 0

1 = 1

1 = 1

19. a)y = 0.35x2 +5.4x +3877.85,000 = 0.35x2 +5.4x +3877.80=0.35x2 + 5.4x - 1122.2a = 0.35 , b = 5.4, c = -1122.2

x=− 5 . 4 ± ( 5 . 4 ) 2 − 4 ( 0 . 35) ( - 1122. 2 )

2 ( 0 . 35)

x = − 5 . 4 ± 29. 16+ 1571. 08

0 . 7

x =− 5 . 4 ± 1600. 24

0 . 7 = − 5 . 4 ± 40

0 . 7 x = 49.4 years; 1985 + 50 = 2035x = -64.9 years; 1985 - 65 = 1920 We will disregard this value since

our model is not valid for years prior to1985.b)2075 - 1985 = 90y = 0.35(90)2 + 5.4•90 + 3877.8y = 0.35(8100) +486 + 3877.8y = 2835 +486+3877.8 y = 7198.8 thousand birthsc)2000 - 1985 = 15y = 0.35(15)2 + 5.4(15) + 3877.8y = 0.35•225 +81 +3877.8y = 78.75 + 81+3877.8y = 4037.55 thousand births in 20004037.55 + 500 = 4537.55


4537.55 = 0.35x 2 +5.4x+3877.80= 0.35x 2+ 5.4x - 659.75a = 0.35, b = 5.4, c =-659.75

x = − 5 . 4 ± ( 5 . 4 ) 2 − 4 ( 0 . 35) ( - 659. 75)

2 ( 0 . 35)

x =− 5 . 4 ± 29. 16 + 923. 65

0 . 7

x =− 5 . 4 ± 952. 81

0 . 7 = − 5 . 4 ± 30. 87

0 . 7 x = 36.4 years; 1985 + 37 = 2022Again we disregard the negativevalues for x, since our model is notvalid for years before 1985.

21. amount = rate • timeamount in = amount outLet x = time to fill75x = 60(x +1.5)75x = 60x + 9015x = 90x = 6 hours75 • 6 = 450 gallons

22.<---|----|----|----|----|----|----|---•----|----|--->

-2 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2

23.

-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8910x

Data #11

24.

n4 − 1

2 n - 7 2 n 5 − 7 n 4 − 2 n + 7 2 n 5 − 7 n 4 Subtract

0 - 2 n + 7 − 2 n + 7 Subtract

0

25. 25x2 -4 = (5x + 2)(5x + 2)

26. (s6t2) -2(s -6t-2) =s-12t-4s -6t-2 = s-18t-

6=1

s 18t 6

27.1 . 08 x10− 5

( 0 . 016) ( 0 . 00003) = 1 . 08 x10− 5

1 . 6 x 10− 2 ( 3 x 10− 5 ) 1 . 08 x10− 5

4 . 8 x 10− 7 = 0 . 225 x 102 =

2 . 25 x10− 1 x 102 = 2 . 25 x10

28.5 y 2

x 2 z •

3 x 5

10y 5 •

4 z 2

9 =

5 y 2

x 2 z •

3 x 5

2 • 5 y 5 •

2 • 2 z 2

3 • 3 = 2 x 3 z

3 y 3

29.2 x 2 + 2 x

x - 2 •

x 2 − 4

3 x 2 + 3 x =

2 x ( x + 1 ) x - 2

• ( x + 2 ) ( x - 2 )

3 x ( x + 1 ) = 2 ( x + 2 )

3

30.x 4 − x 3 − 2 x 2

x 2 + x ÷ ( x - 2 ) =

x 2 ( x 2 − x − 2 ) x ( x + 1 )

• 1

x - 2 =

x 2 ( x - 2 ) ( x + 1 ) x ( x + 1 )

• 1

x - 2 = x

31.1

x - 1 − 5 = 1

x - 1 − 5 •

x - 1 x - 1

=

1 - 5 ( x - 1 ) x - 1

= 1 - 5 x + 5 x - 1

= 6 - 5 x x - 1

32.3

t + 1 1

t

= t ( 3

t + 1 )

t ( 1

t )

= 3 + t 1

= 3 + t

33.4 ab2

9 = 2 b a

3

34. 2 • 3 • 5 2 = 5 2 • 3 = 5 6

35. 18x 7 y 2 = 2 • 3 2 x 7 y 2 = 3 x 3 y 2 x


36.4

6 = 4

6 •

6

6 = 4 6

6 = 2 6

3

37. -80 - 64 ÷22 •2(1-2) = -1-64 ÷4 •2(-1) = -1-16•(-2) =-1+32 =31

38. 3x2 + 5x - 2 = 0(3x - 1)(x + 2) = 03x - 1 = 0 or x + 2 = 03x = 1 or x = -2x = 1/3 or x = -2

39.z - 6 2

+ 1 = 2

z - 6 2

= 1

z - 6 = 2 z - 6 = 4 Square both sidesz = 10Check is left to student.

40. Eq 1: 3x - 4y = 2Eq 2: -6x+ 8y =52 times Eq 1: 6x - 8y = 4Add together 0 = 9This is a contradiction. There is nosolution.

41. Yes, it passed the vertical line test.


Chapter 9 Review Exercises

1. − − 9 The radicand is negative.Not a real number.

2. − 16 = - 4

3. ( 49 − 36 ) 2 = ( 7 - 6 ) 2 = ( 1 ) 2 = 1

4.12 4 − 6 4

4 4 = 6 4

4 4 = 3

2

5. 49x 2 y 2 = 7 xy

6.9 m 2 n

16= 3 m n

4

7.50m 6 m

2 3 = 300m 2

2 3 =

3 • 100m 2

2 3 = 10m 3

2 3 = 5 m

8. 15x 16x = 15• 16x 2 = 4 x 15

9. 72 = 2 • 36 = 6 2

10.30025

= 12 = 3 • 4 = 2 3

11. 1568 = 2 5 • 7 2 = 2 2 • 7 2 = 28 2

12. 1440 = 122 • 10 = 12 10

13. x 2 y 4 z 11 = xy2 z 5 z

14. 4 f 3 h 9 = 4 fh4 fh

15. 108s 4 t 2 = 2 2 • 3 3 s 4 t 2 = 2 • 3 s 2 t 3 = 6 s 2 t 3

16. 700m 3 n 4 p 5 = 7 • 100m 3 n 4 p 5 = 10mn2 p 2 7 mp

17.3 t 294s 7

21s 4 = 3 t 2 • 3 • 7 2 s 7

21s 4 =

3 t•7 s 3 6 s

3 • 7 s 4 = t 6 s

s

18. 2 b 3 32b 5 cd8 = 2 b 3 2 5 b 5 cd8 = 2 b 3 • 2 b 2 d 4 2 bc = 8 b 5 d 4 2 bc

19.4 3 a 4 6 a 3 b 2

3 a 3 b = 4 18a 7 b 2

3 a 3 b =

4 9 • 2 a 7 b 2

3 a 3 b = 4 • 3 a 3 b 2 a

3 a 3 b = 4 2 a

20.5 x 3

2 y 5

20x

y 3 = 5 x 3 • 20x

2 y 5 y 3 =

50x 4

y 8 = 25• 2 x 4

y 8 = 5 x 2 2

y 4

21.4 6

12= 4 6

12•

3

3 = 4 18

36=

4 9 • 2 6

= 4 • 3 2 6

= 2 2

22.3 y

18y = 3 y

18y •

2 y

2 y =


3 y 2 y

36y 2 = 3 y 2 y

6 y = 2 y

2

23. ( 2 x + x ) 2 = ( 2 x ) 2 + 2 ( 2 x ) ( x ) + x 2 = 4 x + 4 x x + x 2

24. 6 t + 3 t − 2 t t = 9 t − 2 t t

25. 14 2 r − 2 14r + 3 14r − 11 2 r = 3 2 r + 14r

26. 12x 3 + 27x − x 75x = 4 • 3 x 3 + 3 3 x − x 25• 3 x =

2 x 3 x + 3 3 x − 5 x 3 x = 3 3 x − 3 x 3 x

27. 28 − 20 − 112 + 3 5 = 4 • 7 − 4 • 5 − 16• 7 + 3 5 =

2 7 − 2 5 − 4 7 + 3 5 = - 2 7 + 5

28. 3 t ( t − t ) + 3 ( t + t t ) =3 t - 3 t t + 3 t + 3 t t = 6 t

29. 2 y - 4 = 6 2y - 4 = 36 Square both sides.2y = 40y = 20

30. 4 16t = 164 • 4 t = 1616 t = 16

t = 1 t = 1 Square both sides.

31. − 4 t + 6 = 3 2 t + 8 -4t +6 =9(2t +8) Square both

sides.-4t + 6 = 18t + 72-22t = 66t = -3Check: − 4 ( - 3 ) + 6 = 3 2 ( - 3 ) + 8

18 = 3 2 9 • 2 = 3 2

3 2 = 3 2 -3 is a solution.

32. 3 - 4 - 3 x = - 2 - 4 - 3 x = - 5 4 - 3x = 25-3x = 21x = -73 - 4 - 3 ( - 7 ) = - 2 3 - 25 = - 2 3 - 5 = -2-2 = -2-7 is a solution.

33. 12 = 2 x + 14- 2 = 2 x 4 = 2xx = 212 = 2 ( 2 ) + 1412 = 2 + 1412 = 16This is a contadiction. There is nosolution.

34. x 2 + 5 + 5 = x

x 2 + 5 = x - 5 x2 + 5 = (x -5)2 Square both sides.x2 + 5 = x2 - 10x + 25-20 = -10xx = 2

2 2 + 5 + 5 = 2 9 + 5 = 2

3 + 5≠ 2This is a contradiction. There is nosolution.

35. -2x2 -3x +2 =0a = -2, b = -3, c = 2

x = − ( - 3 ) ± ( - 3 ) 2 − 4 ( - 2 ) ( 2 )

2 ( - 2 )

x= 3 ± 9 + 16

− 4 = 3 ± 25

− 4 = 3 ± 5

− 4 Solution 1 Solution 2

x = 3 + 5 − 4

x = 3 - 5 − 4

x = -2 x = 1/2

36. 4y2 -8y-3 =0a = 4, b = -8, c = -3

y = − ( - 8 ) ± ( - 8 ) 2 − 4 ( 4 ) ( - 3 )

2 ( 4 )


y =8 ± 64 + 48

8 = 8 ± 112

8

y = 8 ± 4 7

8 = 1 ±

7 2

Solution 1 Solution 2y =

1 + 7 2

y =

1 - 7 2

37. q2 - 2 3

q + 1 9

= 0

a = 1, b = -2/3, c = 1/9

q = - − 2

3 ± ( − 2

3 ) 2 − 4 ( 1 ) ( 1

9 )

2 ( 1 )

q =

2

3 ± 4

9 − 4

9

2 =

2

3 ± 0

2

q = 2

3 •

1

2 =

1

3

38. b2 = b+1.75b2 - b - 1.75 = 0a = 1, b = -1, c = -1.75

b = − ( - 1 ) ± ( - 1 ) 2 − 4 ( 1 ) ( - 1 . 75)

2 ( 1 )

b = 1 ± 1 + 7

2 = 1 ± 8

2

b = 1 ± 2 2

2 = 1

2 ± 2


b = 1 2

+ 2 b = 1 2

− 2

39. distance = rate •timedistance/rate = time

200 feet

100 feetd

(100)2 + (200)2 = d2

10000 +40000= d2

d2 -50000 =0a = 1, b = 0, c = -50000

d = − 0 ± 0 2 − 4 ( 1 ) ( - 50000)

2 ( 1 )

d=0 ± 0 + 200000

2 = ± 447. 2

2 d = 223.6 The negative value isinvalid since we can’t have a

negative distance.100 + 200 - 223.6 = 76.4 more feet76.4 ÷4.4 = 17 extra seconds to

walk the edge.


Chapter 9 Test

1. 16 - 2 16 − ( 2 25 + 1 ) − 3

=

16 - 2 • 4 − ( 2 • 5 + 1 ) − 3

=

16 - 8 − ( 10 + 1 ) − 3

= 16 - 8 − 11− 3

=

− 3 − 3

= 1

2.36t

a 2 = 6 t

a

3.20y 4

5 y = 4 • 5 y 4

5 y =

2 • 2 5 y

5 y = 4

4. 6 5 t 3 + t 45t + 80t 3 = 6 t 5 t + t 9 • 5 t + t 16• 5 t = 6 t 5 t + 3 t 5 t + 4 t 5 t = 13t 5 t

5. 588 = 2 2 3 • 7 2 = 2 • 7 3 = 14 3

6. h 11j 7 = h 5 j 3 hj

7. 15 ( 30 − 6 ) - 3 ( 5 2 + 10 ) =15• 15• 2 − 15• 6 − 15 2 − 3 10 = 152 • 2 − 90 − 15 2 − 3 10 =

15 2 − 9 • 1 0 − 15 2 − 3 10 = − 3 10 − 3 10 = - 6 10

8. 9 n 2 + 12n + 4 = ( 3 n + 2 ) 2 = | 3 n + 2 |

9.15a 2 b

c 3

10b 3

c =

3 • 5 • 2 • 5 a 2 b 4

c 4 = 3 • 2 • 5 2 a 2 b 4

c 4 =

5 ab2 6

c 2

10. − 1 4

2 y + 1 3

2 y + 1 2

2 y =

− 1 4

• 3 3

2 y + 1 3

• 4 4

2 y + 1 2

• 6 6

2 y =

− 3 12

2 y + 4 12

2 y + 6 12

2 y =

7 12

2 y


11. 6 y 42y = 6 y•6 • 7 y = 6 2 • 7 y 2 = 6 y 7

12.3 d

24= 3 d

4 • 6 •

6

6 = 3 d 6

4 • 6 2 =

3 d 6 2 • 6

= d 6 4

13. 3 2 x = 129(2x) = 144 Square both sides18x = 144x = 8Check left for student.

14. x 2 + 4 x + 4 − 1 = 2 x

x 2 + 4 x + 4 = 2 x + 1 x2 + 4x + 4 = (2x + 1)2 Square bothsides.

x2

+4x+4 =4x2

+4x+10 =

3x2 - 30 =

3(x + 1)(x -1)x + 1 = 0 or x - 1 = 0x = -1 or x = 1Check x = -1:

( - 1 ) 2 + 4 ( - 1 ) + 4 − 1 = 2 ( - 1 ) 1 - 4 + 4 − 1 = - 2

1 -1 ≠ -2; -1 is not a solution.Check x = 1

( 1 ) 2 + 4 ( 1 ) + 4 − 1 = 2 ( 1 ) 9 − 1 = 2

3 - 1 = 2 ; 1 is a solution.

15. 3 x + 1 = 2 x - 5 9(x+1) = 2x - 5 Square both

sides.9x + 9 = 2x - 57x = -14x = -2Check: 3 - 2 + 1 = 2 ( − 2 ) - 5 3 − 1 = − 9 The radicand is negative. There is

no real solution.

16. 6x2 +13x+6 =0a = 6, b = 13, c = 6

x = − 13± ( 13) 2 − 4 ( 6 ) ( 6 )

2 ( 6 )

x = − 13± 169 − 14412

= − 13± 2512

x = − 13± 5

12Solution 1 Solution 2

x = − 13 + 5

12 x =

− 13 - 5 12

x = -8/12 = -2/3 x = -18/12 = -3/2

17. 3y2 - 3y -1 = 0a = 3, b = -3, c = -1

y = − ( - 3 ) ± ( - 3 ) 2 − 4 ( 3 ) ( - 1 )

2 ( 3 )

y = 3 ± 9 + 12

6 = 3 ± 21

6

18.4 feet

7 feet

d

42 + 72 = d2

d2 = 16 + 49 d2 = 65d2 - 65 = 0a = 1, b = 0, c = -65

d = 0 ± 0 - 4 ( 1 ) ( - 65)

2 ( 1 )

d = ± 260

2 = ± 16. 12

2 d = 8.06 feet (Disregard the

negative value. It is invalid.)


chapter 9 radical expressions and negative so this is not defined using 4. 25 121 5 …€¦ ·...

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