13.1.9: Chapter 9

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Be Prepared

9.1

$8 \sqrt{2}$

9.2

$\frac{4 \sqrt{10}}{5}$

9.3

${(3 x - 2)}^{2}$

9.4

$x^{2} + 18 x + 81$

9.5

$y - 7^{2}$

9.6

$5 n + 4^{2}$

9.7

$28$

9.8

$6 \sqrt{3}$

9.9

$5 \sqrt{2}$

9.10

$(y^{2} + 4) (y^{2} - 5)$

9.11

$(y - 1) (y + 1)$

9.12

ⓐ $x^{\frac{3}{4}}$ ; ⓑ $x^{\frac{2}{3}}$ ; ⓒ $x^{- 2}$

9.13

$- 51, - 49$

9.14

$x = \frac{2}{3}$

9.15

$13 inches$

9.16

$.$

9.17

$x = \frac{1}{2}, x = - 2$

9.18

$1$

9.19

$.$

9.20

${(y^{- 7})}^{2}$

9.21

$2 {(x^{- 4})}^{2}$

9.22

$x = \frac{3}{2}$

9.23

$y = - 3, y = \frac{5}{2}$

9.24

$(- \infty, - 4) \cup (2, \infty)$

Try It

9.1

$x = 4 \sqrt{3}, x = −4 \sqrt{3}$

9.2

$y = 3 \sqrt{3}, y = −3 \sqrt{3}$

9.3

$x = 7, x = −7$

9.4

$m = 4, m = −4$

9.5

$c = 2 \sqrt{3} i, c = −2 \sqrt{3} i$

9.6

$c = 2 \sqrt{6} i, c = −2 \sqrt{6} i$

9.7

$x = 2 \sqrt{10}, x = −2 \sqrt{10}$

9.8

$y = 2 \sqrt{7}, y = −2 \sqrt{7}$

9.9

$r = \frac{6 \sqrt{5}}{5}, r = - \frac{6 \sqrt{5}}{5}$

9.10

$t = \frac{8 \sqrt{3}}{3}, t = - \frac{8 \sqrt{3}}{3}$

9.11

$a = 3 + 3 \sqrt{2}, a = 3 - 3 \sqrt{2}$

9.12

$b = −2 + 2 \sqrt{10}, b = −2 - 2 \sqrt{10}$

9.13

$x = \frac{1}{2} + \frac{\sqrt{5}}{2}$ , $x = \frac{1}{2} - \frac{\sqrt{5}}{2}$

9.14

$y = - \frac{3}{4} + \frac{\sqrt{7}}{4}, y = - \frac{3}{4} - \frac{\sqrt{7}}{4}$

9.15

$a = 5 + 2 \sqrt{5}, a = 5 - 2 \sqrt{5}$

9.16

$b = −3 + 4 \sqrt{2}, b = −3 - 4 \sqrt{2}$

9.17

$r = - \frac{4}{3} + \frac{2 \sqrt{2} i}{3}, r = - \frac{4}{3} - \frac{2 \sqrt{2} i}{3}$

9.18

$t = 4 + \frac{\sqrt{10} i}{2}, t = 4 - \frac{\sqrt{10} i}{2}$

9.19

$m = \frac{7}{3}, m = −1$

9.20

$n = - \frac{3}{4}, n = - \frac{7}{4}$

9.21

ⓐ ${(a - 10)}^{2}$ ⓑ ${(b - \frac{5}{2})}^{2}$
ⓒ ${(p + \frac{1}{8})}^{2}$

9.22

ⓐ ${(b - 2)}^{2}$ ⓑ ${(n + \frac{13}{2})}^{2}$
ⓒ ${(q - \frac{1}{3})}^{2}$

9.23

$x = −5, x = −1$

9.24

$y = 1, y = 9$

9.25

$y = 5 \pm \sqrt{10} i$

9.26

$z = −4 + \sqrt{3} i, z = −4 - \sqrt{3} i$

9.27

$x = 8 + 4 \sqrt{3}, x = 8 - 4 \sqrt{3}$

9.28

$y = −4 + 3 \sqrt{3}, y = −4 - 3 \sqrt{3}$

9.29

$a = −7, a = 3$

9.30

$b = −10, b = 2$

9.31

$p = \frac{5}{2} + \frac{\sqrt{61}}{2}, p = \frac{5}{2} - \frac{\sqrt{61}}{2}$

9.32

$q = \frac{7}{2} + \frac{\sqrt{37}}{2}, q = \frac{7}{2} - \frac{\sqrt{37}}{2}$

9.33

$c = −9, c = 3$

9.34

$d = 11, d = −7$

9.35

$m = −7, m = −1$

9.36

$n = −2, n = 8$

9.37

$r = - \frac{7}{3}, r = 3$

9.38

$t = - \frac{5}{2}, t = 2$

9.39

$x = - \frac{3}{8} + \frac{\sqrt{41}}{8}, x = - \frac{3}{8} - \frac{\sqrt{41}}{8}$

9.40

$y = \frac{5}{3} + \frac{\sqrt{10}}{3}, y = \frac{5}{3} - \frac{\sqrt{10}}{3}$

9.41

$y = 1, y = \frac{2}{3}$

9.42

$z = 1, z = - \frac{3}{2}$

9.43

$a = −3, a = 5$

9.44

$b = −6, b = −4$

9.45

$m = \frac{−6 + \sqrt{15}}{3}, m = \frac{−6 - \sqrt{15}}{3}$

9.46

$n = \frac{−2 + 2 \sqrt{6}}{5}, n = \frac{−2 - 2 \sqrt{6}}{5}$

9.47

$a = \frac{1}{4} + \frac{\sqrt{31}}{4} i, a = \frac{1}{4} - \frac{\sqrt{31}}{4} i$

9.48

$b = - \frac{1}{5} + \frac{\sqrt{19}}{5} i, b = - \frac{1}{5} - \frac{\sqrt{19}}{5} i$

9.49

$x = −1 + \sqrt{6}, x = −1 - \sqrt{6}$

9.50

$y = 1 + \sqrt{2}, y = 1 - \sqrt{2}$

9.51

$c = \frac{2 + \sqrt{7}}{3}, c = \frac{2 - \sqrt{7}}{3}$

9.52

$d = \frac{9 + \sqrt{33}}{4}, d = \frac{9 - \sqrt{33}}{4}$

9.53

$r = −5$

9.54

$t = \frac{4}{5}$

9.55

ⓐ 2 complex solutions; ⓑ 2 real solutions; ⓒ 1 real solution

9.56

ⓐ 2 real solutions; ⓑ 2 complex solutions; ⓒ 1 real solution

9.57

ⓐ factoring; ⓑ Square Root Property; ⓒ Quadratic Formula

9.58

ⓐ Quadratic Forumula;
ⓑ Factoring or Square Root Property ⓒ Square Root Property

9.59

$x = \sqrt{2}, x = − \sqrt{2}, x = 2, x = −2$

9.60

$x = \sqrt{7}, x = − \sqrt{7}, x = 2, x = −2$

9.61

$x = 3, x = 1$

9.62

$y = −1, y = 1$

9.63

$x = 9, x = 16$

9.64

$x = 4, x = 16$

9.65

$x = −8, x = 343$

9.66

$x = 81, x = 625$

9.67

$x = \frac{4}{3} x = 2$

9.68

$x = \frac{2}{5}, x = \frac{3}{4}$

9.69

The two consecutive odd integers whose product is 99 are 9, 11, and −9, −11

9.70

The two consecutive even integers whose product is 128 are 12, 14 and −12, −14.

9.71

The height of the triangle is 12 inches and the base is 76 inches.

9.72

The height of the triangle is 11 feet and the base is 20 feet.

9.73

The length of the garden is approximately 18 feet and the width 11 feet.

9.74

The length of the tablecloth is approximatel 11.8 feet and the width 6.8 feet.

9.75

The length of the flag pole’s shadow is approximately 6.3 feet and the height of the flag pole is 18.9 feet.

9.76

The distance between the opposite corners is approximately 7.2 feet.

9.77

The arrow will reach 180 feet on its way up after 3 seconds and again on its way down after approximately 3.8 seconds.

9.78

The ball will reach 48 feet on its way up after approximately .6 second and again on its way down after approximately 5.4 seconds.

9.79

The speed of the jet stream was 100 mph.

9.80

The speed of the jet stream was 50 mph.

9.81

Press #1 would take 12 hours, and Press #2 would take 6 hours to do the job alone.

9.82

The red hose take 6 hours and the green hose take 3 hours alone.

9.83

$This figure shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (0, 0).$

9.84

9.85

ⓐ up; ⓑ down

9.86

ⓐ down; ⓑ up

9.87

ⓐ $x = 2;$ ⓑ $(2, −7)$

9.88

ⓐ $x = 1;$ ⓑ $(1, −5)$

9.89

y-intercept: $(0, −8)$ x-intercepts $(−4, 0), (2, 0)$

9.90

y-intercept: $(0, −12)$ x-intercepts $(−2, 0), (6, 0)$

9.91

y-intercept: $(0, 4)$ no x-intercept

9.92

y-intercept: $(0, −5)$ x-intercepts $(−1, 0), (5, 0)$

9.93

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 9). The y-intercept of the parabola is the point (0, negative 8). The x-intercepts of the parabola are the points (negative 4, 0) and (4, 0).$

9.94

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 15. The axis of symmetry, x equals 4, is graphed as a dashed line. The parabola has a vertex at (4, negative 4). The y-intercept of the parabola is the point (0, 12). The x-intercepts of the parabola are the points (2, 0) and (6, 0).$

9.95

$This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (2, 0). The y-intercept (0, negative 12) is plotted as well as the axis of symmetry, x equals 2.$

9.96

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 30 to 20. The y-axis of the plane runs from negative 10 to 40. The parabola has a vertex at (negative 3, 0). The y-intercept (0, 36) is plotted as well as the axis of symmetry, x equals negative 3.$

9.97

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 4. The y-axis of the plane runs from negative 1 to 5. The parabola has a vertex at (1, 2). The y-intercept (0, 3) is plotted as is the line of symmetry, x equals 1.$

9.98

$This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 2. The y-axis of the plane runs from negative 5 to 1. The parabola has a vertex at (negative 1, negative 2). The y-intercept (0, negative 4) is plotted as is the line of symmetry, x equals negative 1.$

9.99

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, negative 2). The y-intercept of the parabola is the point (0, 3). The x-intercepts of the parabola are approximately (negative 1.6, 0) and (negative 0.4, 0).$

9.100

$This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The axis of symmetry, x equals negative 1, is graphed as a dashed line. The parabola has a vertex at (negative 1, 8). The y-intercept of the parabola is the point (0, 5). The x-intercepts of the parabola are approximately (negative 2.6, 0) and (0.6, 0).$

9.101

The minimum value of the quadratic function is −4 and it occurs when x = 4.

9.102

The maximum value of the quadratic function is 5 and it occurs when x = 2.

9.103

It will take 4 seconds for the stone to reach its maximum height of 288 feet.

9.104

It will 6.5 seconds for the rocket to reach its maximum height of 676 feet.

9.105

ⓐ

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.$

ⓑ The graph of $g (x) = x^{2} + 1$ is the same as the graph of $f (x) = x^{2}$ but shifted up 1 unit. The graph of $h (x) = x^{2} - 1$ is the same as the graph of $f (x) = x^{2}$ but shifted down 1 unit.

9.106

ⓐ

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.$

ⓑ The graph of $h (x) = x^{2} + 6$ is the same as the graph of $f (x) = x^{2}$ but shifted up 6 units. The graph of $h (x) = x^{2} - 6$ is the same as the graph of $f (x) = x^{2}$ but shifted down 6 units.

9.107

$This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 5 units.$

9.108

$This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.$

9.109

ⓐ

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 2 units, and the right curve has been moved to the right 2 units.$

ⓑ The graph of $g (x) = {(x + 2)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted left 2 units. The graph of $h (x) = {(x - 2)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shift right 2 units.

9.110

ⓐ

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 5 units, and the right curve has been moved to the right 5 units.$

ⓑ The graph of $g (x) = {(x + 5)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted left 5 units. The graph of $h (x) = {(x - 5)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted right 5 units.

9.111

$This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved right 4 units.$

9.112

$This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The right curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 6 units.$

9.113

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 2 units to the left to produce f of x equals the quantity of x plus 2 squared. The final curve is produced by moving down 3 units to produce f of x equals the quantity of x plus 2 squared minus 3.$

9.114

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 3 units to the right to produce f of x equals the quantity of x minus 3 squared. The final curve is produced by moving up 1 unit to produce f of x equals the quantity of x minus 3squared plus 1.$

9.115

$The graph shows the upward-opening parabola on the x y-coordinate plane of f of x equals x squared that has a vertex of (0, 0). Other points given on the curve are located at (negative 2, 4) (negative 1, 1), (1, 1), and (2,4). Also shown is a downward-opening parabola of f of x equals negative 3 times x squared. It has a vertex of (0,0) with other points at (negative 1, negative 3) and (1, negative 3)$

9.116

$This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).$

9.117

$f (x) = −4 {(x + 1)}^{2} + 5$

9.118

$f (x) = 2 {(x - 2)}^{2} - 5$

9.119

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the left has been moved 1 unit to the left to produce f of x equals the quantity of x plus 1 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x plus 1 squared minus 4.$

9.120

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The curve to the right has been moved 4 units to the right to produce f of x equals the quantity of x minus 4 squared. The third graph has been moved down 4 units to produce f of x equals the quantity of x minus 4 squared minus 4.$

9.121

$This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).$

9.122

$This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).$

9.123

ⓐ $f (x) = 3 {(x - 1)}^{2} + 2$
ⓑ

$The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.$

9.124

ⓐ $f (x) = −2 {(x - 2)}^{2} + 1$
ⓑ

$The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.$

9.125

$f (x) = {(x - 3)}^{2} - 4$

9.126

$f (x) = {(x + 3)}^{2} - 1$

9.127

ⓐ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.$

ⓑ $(−4, 2)$

9.128

ⓐ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).$

ⓑ $(− \infty, 2] \cup [6, \infty)$

9.129

ⓐ

$A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.$

ⓑ $(−5, −1)$

9.130

ⓐ

$A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.$

ⓑ $(− \infty, 2] \cup [8, \infty)$

9.131

$(− \infty, −4] \cup [2, \infty)$

9.132

$[−3, 5]$

9.133

$[−1 - \sqrt{2}, −1 + \sqrt{2}]$

9.134

$(− \infty, 4 - \sqrt{2}) \cup (4 + \sqrt{2}, \infty)$

9.135

ⓐ $(− \infty, \infty)$
ⓑ no solution

9.136

ⓐ no solution
ⓑ $(− \infty, \infty)$

Section 9.1 Exercises

$a = \pm 7$

$r = \pm 2 \sqrt{6}$

$u = \pm 10 \sqrt{3}$

$m = \pm 3$

$x = \pm 6$

11.

$x = \pm 5 i$

13.

$x = \pm 3 \sqrt{7} i$

15.

$x = \pm 9$

17.

$a = \pm 2 \sqrt{5}$

19.

$p = \pm \frac{4 \sqrt{7}}{7}$

21.

$y = \pm \frac{4 \sqrt{10}}{5}$

23.

$u = 14, u = −2$

25.

$m = 6 \pm 2 \sqrt{5}$

27.

$r = \frac{1}{2} \pm \frac{\sqrt{3}}{2}$

29.

$y = - \frac{2}{3} \pm \frac{2 \sqrt{2}}{9}$

31.

$a = 7 \pm 5 \sqrt{2}$

33.

$x = −3 \pm 2 \sqrt{2}$

35.

$c = - \frac{1}{5} \pm \frac{3 \sqrt{3}}{5} i$

37.

$x = \frac{3}{4} \pm \frac{\sqrt{7}}{2} i$

39.

$m = 2 \pm 2 \sqrt{2}$

41.

$x = 3 \pm 2 \sqrt{3}$

43.

$x = - \frac{3}{5}, x = \frac{9}{5}$

45.

$x = - \frac{7}{6}, x = \frac{11}{6}$

47.

$r = \pm 4$

49.

$a = 4 \pm 2 \sqrt{7}$

51.

$w = 1, w = \frac{5}{3}$

53.

$a = \pm 3 \sqrt{2}$

55.

$p = \frac{1}{3} \pm \frac{\sqrt{7}}{3}$

57.

$m = \pm 2 \sqrt{3} i$

59.

$u = 7 \pm 6 \sqrt{2}$

61.

$m = 4 \pm 2 \sqrt{3}$

63.

$x = −3, x = −7$

65.

$c = \pm \frac{5 \sqrt{6}}{6}$

67.

$x = 6 \pm 2 i$

69.

Answers will vary.

Section 9.2 Exercises

71.

ⓐ ${(m - 12)}^{2}$ ⓑ ${(x - \frac{11}{2})}^{2}$
ⓒ ${(p - \frac{1}{6})}^{2}$

73.

ⓐ ${(p - 11)}^{2}$ ⓑ ${(y + \frac{5}{2})}^{2}$
ⓒ ${(m + \frac{1}{5})}^{2}$

75.

$u = −3, u = 1$

77.

$x = −1, x = 21$

79.

$m = −2 \pm 2 \sqrt{10} i$

81.

$r = −3 \pm \sqrt{2} i$

83.

$a = 5 \pm 2 \sqrt{5}$

85.

$x = - \frac{5}{2} \pm \frac{\sqrt{33}}{2}$

87.

$u = 1, u = 13$

89.

$r = −2, r = 6$

91.

$v = \frac{9}{2} \pm \frac{\sqrt{89}}{2}$

93.

$x = 5 \pm \sqrt{30}$

95.

$x = −7, x = 3$

97.

$x = −5, x = −1$

99.

$m = −11, m = 1$

101.

$n = - 1 \pm \sqrt{14}$

103.

$c = −2, c = \frac{3}{2}$

105.

$x = −5, x = \frac{3}{2}$

107.

$p = - \frac{7}{4} \pm \frac{\sqrt{161}}{4}$

109.

$x = \frac{3}{10} \pm \frac{\sqrt{191}}{10} i$

111.

Answers will vary.

Section 9.3 Exercises

113.

$m = −1, m = \frac{3}{4}$

115.

$p = \frac{1}{2}, p = 3$

117.

$p = −4, p = −3$

119.

$r = −3, r = 11$

121.

$u = \frac{−7 \pm \sqrt{73}}{6}$

123.

$a = \frac{3 \pm \sqrt{3}}{2}$

125.

$x = −4 \pm 2 \sqrt{5}$

127.

$y = −2, y = \frac{1}{3}$

129.

$x = - \frac{3}{4} \pm \frac{\sqrt{15}}{4} i$

131.

$x = \frac{3}{8} \pm \frac{\sqrt{7}}{8} i$

133.

$v = 2 \pm \sqrt{13}$

135.

$y = \frac{3 \pm \sqrt{193}}{2}$

137.

$m = −1, m = \frac{3}{4}$

139.

$b = \frac{−2 \pm \sqrt{22}}{6}$

141.

$c = - \frac{3}{4}$

143.

$q = - \frac{3}{5}$

145.

ⓐ $no real solutions$ ⓑ $1$
ⓒ $2$

147.

ⓐ $1$ ⓑ $no real solutions$
ⓒ $2$

149.

ⓐ $factor$
ⓑ $square root$
ⓒ $Quadratic Formula$

151.

ⓐ $Quadratic Formula$
ⓑ $square root$
ⓒ $factor$

153.

Answers will vary.

Section 9.4 Exercises

155.

$x = \pm \sqrt{3}, x = \pm 2$

157.

$x = \pm \sqrt{15}, x = \pm \sqrt{2} i$

159.

$x = \pm 1, x = \frac{\pm \sqrt{6}}{2}$

161.

$x = \pm \sqrt{3}, x = \pm \frac{\sqrt{2}}{2}$

163.

$x = −1, x = 12$

165.

$x = - \frac{5}{3}, x = 0$

167.

$x = 0, x = \pm \sqrt{3}$

169.

$x = \pm \frac{\sqrt{22}}{2}, x = \pm \sqrt{7}$

171.

$x = 25$

173.

$x = 4$

175.

$x = \frac{1}{4}$

177.

$x = \frac{1}{25}, x = \frac{9}{4}$

179.

$x = −1, x = −512$

181.

$x = 8, x = −216$

183.

$x = \frac{27}{8}, x = - \frac{64}{27}$

185.

$x = \frac{27}{512}, x = 125$

187.

$x = 1, x = 49$

189.

$x = −2, x = - \frac{3}{5}$

191.

$x = −2, x = \frac{4}{3}$

193.

Answers will vary.

Section 9.5 Exercises

195.

Two consecutive odd numbers whose product is 255 are 15 and 17, and −15 and −17.

197.

The first and second consecutive odd numbers are 24 and 26, and −26 and −24.

199.

Two consecutive odd numbers whose product is 483 are 21 and 23, and −21 and −23.

201.

The width of the triangle is 5 inches and the height is 18 inches.

203.

The base is 24 feet and the height of the triangle is 10 feet.

205.

The length of the driveway is 15.0 feet and the width is 3.3 feet.

207.

The length of table is 8 feet and the width is 3 feet.

209.

The lengths of the three sides of the triangle are 1.7, 3, and 3.5 ft.

211.

The length of the diagonal fencing is 7.3 yards.

213.

The ladder will reach 24.5 feet on the side of the house.

215.

The rocket will reach 1200 feet on its way up at 1.97 seconds and on its way down at 38.03 seconds.

217.

The bullet will take 70 seconds to hit the ground.

219.

The speed of the wind was 49 mph.

221.

The speed of the current was 4.3 mph.

223.

The less experienced painter takes 6 hours and the experienced painter takes 3 hours to do the job alone.

225.

Machine #1 takes 3.6 hours and Machine #2 takes 4.6 hours to do the job alone.

227.

Answers will vary.

Section 9.6 Exercises

229.

231.

233.

ⓐ down ⓑ up

235.

ⓐ down ⓑ up

237.

ⓐ $x = −4$ ; ⓑ $(−4, −17)$

239.

ⓐ $x = 1$ ; ⓑ $(1, 6)$

241.

y-intercept: $(0, 6);$ x-intercept $(−1, 0), (−6, 0)$

243.

y-intercept: $(0, 12);$ x-intercept $(−2, 0), (−6, 0)$

245.

y-intercept: $(0, −19);$ x-intercept: none

247.

y-intercept: $(0, 13);$ x-intercept: none

249.

y-intercept: $(0, 25);$ x-intercept $(\frac{5}{2}, 0)$

251.

y-intercept: $(0, −9);$ x-intercept $(−3, 0)$

253.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 4). The y-intercept, point (0, 5), is plotted as are the x-intercepts, (negative 5, 0) and (negative 1, 0).$

255.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, negative 1). The y-intercept, point (0, 3), is plotted as are the x-intercepts, (negative 3, 0) and (negative 1, 0).$

257.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (negative 2 thirds, 0). The y-intercept, point (0, 4), is plotted. The axis of symmetry, x equals negative 2 thirds, is plotted as a dashed vertical line.$

259.

$This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 15 to 10. The parabola has a vertex at (1, negative 6). The y-intercept, point (0, negative 7), is plotted. The axis of symmetry, x equals 1, is plotted as a dashed vertical line.$

261.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, negative 1). The y-intercept, point (0, 1), is plotted as are the x-intercepts, approximately (0.3, 0) and (1.7, 0). The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.$

263.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 0). This point is the only x-intercept. The y-intercept, point (0, 2), is plotted. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.$

265.

$This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 2, 6). The y-intercept, point (0, 2), is plotted as are the x-intercepts, approximately (negative 4.4, 0) and (0.4, 0). The axis of symmetry is the vertical line x equals 2, plotted as a dashed line.$

267.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (1, 3). The y-intercept, point (0, 8), is plotted; there are no x-intercepts. The axis of symmetry is the vertical line x equals 1, plotted as a dashed line.$

269.

$This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The parabola has a vertex at (negative 3, negative 7). The x-intercepts are plotted at the approximate points (negative 4.5, 0) and (negative 1.5, 0). The axis of symmetry is the vertical line x equals negative 3, plotted as a dashed line.$

271.

The minimum value is $- \frac{9}{8}$ when $x = - \frac{1}{4} .$

273.

The minimum value is 6 when x = 3.

275.

The maximum value is 16 when x = 0.

277.

In 5.3 sec the arrow will reach maximum height of 486 ft.

279.

In 3.4 seconds the ball will reach its maximum height of 185.6 feet.

281.

A selling price of $20 per computer will give the maximum revenue of $400.

283.

A selling price of $35 per pair of boots will give a maximum revenue of $1,225.

285.

The length of one side along the river is 120 feet and the maximum are is 7,200 square feet.

287.

The maximum area of the patio is 800 feet.

289.

Answers will vary.

291.

Answers will vary.

Section 9.7 Exercises

293.

ⓐ

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.$

ⓑ The graph of $g (x) = x^{2} + 4$ is the same as the graph of $f (x) = x^{2}$ but shifted up 4 units. The graph of $h (x) = x^{2} - 4$ is the same as the graph of $f (x) = x^{2}$ but shift down 4 units.

295.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).$

297.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).$

299.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).$

301.

ⓐ

$This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The graph to the right is shifted 3 units to the right to produce g of x equals the quantity of x minus 3 squared. The graph the left is shifted 3 units to the left to produce h of x equals the quantity of x plus 3 squared.$

ⓑ The graph of $g (x) = {(x - 3)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted right 3 units. The graph of $h (x) = {(x + 3)}^{2}$ is the same as the graph of $f (x) = x^{2}$ but shifted left 3 units.

303.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).$

305.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).$

307.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).$

309.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).$

311.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).$

313.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).$

315.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).$

317.

$This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).$

319.

$This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).$

321.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).$

323.

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).$

325.

$f (x) = −3 {(x + 2)}^{2} + 7$

327.

$f (x) = 3 {(x + 1)}^{2} - 4$

329.

ⓐ $f (x) = {(x + 3)}^{2} - 4$
ⓑ

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.$

331.

ⓐ $f (x) = {(x + 2)}^{2} - 1$
ⓑ

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.$

333.

ⓐ $f (x) = {(x - 3)}^{2} + 6$
ⓑ

$This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.$

335.

ⓐ $f (x) = − {(x - 4)}^{2} + 0$
ⓑ

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.$

337.

ⓐ $f (x) = − {(x + 2)}^{2} + 6$
ⓑ

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.$

339.

ⓐ $f (x) = 5 {(x - 1)}^{2} + 3$
ⓑ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.$

341.

ⓐ $f (x) = 2 {(x - 1)}^{2} - 1$
ⓑ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.$

343.

ⓐ $f (x) = −2 {(x - 2)}^{2} - 2$
ⓑ

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.$

345.

ⓐ $f (x) = 2 {(x + 1)}^{2} + 4$
ⓑ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.$

347.

ⓐ $f (x) = − {(x - 1)}^{2} - 3$
ⓑ

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.$

349.

ⓒ

351.

ⓔ

353.

ⓓ

355.

ⓖ

357.

$f (x) = {(x + 1)}^{2} - 5$

359.

$f (x) = 2 {(x - 1)}^{2} - 3$

361.

Answers will vary.

Section 9.8 Exercises

363.

ⓐ

$The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).$

ⓑ $(− \infty, −5) \cup (−1, \infty)$

365.

ⓐ

$The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).$

ⓑ $[−3, −1]$

367.

ⓐ

$The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).$

ⓑ $(− \infty, −6] \cup [3, \infty)$

369.

ⓐ

$The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).$

ⓑ $[−3, 4]$

371.

$(− \infty, −4] \cup [1, \infty)$

373.

$(2, 5)$

375.

$(− \infty, −5) \cup (−3, \infty)$

377.

$[2 - \sqrt{2}, 2 + \sqrt{2}]$

379.

$(− \infty, 5 - \sqrt{6}) \cup (5 + \sqrt{6}, \infty)$

381.

$[\frac{2}{3}, \frac{5}{2}]$

383.

$[− \frac{1}{2}, 4]$

385.

$(− \infty, \infty) .$

387.

no solution

389.

$(− \infty, \infty) .$

391.

Answers will vary.

393.

Answers will vary.

Review Exercises

395.

$y = \pm 12$

397.

$a = \pm 5$

399.

$r = \pm 4 \sqrt{2} i$

401.

$w = \pm 5 \sqrt{3}$

403.

$p = 1, p = 9$

405.

$x = \frac{1}{4} \pm \frac{\sqrt{3}}{4}$

407.

$n = 4 \pm 10 \sqrt{2}$

409.

$n = −5 \pm 2 \sqrt{3}$

411.

${(x + 11)}^{2}$

413.

${(a - \frac{3}{2})}^{2}$

415.

$d = −13, −1$

417.

$m = −3 \pm 10 i$

419.

$v = 7 \pm 3 \sqrt{2}$

421.

$m = −9, −1$

423.

$a = \frac{3}{2} \pm \frac{\sqrt{41}}{2}$

425.

$u = −6 \pm 3 \sqrt{2}$

427.

$p = 0, 6$

429.

$y = - \frac{1}{2}, 2$

431.

$c = - \frac{1}{3} \pm \frac{2 \sqrt{7}}{3}$

433.

$x = \frac{3}{2} \pm \frac{1}{2} i$

435.

$x = \frac{1}{4}, 1$

437.

$r = −6, 7$

439.

$v = - \frac{5}{4}, 1$

441.

$m = \frac{−4 \pm \sqrt{10}}{3}$

443.

$a = \frac{5}{12} \pm \frac{\sqrt{23}}{12} i$

445.

$u = 5 \pm \sqrt{22}$

447.

$p = \frac{4 \pm \sqrt{6}}{5}$

449.

$c = - \frac{1}{2}$

451.

453.

455.

$x = \pm \sqrt{2}, \pm 2 \sqrt{3}$

457.

$x = \pm 1, \pm \frac{1}{2}$

459.

$x = 16$

461.

$x = 64, 216$

463.

$x = −2, \frac{4}{3}$

465.

Two consecutive even numbers whose product is 624 are 24 and 26, and −24 and −26.

467.

The height is 14 inches and the width is 10 inches.

469.

The length of the diagonal is 3.6 feet.

471.

The width of the serving table is 4.7 feet and the length is 16.1 feet.

$Four tables arranged end-to-end are shown. Together, they have an area of 75 feet. The short side measures w and the long side measures 3 times w plus 2.$

473.

The speed of the wind was 30 mph.

475.

One man takes 3 hours and the other man 6 hours to finish the repair alone.

477.

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).$

479.

ⓐ up ⓑ down

481.

$x = 2; (2, −7)$

483.

$\begin{matrix} y : (0, 15) \\ x : (3, 0), (5, 0) \end{matrix}$

485.

$\begin{matrix} y : (0, −46) \\ x : none \end{matrix}$

487.

$\begin{matrix} y : (0, 64) \\ x : (−8, 0) \end{matrix}$

489.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).$

491.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).$

493.

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).$

495.

The maximum value is 2 when x = 2.

497.

The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.

499.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).$

501.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).$

503.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).$

505.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).$

507.

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).$

509.

$f (x) = 2 {(x - 1)}^{2} - 6$

511.

ⓐ $f (x) = 3 {(x - 1)}^{2} - 4$
ⓑ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).$

513.

ⓐ $f (x) = 2 {(x + 1)}^{2} + 4$
ⓑ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).$

515.

ⓐ $f (x) = −3 {(x + 2)}^{2} + 7$
ⓑ

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).$

517.

$f (x) = {(x + 1)}^{2} - 5$

519.

ⓐ

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).$

ⓑ $(- \infty, −2) \cup (3, \infty)$

521.

ⓐ

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).$

ⓑ $[−2, 1]$

523.

$(2, 4)$

525.

$[3 - \sqrt{5}, 3 + \sqrt{5}]$

527.

no solution

Practice Test

529.

$w = −2, w = −8$

531.

$m = 1, m = \frac{3}{2}$

533.

$y = \frac{2}{3}$

535.

2 complex

537.

$y = 1, y = −27$

539.

541.

$This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).$

543.

$This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).$

$f (x) = 2 {(x - 1)}^{2} - 6$

545.

$(− \infty, - \frac{5}{2}) \cup (2, \infty)$

547.

The diagonal is 4.4 units long.

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Section 9.7 Exercises

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