Completing the Square
4-6
Content Standard
Completing the Square
Reviews A.REI.4.b Solve quadratic equations by . . . completing the square . . .
Objectives To solve equations by completing the square To rewrite functions by completing the square
How can you use pieces like these to form a square with side length x 1 3 (and no overlapping pieces)? Show a sketch of your solution. How many of each piece do you need? Explain. Can you write the area of your square in two ways? MATHEMATICAL
PRACTICES Lesson Vocabulary V • ccompleting the square
Forming a square with model pieces provides a useful geometric image for completing a square algebraically.
Essential Understanding Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial. You can solve an equation that contains a perfect square by finding square roots. The simplest of this type of equation has the form ax 2 5 c.
Problem 1 Solving by Finding Square Roots What is the solution of each equation? How is solving this equation like solving a linear equation? You isolate the variable term.
A 4x2 1 10 5 46
B 3x2 2 5 5 25
4x2 5 36
d
Rewrite in ax2 5 c form.
S
3x2 5 30
4x2 36 4 5 4
d
Isolate x2.
S
3x2 30 3 5 3
x2 5 9 x 5 43 d
x2 5 10 Find square roots.
S
x 5 4 !10
Got It? 1. What is the solution of each equation? a. 7x2 2 10 5 25
b. 2x2 1 9 5 13
Lesson 4-6 Completing the Square
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Problem 2 Determining Dimensions D
IC
AC
ES
AM YN
TIVITI
Dynamic Activity Completing the Square
STEM
Architecture While designing a house, an architect used windows like the one shown here. What are the dimensions of the window if it has 2766 square inches of glass? Step 1 Find the area of the window. The area of the rectangular part is (2x)(x) 5 2x 2 in.2. The area of the semicircular part is p 2 2 1 2 1 x 2 1 x2 2 pr 5 2 p Q 2 R 5 2 p 4 5 8 x in. .
So, the total amount of glass used is p 2x 2 1 8 x 2 5 2766 in.2. Step 2 Solve for x. p Q 2 1 8 R x 2 5 2766 Is the answer reasonable? Yes; the rectangular part is about 30 3 70 5 2100 in.2. This leaves enough glass for the semicircle.
x2 5
2766 2 1 p8
x < 434
Write the equation in ax2 5 c form. Isolate x2. Find square roots. Use a calculator.
Length cannot be negative. So the rectangular portion of the window is 34 in. wide by 68 in. long. The semicircular top has a radius of 17 in.
Got It? 2. The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 2822.4 in.2. What are the window dimensions?
Sometimes an equation shows a perfect square trinomial equal to a constant. To solve, factor the perfect square trinomial into the square of a binomial. Then find square roots.
Problem 3 Solving a Perfect Square Trinomial Equation What is the solution of x 2 1 4x 1 4 5 25?
Factor the perfect square trinomial. Find square roots.
Rewrite as two equations. Solve for x.
x2 1 4x 1 4 5 25 (x 1 2)2 5 25 x 1 2 5 w5 x 1 2 5 5 or x 1 2 5 25 x 5 3 or x 5 27
Got It? 3. What is the solution of x 2 2 14x 1 49 5 25? 234
Chapter 4
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Quadratic Functions and Equations
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If x 2 1 bx is not part of a perfect square trinomial, you can use the coefficient b to find a constant c so that x 2 1 bx 1 c is a perfect square. When you do this, you are completing the square. The diagram models this process. b 2
b 2
b 2
b 2
x
x
x x x2
b 2
b
x
bx
x2
bx b 2
x 2
x
b 2
2
b 2
Key Concept Completing the Square 2 You can form a perfect square trinomial from x2 1 bx by adding Q b2 R . 2
x 2 1 bx 1 Q b2 R 5 Q x 1 b2 R
2
Problem 4 Completing the Square What value completes the square for x 2 2 10x? Justify your answer. Why do you want a perfect square trinomial? You can factor a perfect square trinomial into the square of a binomial.
x 2 2 10x
Identify b; b 5 210
210 2 b 2 Q 2 R 5 Q 2 R 5 (25)2 5 25
2 Find Q b R .
x2
2 10x 1 25
x 2 2 10x 1 25 5 (x 2 5)2
2
2 Add the value of Q b R to complete the square.
2
Rewrite as the square of a binomial.
Got It? 4. a. What value completes the square for x 2 1 6x? b. Reasoning Is it possible for more than one value to complete the square for an expression? Explain.
Key Concept Solving an Equation by Completing the Square 1. Rewrite the equation in the form x 2 1 bx 5 c. To do this, get all terms with the variable on one side of the equation and the constant on the other side. Divide all the terms of the equation by the coefficient of x2 if it is not 1. 2 2. Complete the square by adding Q b2 R to each side of the equation.
3. Factor the trinomial. 4. Find square roots. 5. Solve for x.
Lesson 4-6 Completing the Square
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Problem 5 Solving by Completing the Square What is the solution of 3x 2 2 12x 1 6 5 0? 3x2 2 12x 1 6 5 0 3x2 2 12x 5 26
Rewrite. Get all terms with x on one side of the equation.
12x 26 3 2 3 5 3
3x2
Divide each side by 3 so the coefficient of x2 will be 1.
x2 2 4x 5 22 b 2
Simplify.
24 2
2 Find Q b2 R 5 4.
Q 2 R 5 Q 2 R 5 (22)2 5 4 x2 2 4x 1 4 5 22 1 4 (x 2 Is there a way to check without a calculator? Yes; you can check that your solutions are reasonable by estimating.
2)2
52
Add 4 to each side. Factor the trinomial.
x 2 2 5 4!2 x 5 2 4 !2
Find square roots. Solve for x.
Check your results on your calculator. Replace x in the original equation with 2 1 !2 and 2 2 !2.
Got It? 5. What is the solution of 2x 2 2 x 1 3 5 x 1 9?
You can complete a square to change a quadratic function to vertex form.
Problem 6 Writing in Vertex Form What should be your first step? Complete the square.
What is y 5 x 2 1 4x 2 6 in vertex form? Name the vertex and y-intercept. y 5 x 2 1 4x 2 6 2
y 5 x 2 1 4x 1 22 2 6 2 22
Add Q 42 R 5 22 to complete the square. Also, subtract 22 to leave the function unchanged.
y 5 (x 1 2)2 2 6 2 22
Factor the perfect square trinomial.
y 5 (x 1 2)2 2 10
Simplify.
The vertex is (22, 210). The y-intercept is (0, 26). Check with a graphing calculator. Plot1 Plot2 \Y1 = X 2+4X–6 \Y2 = \Y3 = \Y4 = \Y5 = \Y6 = \Y7 =
Plot3
X
Minimum X=–2
Y=–10
–4 –3 –2 –1 0 1 2 Y1=–10
Y1 –6 –9 –10 –9 –6 –1 6
Got It? 6. What is y 5 x 2 1 3x 2 6 in vertex form? Name the vertex and y-intercept.
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Quadratic Functions and Equations
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Lesson Check Do you know HOW?
Do you UNDERSTAND?
9. Vocabulary Explain the process of completing the square.
Solve each equation by finding square roots. 1. 2x 2 5 72
2. 6x 2 5 54
10. How can you rewrite the equation x 2 1 12x 1 5 5 3 so the left side of the equation is in the form (x 1 a)2 ?
Complete the square. 3. x 2 1 2x 1 j 5.
x2
4. x 2 1 10x 1 j
2 4x 1 j
6.
7. x 2 1 100x 1 j
x2
11. Error Analysis Your friend completed the square and wrote the expression shown. Explain your friend’s error and write the expression correctly.
1 12x 1 j
8. x 2 2 32x 1 j
Practice and Problem-Solving Exercises
A
Practice
MATHEMATICAL
PRACTICES
x2 - 14x + 36 x2 - 14x + 49 + 36 (x - 7)2 + 36
MATHEMATICAL
PRACTICES
See Problem 1.
Solve each equation by finding square roots. 12. 5x 2 5 80
13. x 2 2 4 5 0
14. 2x 2 5 32
15. 9x 2 5 25
16. 3x 2 2 15 5 0
17. 5x 2 2 40 5 0
18. Fitness A rectangular swimming pool is 6 ft deep. One side of the pool is 2.5 times longer than the other. The amount of water needed to fill the swimming pool is 2160 cubic feet. Find the dimensions of the pool.
See Problem 2.
Solve each equation.
See Problem 3.
19. x 2 1 6x 1 9 5 1
20. x 2 2 4x 1 4 5 100
21. x 2 2 2x 1 1 5 4
16 22. x 2 1 8x 1 16 5 9 25. 25x 2 1 10x 1 1 5 9
23. 4x 2 1 4x 1 1 5 49
24. x 2 2 12x 1 36 5 25
26. x 2 2 30x 1 225 5 400
27. 9x 2 1 24x 1 16 5 36 See Problem 4.
Complete the square. 28.
x2
1 18x 1 j
31. x 2 1 20x 1 j
29.
x2
2x1j
32. m 2 2 3m 1 j
30.
x 2 2 24x
1j
33. x 2 1 4x 1 j See Problem 5.
Solve each quadratic equation by completing the square. 34. x2 1 6x 2 3 5 0
35. x2 2 12x 1 7 5 0
36. x2 1 4x 1 2 5 0
37. x2 2 2x 5 5
38. x2 1 8x 5 11
39. x2 1 12 5 10x
40. x2 2 3x 5 x 2 1
41. x2 1 2 5 6x 1 4
42. 2x2 1 2x 2 5 5 x2
43. 4x2 1 10x 2 3 5 0
44. 9x2 2 12x 2 2 5 0
45. 25x2 1 30x 5 12
Lesson 4-6 Completing the Square
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See Problem 6.
Rewrite each equation in vertex form. 46. y 5
x2
1 4x 1 1
49. y 5 x 2 1 4x 2 7
B
Apply
47. y 5
2x 2
2 8x 1 1
50. y 5 2x 2 2 6x 2 1
48. y 5
2x 2
2 2x 1 3
51. y 5 2x 2 1 4x 2 1
52. Think About a Plan The area of the rectangle shown is 80 square inches. What is the value of x? • How can you write an equation to represent 80 in terms of x ? • How can you find the value of x by completing the square?
x3 2x
Find the value of k that would make the left side of each equation a perfect square trinomial. 53. x 2 1 kx 1 25 5 0
54. x 2 2 kx 1 100 5 0
55. x 2 2 kx 1 121 5 0
56. x 2 1 kx 1 64 5 0
57. x 2 2 kx 1 81 5 0
58. 25x 2 2 kx 1 1 5 0
59. x 2 1 kx 1 14 5 0
60. 9x 2 2 kx 1 4 5 0
61. 36x 2 2 kx 1 49 5 0
62. Geometry The table shows some possible dimensions of rectangles with a perimeter of 100 units. Copy and complete the table. a. Plot the points (width, area). Find a model for the data set. b. What is another point in the data set? Use it to verify your model. c. What is a reasonable domain for this function? Explain. d. Find the maximum possible area. What dimensions yield this area? e. Find a function for area in terms of width without using the table. Do you get the same model as in part (a)? Explain.
Width
Length
Area
1
49
49
2
48
O
3
O
O
4
O
O
5
O
O
Solve each quadratic equation by completing the square. 63. x2 1 5x 2 3 5 0
64. x2 1 3x 5 2
65. x2 2 x 5 5
66. x2 1 x 2 1 5 0
67. 3x2 2 4x 5 2
68. 5x2 2 x 5 4
3 69. x2 1 4 x 5 12 71. 3x2 1 x 5 23
70. 2x2 2 12 x 5 18 72. 2x2 1 2x 1 4 5 0
73. 2x2 2 6x 5 2
74. 20.25x2 2 0.6x 1 0.3 5 0
75. Football The quadratic function h 5 20.01x 2 1 1.18x 1 2 models the height of a punted football. The horizontal distance in feet from the point of impact with the kicker’s foot is x, and h is the height of the ball in feet. a. Write the function in vertex form. What is the maximum height of the punt? b. The nearest defensive player is 5 ft horizontally from the point of impact. How high must the player reach to block the punt? c. Suppose the ball was not blocked but continued on its path. How far down the field would the ball go before it hit the ground?
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Quadratic Functions and Equations
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C
Challenge
Solve for x in terms of a. 76. 2x 2 2 ax 5 6a 2
77. 3x 2 1 ax 5 a 2
78. 2a 2x 2 2 8ax 5 26
79. 4a 2x 2 1 8ax 1 3 5 0
80. 3x 2 1 ax 2 5 9x 1 9a
81. 6a 2x 2 2 11ax 5 10
82. Solve x 2 5 (6 !2)x 1 7 by completing the square. Rewrite each equation in vertex form. Then find the vertex of the graph. 84. y 5 12 x 2 2 5x 1 12
83. y 5 24x 2 2 5x 1 3
85. y 5 215 x 2 1 45x 1 11 5
Standardized Test Prep SAT/ACT
86. The graph of which inequality has its vertex at Q 2 12, 25 R ? y , u 2x 2 5 u 1 5
y . u 2x 1 5 u 2 5
y , u 2x 1 5 u 2 5
y . u 2x 2 5 u 2 5
87. Which number is a solution of u 9 2 x u 5 9 1 x? 23
0
3
6
88. Joanne tosses an apple seed on the ground. It travels along a parabola with the equation y 5 2x2 1 4. Assume the seed was thrown from a height of 4 ft. How many feet away from Joanne will the apple seed land? 1 ft Extended Response
2 ft
4 ft
8 ft
89. List the steps for solving the equation x 2 2 9 5 28x by the completing the square method. Explain each step.
Mixed Review See Lesson 4-5.
Solve each equation by factoring. Check your answers. 90. 2x 2 2 3x 1 1 5 0
91. x 2 2 4 5 23x
92. 16 1 22x 5 3x 2
Determine whether a quadratic model exists for each set of values. If so, write the model. 93. (24, 3), (23, 3), (22, 4)
94. Q 21, 12 R , (0, 2), (2, 2)
95. (0, 2), (1, 0), (2, 4) See Lesson 3-2.
Solve each system by elimination. 96. e
2x 1 y 5 4 3x 2 y 5 6
See Lesson 4-3.
97. e
2x 1 y 5 7 22x 1 5y 5 21
98. e
2x 1 4y 5 10 3x 1 5y 5 14
Get Ready! To prepare for Lesson 4-7, do Exercises 99–100. Evaluate each expression for the given values of the variables. 99. b 2 2 4ac; a 5 1, b 5 6, c 5 3
100. b 2 2 4ac; a 5 25, b 5 2, c 5 4 Lesson 4-6 Completing the Square
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See Lesson 1-3.
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Content Standard
Completing the Square
Reviews A.REI.4.b Solve quadratic equations by . . . completing the square . . .
Objectives To solve equations by completing the square To rewrite functions by completing the square
How can you use pieces like these to form a square with side length x 1 3 (and no overlapping pieces)? Show a sketch of your solution. How many of each piece do you need? Explain. Can you write the area of your square in two ways? MATHEMATICAL
PRACTICES Lesson Vocabulary V • ccompleting the square
Forming a square with model pieces provides a useful geometric image for completing a square algebraically.
Essential Understanding Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial. You can solve an equation that contains a perfect square by finding square roots. The simplest of this type of equation has the form ax 2 5 c.
Problem 1 Solving by Finding Square Roots What is the solution of each equation? How is solving this equation like solving a linear equation? You isolate the variable term.
A 4x2 1 10 5 46
B 3x2 2 5 5 25
4x2 5 36
d
Rewrite in ax2 5 c form.
S
3x2 5 30
4x2 36 4 5 4
d
Isolate x2.
S
3x2 30 3 5 3
x2 5 9 x 5 43 d
x2 5 10 Find square roots.
S
x 5 4 !10
Got It? 1. What is the solution of each equation? a. 7x2 2 10 5 25
b. 2x2 1 9 5 13
Lesson 4-6 Completing the Square
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Problem 2 Determining Dimensions D
IC
AC
ES
AM YN
TIVITI
Dynamic Activity Completing the Square
STEM
Architecture While designing a house, an architect used windows like the one shown here. What are the dimensions of the window if it has 2766 square inches of glass? Step 1 Find the area of the window. The area of the rectangular part is (2x)(x) 5 2x 2 in.2. The area of the semicircular part is p 2 2 1 2 1 x 2 1 x2 2 pr 5 2 p Q 2 R 5 2 p 4 5 8 x in. .
So, the total amount of glass used is p 2x 2 1 8 x 2 5 2766 in.2. Step 2 Solve for x. p Q 2 1 8 R x 2 5 2766 Is the answer reasonable? Yes; the rectangular part is about 30 3 70 5 2100 in.2. This leaves enough glass for the semicircle.
x2 5
2766 2 1 p8
x < 434
Write the equation in ax2 5 c form. Isolate x2. Find square roots. Use a calculator.
Length cannot be negative. So the rectangular portion of the window is 34 in. wide by 68 in. long. The semicircular top has a radius of 17 in.
Got It? 2. The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 2822.4 in.2. What are the window dimensions?
Sometimes an equation shows a perfect square trinomial equal to a constant. To solve, factor the perfect square trinomial into the square of a binomial. Then find square roots.
Problem 3 Solving a Perfect Square Trinomial Equation What is the solution of x 2 1 4x 1 4 5 25?
Factor the perfect square trinomial. Find square roots.
Rewrite as two equations. Solve for x.
x2 1 4x 1 4 5 25 (x 1 2)2 5 25 x 1 2 5 w5 x 1 2 5 5 or x 1 2 5 25 x 5 3 or x 5 27
Got It? 3. What is the solution of x 2 2 14x 1 49 5 25? 234
Chapter 4
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Quadratic Functions and Equations
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If x 2 1 bx is not part of a perfect square trinomial, you can use the coefficient b to find a constant c so that x 2 1 bx 1 c is a perfect square. When you do this, you are completing the square. The diagram models this process. b 2
b 2
b 2
b 2
x
x
x x x2
b 2
b
x
bx
x2
bx b 2
x 2
x
b 2
2
b 2
Key Concept Completing the Square 2 You can form a perfect square trinomial from x2 1 bx by adding Q b2 R . 2
x 2 1 bx 1 Q b2 R 5 Q x 1 b2 R
2
Problem 4 Completing the Square What value completes the square for x 2 2 10x? Justify your answer. Why do you want a perfect square trinomial? You can factor a perfect square trinomial into the square of a binomial.
x 2 2 10x
Identify b; b 5 210
210 2 b 2 Q 2 R 5 Q 2 R 5 (25)2 5 25
2 Find Q b R .
x2
2 10x 1 25
x 2 2 10x 1 25 5 (x 2 5)2
2
2 Add the value of Q b R to complete the square.
2
Rewrite as the square of a binomial.
Got It? 4. a. What value completes the square for x 2 1 6x? b. Reasoning Is it possible for more than one value to complete the square for an expression? Explain.
Key Concept Solving an Equation by Completing the Square 1. Rewrite the equation in the form x 2 1 bx 5 c. To do this, get all terms with the variable on one side of the equation and the constant on the other side. Divide all the terms of the equation by the coefficient of x2 if it is not 1. 2 2. Complete the square by adding Q b2 R to each side of the equation.
3. Factor the trinomial. 4. Find square roots. 5. Solve for x.
Lesson 4-6 Completing the Square
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Problem 5 Solving by Completing the Square What is the solution of 3x 2 2 12x 1 6 5 0? 3x2 2 12x 1 6 5 0 3x2 2 12x 5 26
Rewrite. Get all terms with x on one side of the equation.
12x 26 3 2 3 5 3
3x2
Divide each side by 3 so the coefficient of x2 will be 1.
x2 2 4x 5 22 b 2
Simplify.
24 2
2 Find Q b2 R 5 4.
Q 2 R 5 Q 2 R 5 (22)2 5 4 x2 2 4x 1 4 5 22 1 4 (x 2 Is there a way to check without a calculator? Yes; you can check that your solutions are reasonable by estimating.
2)2
52
Add 4 to each side. Factor the trinomial.
x 2 2 5 4!2 x 5 2 4 !2
Find square roots. Solve for x.
Check your results on your calculator. Replace x in the original equation with 2 1 !2 and 2 2 !2.
Got It? 5. What is the solution of 2x 2 2 x 1 3 5 x 1 9?
You can complete a square to change a quadratic function to vertex form.
Problem 6 Writing in Vertex Form What should be your first step? Complete the square.
What is y 5 x 2 1 4x 2 6 in vertex form? Name the vertex and y-intercept. y 5 x 2 1 4x 2 6 2
y 5 x 2 1 4x 1 22 2 6 2 22
Add Q 42 R 5 22 to complete the square. Also, subtract 22 to leave the function unchanged.
y 5 (x 1 2)2 2 6 2 22
Factor the perfect square trinomial.
y 5 (x 1 2)2 2 10
Simplify.
The vertex is (22, 210). The y-intercept is (0, 26). Check with a graphing calculator. Plot1 Plot2 \Y1 = X 2+4X–6 \Y2 = \Y3 = \Y4 = \Y5 = \Y6 = \Y7 =
Plot3
X
Minimum X=–2
Y=–10
–4 –3 –2 –1 0 1 2 Y1=–10
Y1 –6 –9 –10 –9 –6 –1 6
Got It? 6. What is y 5 x 2 1 3x 2 6 in vertex form? Name the vertex and y-intercept.
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Quadratic Functions and Equations
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Lesson Check Do you know HOW?
Do you UNDERSTAND?
9. Vocabulary Explain the process of completing the square.
Solve each equation by finding square roots. 1. 2x 2 5 72
2. 6x 2 5 54
10. How can you rewrite the equation x 2 1 12x 1 5 5 3 so the left side of the equation is in the form (x 1 a)2 ?
Complete the square. 3. x 2 1 2x 1 j 5.
x2
4. x 2 1 10x 1 j
2 4x 1 j
6.
7. x 2 1 100x 1 j
x2
11. Error Analysis Your friend completed the square and wrote the expression shown. Explain your friend’s error and write the expression correctly.
1 12x 1 j
8. x 2 2 32x 1 j
Practice and Problem-Solving Exercises
A
Practice
MATHEMATICAL
PRACTICES
x2 - 14x + 36 x2 - 14x + 49 + 36 (x - 7)2 + 36
MATHEMATICAL
PRACTICES
See Problem 1.
Solve each equation by finding square roots. 12. 5x 2 5 80
13. x 2 2 4 5 0
14. 2x 2 5 32
15. 9x 2 5 25
16. 3x 2 2 15 5 0
17. 5x 2 2 40 5 0
18. Fitness A rectangular swimming pool is 6 ft deep. One side of the pool is 2.5 times longer than the other. The amount of water needed to fill the swimming pool is 2160 cubic feet. Find the dimensions of the pool.
See Problem 2.
Solve each equation.
See Problem 3.
19. x 2 1 6x 1 9 5 1
20. x 2 2 4x 1 4 5 100
21. x 2 2 2x 1 1 5 4
16 22. x 2 1 8x 1 16 5 9 25. 25x 2 1 10x 1 1 5 9
23. 4x 2 1 4x 1 1 5 49
24. x 2 2 12x 1 36 5 25
26. x 2 2 30x 1 225 5 400
27. 9x 2 1 24x 1 16 5 36 See Problem 4.
Complete the square. 28.
x2
1 18x 1 j
31. x 2 1 20x 1 j
29.
x2
2x1j
32. m 2 2 3m 1 j
30.
x 2 2 24x
1j
33. x 2 1 4x 1 j See Problem 5.
Solve each quadratic equation by completing the square. 34. x2 1 6x 2 3 5 0
35. x2 2 12x 1 7 5 0
36. x2 1 4x 1 2 5 0
37. x2 2 2x 5 5
38. x2 1 8x 5 11
39. x2 1 12 5 10x
40. x2 2 3x 5 x 2 1
41. x2 1 2 5 6x 1 4
42. 2x2 1 2x 2 5 5 x2
43. 4x2 1 10x 2 3 5 0
44. 9x2 2 12x 2 2 5 0
45. 25x2 1 30x 5 12
Lesson 4-6 Completing the Square
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See Problem 6.
Rewrite each equation in vertex form. 46. y 5
x2
1 4x 1 1
49. y 5 x 2 1 4x 2 7
B
Apply
47. y 5
2x 2
2 8x 1 1
50. y 5 2x 2 2 6x 2 1
48. y 5
2x 2
2 2x 1 3
51. y 5 2x 2 1 4x 2 1
52. Think About a Plan The area of the rectangle shown is 80 square inches. What is the value of x? • How can you write an equation to represent 80 in terms of x ? • How can you find the value of x by completing the square?
x3 2x
Find the value of k that would make the left side of each equation a perfect square trinomial. 53. x 2 1 kx 1 25 5 0
54. x 2 2 kx 1 100 5 0
55. x 2 2 kx 1 121 5 0
56. x 2 1 kx 1 64 5 0
57. x 2 2 kx 1 81 5 0
58. 25x 2 2 kx 1 1 5 0
59. x 2 1 kx 1 14 5 0
60. 9x 2 2 kx 1 4 5 0
61. 36x 2 2 kx 1 49 5 0
62. Geometry The table shows some possible dimensions of rectangles with a perimeter of 100 units. Copy and complete the table. a. Plot the points (width, area). Find a model for the data set. b. What is another point in the data set? Use it to verify your model. c. What is a reasonable domain for this function? Explain. d. Find the maximum possible area. What dimensions yield this area? e. Find a function for area in terms of width without using the table. Do you get the same model as in part (a)? Explain.
Width
Length
Area
1
49
49
2
48
O
3
O
O
4
O
O
5
O
O
Solve each quadratic equation by completing the square. 63. x2 1 5x 2 3 5 0
64. x2 1 3x 5 2
65. x2 2 x 5 5
66. x2 1 x 2 1 5 0
67. 3x2 2 4x 5 2
68. 5x2 2 x 5 4
3 69. x2 1 4 x 5 12 71. 3x2 1 x 5 23
70. 2x2 2 12 x 5 18 72. 2x2 1 2x 1 4 5 0
73. 2x2 2 6x 5 2
74. 20.25x2 2 0.6x 1 0.3 5 0
75. Football The quadratic function h 5 20.01x 2 1 1.18x 1 2 models the height of a punted football. The horizontal distance in feet from the point of impact with the kicker’s foot is x, and h is the height of the ball in feet. a. Write the function in vertex form. What is the maximum height of the punt? b. The nearest defensive player is 5 ft horizontally from the point of impact. How high must the player reach to block the punt? c. Suppose the ball was not blocked but continued on its path. How far down the field would the ball go before it hit the ground?
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Chapter 4
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Quadratic Functions and Equations
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C
Challenge
Solve for x in terms of a. 76. 2x 2 2 ax 5 6a 2
77. 3x 2 1 ax 5 a 2
78. 2a 2x 2 2 8ax 5 26
79. 4a 2x 2 1 8ax 1 3 5 0
80. 3x 2 1 ax 2 5 9x 1 9a
81. 6a 2x 2 2 11ax 5 10
82. Solve x 2 5 (6 !2)x 1 7 by completing the square. Rewrite each equation in vertex form. Then find the vertex of the graph. 84. y 5 12 x 2 2 5x 1 12
83. y 5 24x 2 2 5x 1 3
85. y 5 215 x 2 1 45x 1 11 5
Standardized Test Prep SAT/ACT
86. The graph of which inequality has its vertex at Q 2 12, 25 R ? y , u 2x 2 5 u 1 5
y . u 2x 1 5 u 2 5
y , u 2x 1 5 u 2 5
y . u 2x 2 5 u 2 5
87. Which number is a solution of u 9 2 x u 5 9 1 x? 23
0
3
6
88. Joanne tosses an apple seed on the ground. It travels along a parabola with the equation y 5 2x2 1 4. Assume the seed was thrown from a height of 4 ft. How many feet away from Joanne will the apple seed land? 1 ft Extended Response
2 ft
4 ft
8 ft
89. List the steps for solving the equation x 2 2 9 5 28x by the completing the square method. Explain each step.
Mixed Review See Lesson 4-5.
Solve each equation by factoring. Check your answers. 90. 2x 2 2 3x 1 1 5 0
91. x 2 2 4 5 23x
92. 16 1 22x 5 3x 2
Determine whether a quadratic model exists for each set of values. If so, write the model. 93. (24, 3), (23, 3), (22, 4)
94. Q 21, 12 R , (0, 2), (2, 2)
95. (0, 2), (1, 0), (2, 4) See Lesson 3-2.
Solve each system by elimination. 96. e
2x 1 y 5 4 3x 2 y 5 6
See Lesson 4-3.
97. e
2x 1 y 5 7 22x 1 5y 5 21
98. e
2x 1 4y 5 10 3x 1 5y 5 14
Get Ready! To prepare for Lesson 4-7, do Exercises 99–100. Evaluate each expression for the given values of the variables. 99. b 2 2 4ac; a 5 1, b 5 6, c 5 3
100. b 2 2 4ac; a 5 25, b 5 2, c 5 4 Lesson 4-6 Completing the Square
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See Lesson 1-3.
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