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Elementary Algebra

Chapter 10

Elementary AlgebraChapter 10
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Be Prepared

Try It

10.1

x=9,x=−9x=9,x=−9

10.2

y=11,y=−11y=11,y=−11

10.3

x=52,x=−52x=52,x=−52

10.4

y=33,y=−33y=33,y=−33

10.5

x=7,x=−7x=7,x=−7

10.6

z=6,z=−6z=6,z=−6

10.7

no real solution

10.8

no real solution

10.9

x=210,x=−210x=210,x=−210

10.10

y=27,y=−27y=27,y=−27

10.11

r=655,r=655r=655,r=655

10.12

t=833,t=833t=833,t=833

10.13

q=−6,q=−4q=−6,q=−4

10.14

r=8,r=−2r=8,r=−2

10.15

a=3+32,a=332a=3+32,a=332

10.16

b=−2+210,b=−2210b=−2+210,b=−2210

10.17

x=13+53,x=1353x=13+53,x=1353

10.18

y=34+74,y=3474y=34+74,y=3474

10.19

a=5+25,a=525a=5+25,a=525

10.20

b=3+42,b=342b=3+42,b=342

10.21

no real solution

10.22

no real solution

10.23

x=3+23,x=323x=3+23,x=323

10.24

y=−6+42,y=−642y=−6+42,y=−642

10.25

m=7,m=−3m=7,m=−3

10.26

n=34,n=74n=34,n=74

10.27

(y+6)2(y+6)2

10.28

(z+4)2(z+4)2

10.29

(a10)2(a10)2

10.30

(b2)2(b2)2

10.31

(m52)2(m52)2

10.32

(n+132)2(n+132)2

10.33

(p+18)2(p+18)2

10.34

(q13)2(q13)2

10.35

c=−5,c=1c=−5,c=1

10.36

d=−9,d=−1d=−9,d=−1

10.37

r=−2,r=6r=−2,r=6

10.38

t=−1,t=11t=−1,t=11

10.39

no real solution

10.40

no real solution

10.41

x=8±43x=8±43

10.42

y=−4±33y=−4±33

10.43

a=−7,a=3a=−7,a=3

10.44

b=−10,b=−2b=−10,b=−2

10.45

p=52±612p=52±612

10.46

q=72±372q=72±372

10.47

c=−3±42c=−3±42

10.48

d=−7,d=11d=−7,d=11

10.49

m=−4±25m=−4±25

10.50

n=−2,8n=−2,8

10.51

r=73,r=3r=73,r=3

10.52

t=52,t=2t=52,t=2

10.53

x=38±2018x=38±2018

10.54

y=310±20910y=310±20910

10.55

y=23,y=1y=23,y=1

10.56

z=32,z=1z=32,z=1

10.57

a=−3,a=5a=−3,a=5

10.58

b=−6,b=−4b=−6,b=−4

10.59

p=−4±62p=−4±62

10.60

q=11±6110q=11±6110

10.61

m=−6±153m=−6±153

10.62

n=−2±265n=−2±265

10.63

no real solution

10.64

no real solution

10.65

x=−1±6x=−1±6

10.66

y=23,y=1y=23,y=1

10.67

c=2±73c=2±73

10.68

d=23,d=0d=23,d=0

10.69

r=−5r=−5

10.70

t=45t=45

10.71

no real solutions 2 1 no real solutions

10.72

2 no real solutions 1 2

10.73

factor Square Root Property Quadratic Formula

10.74

Quadratic Formula factoring Square Root Property

10.75

Two consecutive odd numbers whose product is 99 are 9 and 11, and −9−9 and −11−11.

10.76

Two consecutive even numbers whose product is 168 are 12 and 14,and −12−12 and −14−14.

10.77

The height of the triangle is 8 inches and the width is 52 inches.

10.78

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

10.79

The length of the shadow is 6.3 feet and the length of the flag pole is 18.9 ft.

10.80

The distance to the opposite corner is 3.2.

10.81

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

10.82

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

10.83

The arrow will reach 180 on its way up in 3 seconds, and on the way down in 3.8 seconds.

10.84

The ball will reach 48 feet on its way up in .6 seconds and on the way down in 5.5 seconds.

10.87

up down

10.88

down up

10.89

x=2x=2 (2,−7)(2,−7)

10.90

x=1x=1 (1,−5)(1,−5)

10.91

y:(0,−8);x:(−4,0),(2,0)y:(0,−8);x:(−4,0),(2,0)

10.92

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

10.93

y:(0,4);x:noney:(0,4);x:none

10.94

y:(0,−5);x:(5,0)(−1,0)y:(0,−5);x:(5,0)(−1,0)

10.95

y:(0,−36);x:(−6,0)y:(0,−36);x:(−6,0)

10.96

y:(0,4);x:(23,0)y:(0,4);x:(23,0)

10.97

y:(0,−8)y:(0,−8); x:(2,0),(−4,0)x:(2,0),(−4,0);
axis: x=−1x=−1; vertex: (−1,−9)(−1,−9);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, -9). Three points are plotted on the curve at (0, -8), (2, 0) and (-4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.
10.98

y:(0,12);x:(2,0),(6,0);y:(0,12);x:(2,0),(6,0);
axis: x=4;vertex:(4,−4)x=4;vertex:(4,−4);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (4, -4). Three points are plotted on the curve at (0, 12), (2, 0) and (6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.
10.99

y:(0,−12);x:(2,0);y:(0,−12);x:(2,0);
axis: x=2;vertex:(2,0)x=2;vertex:(2,0);

The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -1 to 10. The vertex is at the point (2, 0). One other point is plotted on the curve at (0, -12). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.
10.100

y:(0,1);x:(15,0);y:(0,1);x:(15,0);
axis: x=15;vertex:(15,0)x=15;vertex:(15,0);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (-1 fifth, 0). One other point is plotted on the curve at (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1 fifth.
10.101

y:(0,5);x:none;y:(0,5);x:none;
axis: x=32;vertex:(32,12)x=32;vertex:(32,12);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (3 halves, 1 half). One other point is plotted on the curve at (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 halves.
10.102

y:(0,−1);x:none;y:(0,−1);x:none;
axis: x=0;vertex:(0,−1)x=0;vertex:(0,−1);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (0, -1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 0.
10.103

y:(0,3);x:(−1.6,0),(−0.4,0);y:(0,3);x:(−1.6,0),(−0.4,0);
axis: x=−1;vertex:(−1,−2)x=−1;vertex:(−1,−2);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The vertex is at the point (-1,-2). Three other points are plotted on the curve at (0, 3), (-1.6, 0), (-0.4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.
10.104

y:(0,5);x:(0.6,0),(−2.6,0);y:(0,5);x:(0.6,0),(−2.6,0);
axis: x=−1;vertex:(−1,8)x=−1;vertex:(−1,8);

The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, 8). Three other points are plotted on the curve at (0, 5), (0.6, 0) and (-2.6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.
10.105

The minimum value is −4−4 when x=4x=4.

10.106

The maximum value is 5 when x=2x=2.

10.107

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

10.108

It will take 6.5 seconds to reach the maximum height of 676 feet.

Section 10.1 Exercises

1.

a=±7a=±7

3.

r=±26r=±26

5.

u=±103u=±103

7.

m=±3m=±3

9.

no real solution

11.

a=±25a=±25

13.

p=±477p=±477

15.

x=1,x=−5x=1,x=−5

17.

u=14,u=−2u=14,u=−2

19.

m=6±25m=6±25

21.

r=12±32r=12±32

23.

a=7±52a=7±52

25.

no real solution

27.

m=2±22m=2±22

29.

x=35,x=95x=35,x=95

31.

r=±4r=±4

33.

a=4±27a=4±27

35.

w=1,w=53w=1,w=53

37.

a=±32a=±32

39.

p=13±73p=13±73

41.

no real solution

43.

u=7±62u=7±62

45.

m=4±23m=4±23

47.

x=−3,x=−7x=−3,x=−7

49.

c=±566c=±566

51.

no real solution

53.

4 feet

55.

Answers will vary.

Section 10.2 Exercises

57.

(a+5)2(a+5)2

59.

(m+9)2(m+9)2

61.

(m12)2(m12)2

63.

(p11)2(p11)2

65.

(x92)2(x92)2

67.

(p16)2(p16)2

69.

v=−10,v=4v=−10,v=4

71.

u=−3,u=1u=−3,u=1

73.

c=−1,c=13c=−1,c=13

75.

x=−1,x=21x=−1,x=21

77.

no real solution

79.

no real solution

81.

a=5±25a=5±25

83.

u=1,u=13u=1,u=13

85.

v=92±892v=92±892

87.

x=−7,x=3x=−7,x=3

89.

m=−11,m=1m=−11,m=1

91.

c=−2,c=32c=−2,c=32

93.

p=74±1614p=74±1614

95.

16 feet, 20 feet

97.

−5−5 −5−5 Answers will vary.

Section 10.3 Exercises

99.

m=−1,m=34m=−1,m=34

101.

p=12,p=3p=12,p=3

103.

p=−4,p=−3p=−4,p=−3

105.

r=−3,r=11r=−3,r=11

107.

u=−7±736u=−7±736

109.

a=3±32a=3±32

111.

no real solution

113.

v=−5±652v=−5±652

115.

m=−1,m=34m=−1,m=34

117.

c=34c=34

119.

m=75,m=1m=75,m=1

121.

p=−3,p=9p=−3,p=9

123.

r=−3±898r=−3±898

125.

a=−6±262a=−6±262

127.

b=−2±116b=−2±116

129.

x=−6±424x=−6±424

131.

no real solutions 1
2 no real solutions

133.

1 no real solutions
1 2

135.

factor square root
Quadratic Formula

137.

factor square root
factor

139.

5 seconds, 8 seconds

141.

−20,10−20,10 −20,10−20,10
answers will vary

Section 10.4 Exercises

143.

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

145.

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

147.

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

149.

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

151.

The leg of the right triangle is 1.7 feet and the hypotenuse is 3.4 feet.

153.

The length of the fence is 7.1 units.

155.

The width of the driveway is 10 feet and its length is 35 feet.

157.

The rocket will reach 1,200 feet on its way up in 2 seconds and on the way down in 38 seconds.

159.

70 seconds

161.

answers will vary
answers will vary answers will vary answers will vary

Section 10.5 Exercises

163.
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, 3) and goes through the point (1, 4).
165.

down

167.

up

169.

x=−4x=−4 (−4,−17)(−4,−17)

171.

x=1x=1 (1,6)(1,6)

173.

y:(0,6);x:(−1,0),(−6,0)y:(0,6);x:(−1,0),(−6,0)

175.

y:(0,19);x:noney:(0,19);x:none

177.

y:(0,25);x:(52,0)y:(0,25);x:(52,0)

179.

y:(0,5);x:(−1,0),(−5,0);y:(0,5);x:(−1,0),(−5,0);
axis: x=−3;vertex:(−3,−4)x=−3;vertex:(−3,−4)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -4) and the intercepts (-5, 0), (-1, 0) and (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.
181.

y:(0,3);x:(−1,0),(−3,0);y:(0,3);x:(−1,0),(−3,0);
axis: x=−2;vertex:(−2,−1)x=−2;vertex:(−2,−1)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, -1) and the intercepts (-1, 0), (-3, 0) and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.
183.

y:(0,4)x:(23,0);y:(0,4)x:(23,0);
axis: x=23;vertex:(23,0)x=23;vertex:(23,0)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The parabola has points plotted at the vertex (-2 thirds, 0) and the intercept (0, 4). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2 thirds.
185.

y:(0,−7);x:none;y:(0,−7);x:none;
axis: x=1;vertex:(1,−6)x=1;vertex:(1,−6)

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -15 to 5. The parabola has points plotted at the vertex (1, -6) and the intercept (0, -7). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.
187.

y:(0,1);x:(1.7,0),(0.3,0);y:(0,1);x:(1.7,0),(0.3,0);
axis: x=1;vertex:(1,−1)x=1;vertex:(1,−1)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, -1) and the intercepts (1.7, 0), (0.3, 0) and (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.
189.

y:(0,2)x:(1,0);y:(0,2)x:(1,0);
axis: x=1;vertex:(1,0)x=1;vertex:(1,0)

 This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 0) and the intercept (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.
191.

y:(0,2)x:(−4.4,0),(0.4,0);y:(0,2)x:(−4.4,0),(0.4,0);
axis: x=−2;vertex:(−2,6)x=−2;vertex:(−2,6)

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, 6) and the intercepts (-4.4, 0), (0.4, 0) and (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.
193.

y:(0,8);x:none;y:(0,8);x:none;
axis: x=1;vertex:(1,3)x=1;vertex:(1,3)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 3) and the intercept(0, 8). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.
195.

y:(0,20)x:(−4.5,0),(−1.5,0);y:(0,20)x:(−4.5,0),(−1.5,0);
axis: x=−3;vertex:(−3,−7)x=−3;vertex:(−3,−7)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -7) and the intercepts (-4.5, 0) and (-1.5, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.
197.

The minimum value is 9898 when x=14x=14.

199.

The minimum value is 6 when x=3x=3.

201.

The maximum value is 16 when x=0x=0.

203.

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

205.

20 computers will give the maximum of $400 in receipts.

207.

The length of the side along the river of the corral is 120 feet and the maximum area is 7,200 sq ft.

209.

  1. This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 60. The y-axis of the plane runs from -50 to 500. The parabola has a vertex at (20, 400) and also goes through the points (0, 0) and (40, 0).
  2. (0,0),(40,0)(0,0),(40,0)
211.

Answers will vary.

Review Exercises

213.

x=±10x=±10

215.

m=±210m=±210

217.

a=±5a=±5

219.

no solution

221.

v=±32v=±32

223.

c=±455c=±455

225.

p=1,9p=1,9

227.

u=−1±35u=−1±35

229.

x=14±34x=14±34

231.

m=7±26m=7±26

233.

no solution

235.

m=3±43m=3±43

237.

a=32,34a=32,34

239.

(x+11)2(x+11)2

241.

(m4)2(m4)2

243.

(a32)2(a32)2

245.

(p+25)2(p+25)2

247.

c=1,−21c=1,−21

249.

x=−4,8x=−4,8

251.

no solution

253.

v=7±32v=7±32

255.

m=−9,−1m=−9,−1

257.

a=32±412a=32±412

259.

u=−6±22u=−6±22

261.

p=0,6p=0,6

263.

y=12,2y=12,2

265.

c=13±273c=13±273

267.

x=14,1x=14,1

269.

r=−6,7r=−6,7

271.

v=54,1v=54,1

273.

m=−4±103m=−4±103

275.

no real solution

277.

u=5±22u=5±22

279.

p=4±65p=4±65

281.

c=12c=12

283.

1 2 2 none

285.

factor Quadratic Formula square root

287.

Two consecutive odd numbers whose product is 323 are 17 and 19, and −17−17 and −19.−19.

289.

The height of the banner is 13 cm and the length of the side is 54 cm.

291.

The lengths of the sides of the mosaic are 2.2 and 4.4 feet.

293.

The width of the front walk is 8.1 feet and its length is 30.8 feet.

295.

The ball will reach 384 feet on its way up in 4 seconds and on the way down in 6 seconds.

297.
This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, -2) and goes through the point (1, -1).
299.

down

301.

up

303.

x=3x=3 (3,17)(3,17)

305.

y:(0,5);x:(5,0),(−1,0)y:(0,5);x:(5,0),(−1,0)

307.

y:(0,10);x:noney:(0,10);x:none

309.

y:(0,1);x:(14,0)y:(0,1);x:(14,0)

311.

y:(0,15);x:(−3,0),(−5,0);y:(0,15);x:(−3,0),(−5,0);
axis: x=−4;vertex:(−4,−1)x=−4;vertex:(−4,−1)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 17. The parabola has points plotted at the vertex (-4, -1) and the intercepts (-3, 0), (-5, 0) and (0, 15). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -4.
313.

y:(0,−16);x:(4,0);y:(0,−16);x:(4,0);
axis: x=4;vertex:(4,0)x=4;vertex:(4,0)

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -15 to 12. The y-axis of the plane runs from -20 to 2. The parabola has points plotted at the vertex (4, 0) and the intercept (0, -16). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.
315.

y:(0,13);x:none;y:(0,13);x:none;
axis: x=−3;vertex:(−3,4)x=−3;vertex:(−3,4)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 18. The parabola has points plotted at the vertex (-3, 4) and the intercept (0, 13). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.
317.

y:(0,−11)x:(3.1,0),(0.9,0);y:(0,−11)x:(3.1,0),(0.9,0);
axis: x=2;vertex:(2,5)x=2;vertex:(2,5)

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (2, 5) and the intercepts (3.1, 0) and (0.9, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.
319.

The minimum value is −1−1 when x=−1x=−1.

321.

In 3.5 seconds the ball is at its maximum height of 196 feet.

Practice Test

323.

w=−2,−8w=−2,−8

325.

m=1,32m=1,32

327.

n=−4±73n=−4±73

329.

no real solution

331.

2

333.

Two consecutive even number are −20−20 and −18−18 and 18 and 20.

335.

up x=−1x=−1 (−1,5)(−1,5) y:(0,8);x:noney:(0,8);x:none minimum value of 5 when x=−1x=−1

337.

up x=−5x=−5 (−5,−1)(−5,−1) y;(0,24);x:(−6,0),(−4,0)y;(0,24);x:(−6,0),(−4,0) minimum value of −5−5 when x=−1x=−1

339.

down x=−4x=−4
(−4,32)(−4,32) y;(0,16);x:(−9.7,0),(1.7,0)y;(0,16);x:(−9.7,0),(1.7,0)
maximum value of 3232 when x=−4x=−4

341.

y:(0,9);x:(34,0);y:(0,9);x:(34,0);
axis: x=34;vertex:(34,0)x=34;vertex:(34,0)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (3 fourths, 0) and the intercept (0, 9). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 fourths.
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