Solve Quadratic Equations by Completing the Square

10.5 Solve Quadratic Equations by Completing the Square You solved quadratic equations by finding square roots. Before You will solve quadratic equ...
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10.5

Solve Quadratic Equations by Completing the Square You solved quadratic equations by finding square roots.

Before

You will solve quadratic equations by completing the square.

Now

So you can solve a problem about snowboarding, as in Ex. 50.

Why?

For an expression of the form x2 1 bx, you can add a constant c to the expression so that the expression x2 1 bx 1 c is a perfect square trinomial. This process is called completing the square.

Key Vocabulary • completing the square • perfect square trinomial, p. 601

For Your Notebook

KEY CONCEPT Completing the Square

Words To complete the square for the expression x2 1 bx, add the

square of half the coefficient of the term bx.

1 b2 2

Algebra x2 1 bx 1 }

EXAMPLE 1

2

1

b 5 x1} 2

2

2

Complete the square

Find the value of c that makes the expression x 2 1 5x 1 c a perfect square trinomial. Then write the expression as the square of a binomial.

STEP 1 Find the value of c. For the expression to be a perfect square trinomial, c needs to be the square of half the coefficient of bx. 2

5 25 c 5 1} 2 5} 2

4

Find the square of half the coefficient of bx.

STEP 2 Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial. 25 x2 1 5x 1 c 5 x2 1 5x 1 } 4

5 2

5 1x 1 } 22



GUIDED PRACTICE

25 4

Substitute } for c. Square of a binomial

for Example 1

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 1. x2 1 8x 1 c

2. x2 2 12x 1 c

3. x2 1 3x 1 c

10.5 Solve Quadratic Equations by Completing the Square

663

SOLVING EQUATIONS The method of completing the square can be used

to solve any quadratic equation. To use completing the square to solve a quadratic equation, you must write the equation in the form x2 1 bx 5 d.

EXAMPLE 2

Solve a quadratic equation

Solve x 2 2 16x 5 215 by completing the square. Solution x2 2 16x 5 215

Write original equation.

x2 2 16x 1 (28)2 5 215 1 (28)2

AVOID ERRORS When completing the square to solve an equation, be sure you add the term b2 } to both sides of the 2

1 2

2

1 216 2 2

Add } , or (28) 2 , to each side.

(x 2 8)2 5 215 1 (28)2

Write left side as the square of a binomial.

(x 2 8)2 5 49

Simplify the right side.

x 2 8 5 67

equation.

Take square roots of each side.

x5867

Add 8 to each side.

c The solutions of the equation are 8 1 7 5 15 and 8 2 7 5 1.

CHECK

You can check the solutions in the original equation. If x 5 15:

If x 5 1:

(15)2 2 16(15) 0 215

(1)2 2 16(1) 0 215

215 5 215 ✓

EXAMPLE 3

215 5 215 ✓

Solve a quadratic equation in standard form

Solve 2x 2 1 20x 2 8 5 0 by completing the square. Solution 2x2 1 20x 2 8 5 0

Write original equation.

2x2 1 20x 5 8

Add 8 to each side.

2

x 1 10x 5 4

Divide each side by 2. 2

1 10 2 2

x2 1 10x 1 52 5 4 1 52

AVOID ERRORS Be sure that the coefficient of x2 is 1 before you complete the square.

Add } , or 52 , to each side.

(x 1 5)2 5 29

Write left side as the square of a binomial. }

x 1 5 5 6Î29

Take square roots of each side. }

x 5 25 6 Ï 29

Subtract 5 from each side.

}

}

c The solutions are 25 1 Ï29 ø 0.39 and 25 2 Ï 29 ø 210.39.



GUIDED PRACTICE

for Examples 2 and 3

Solve the equation by completing the square. Round your solutions to the nearest hundredth, if necessary. 4. x2 2 2x 5 3

664

Chapter 10 Quadratic Equations and Functions

5. m2 1 10m 5 28

6. 3g 2 2 24g 1 27 5 0

EXAMPLE 4

Solve a multi-step problem

CRAFTS You decide to use chalkboard paint to

create a chalkboard on a door. You want the chalkboard to have a uniform border as shown. You have enough chalkboard paint to cover 6 square feet. Find the width of the border to the nearest inch.

7IDTHOF BORDER XFT FT

Solution

#HALKBOARD

STEP 1 Write a verbal model. Then write an equation. Let x be the width (in feet) of the border. Area of chalkboard (square feet) WRITE EQUATION The width of the border is subtracted twice because it is at the top and the bottom of the door, as well as at the left and the right.

5

Length of chalkboard

5

(7 2 2x)

6

(feet)

FT

p

Width of chalkboard

p

(3 2 2x)

(feet)

STEP 2 Solve the equation. 6 5 (7 2 2x)(3 2 2x) 6 5 21 2 20x 1 4x

Write equation.

2

Multiply binomials.

215 5 4x2 2 20x

Subtract 21 from each side.

15

2} 5 x2 2 5x 4

Divide each side by 4. 2

1 52 2

15 25 25 2} 1} 5 x2 2 5x 1 } 4

4

15

25 5 2} 1} 5 1x 2 } 2 4 4

2

Write right side as the square of a binomial.

2

5 2 2

1

5 2

}5 x2}

Î2 5 5 } 6 Î} 5 x 2 2

25 4

Add 2} , or }, to each side.

4

2

Simplify left side.

}

5 5 6 } 5x2}

Take square roots of each side.

2

}

5 2

Add } to each side.

Î2

}

Î2

}

5 5 5 5 The solutions of the equation are } 1 } ø 4.08 and } 2 } ø 0.92. 2

2

It is not possible for the width of the border to be 4.08 feet because the width of the door is 3 feet. So, the width of the border is 0.92 foot. Convert 0.92 foot to inches. 12 in. 0.92 ft p } 5 11.04 in. 1 ft

Multiply by conversion factor.

c The width of the border should be about 11 inches.



GUIDED PRACTICE

for Example 4

7. WHAT IF? In Example 4, suppose you have enough chalkboard paint to

cover 4 square feet. Find the width of the border to the nearest inch. 10.5 Solve Quadratic Equations by Completing the Square

665

10.5

EXERCISES

HOMEWORK KEY

5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 19 and 47

★ 5 STANDARDIZED TEST PRACTICE Exs. 2, 24, 25, 42, and 49

5 MULTIPLE REPRESENTATIONS Ex. 47

SKILL PRACTICE 1. VOCABULARY Copy and complete: The process of writing an expression

of the form x2 1 bx as a perfect square trinomial is called ? .

2.

EXAMPLE 1 on p. 663 for Exs. 3–11

★ WRITING Give an example of an expression that is a perfect square trinomial. Explain why the expression is a perfect square trinomial.

COMPLETING THE SQUARE Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.

3. x2 1 6x 1 c

4. x2 1 12x 1 c

5. x2 2 4x 1 c

6. x2 2 8x 1 c

7. x2 2 3x 1 c

8. x2 1 5x 1 c

1 10. x2 2 } x1c

4 11. x2 2 } x1c

9. x2 1 2.4x 1 c

2

3

EXAMPLES 2 and 3

SOLVING EQUATIONS Solve the equation by completing the square. Round

on p. 664 for Exs. 12–27

12. x2 1 2x 5 3

13. x2 1 10x 5 24

14. c 2 2 14c 5 15

15. n2 2 6n 5 72

16. a2 2 8a 1 15 5 0

17. y 2 1 4y 2 21 5 0

11 18. w 2 2 5w 5 }

19. z2 1 11z 5 2} 4

your solutions to the nearest hundredth, if necessary.

21

4

2



1



5 4

2

22. v 2 7v 1 1 5 0

23. m 1 3m 1 } 5 0

MULTIPLE CHOICE What are the solutions of 4x2 1 16x 5 9? 9

1 9 B 2} ,}

A 2} , 2} 2 2 25.

3

2

21. k 2 8k 2 7 5 0 24.

2 20. g 2 2 } g57

9 1 C } , 2}

2 2

2

1 9 D } ,}

2

2 2

MULTIPLE CHOICE What are the solutions of x2 1 12x 1 10 5 0? }

A 26 6 Ï46

}

B 26 6 Ï 26

}

}

C 6 6 Ï 26

D 6 6 Ï 46

ERROR ANALYSIS Describe and correct the error in solving the given

equation. 26. x2 2 14x 5 11

27. x2 2 2x 2 4 5 0

x2 2 14x 5 11

x2 2 2x 2 4 5 0

x2 2 14x 1 49 5 11

x2 2 2x 5 4

(x 2 7) 2 5 11

x2 2 2x 1 1 5 4 1 1 }

x 2 7 5 6Ï11

(x 1 1) 2 5 5 }

x 5 7 6 Ï11

}

x 1 1 5 6Ï 5

}

x 5 1 6 Ï5

666

Chapter 10 Quadratic Equations and Functions

SOLVING EQUATIONS Solve the equation by completing the square. Round

your solutions to the nearest hundredth, if necessary. 28. 2x2 2 8x 2 14 5 0

29. 2x2 1 24x 1 10 5 0

30. 3x2 2 48x 1 39 5 0

31. 4y 2 1 4y 2 7 5 0

32. 9n2 1 36n 1 11 5 0

33. 3w 2 2 18w 2 20 5 0

34. 3p2 2 30p 2 11 5 6p

35. 3a2 2 12a 1 3 5 2a2 2 4

36. 15c 2 2 51c 2 30 5 9c 1 15

37. 7m2 1 24m 2 2 5 m2 2 9 38. g 2 1 2g 1 0.4 5 0.9g 2 1 g

39. 11z2 2 10z 2 3 5 29z2 1 }

3 4

GEOMETRY Find the value of x. Round your answer to the nearest hundredth, if necessary.

40. Area of triangle 5 108 m 2

41. Area of rectangle 5 288 in.2

3x in.

xm (2x 1 10) in.

(x 1 6) m

42.



1 2

b 2 WRITING How many solutions does x2 1 bx 5 c have if c < 2 } ? Explain. 2

43. CHALLENGE The product of two consecutive negative integers is 210. Find

the integers. 44. CHALLENGE The product of two consecutive positive even integers is 288.

Find the integers.

PROBLEM SOLVING EXAMPLE 4

45. LANDSCAPING You are building a rectangular brick patio

on p. 665 for Exs. 45–46

surrounded by crushed stone in a rectangular courtyard as shown. The crushed stone border has a uniform width x (in feet). You have enough money in your budget to purchase patio bricks to cover 140 square feet. Solve the equation 140 5 (20 2 2x)(16 2 2x) to find the width of the border. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

46. TRAFFIC ENGINEERING The distance d (in feet) that it takes a car to come

to a complete stop on dry asphalt can be modeled by d 5 0.05s2 1 1.1s where s is the speed of the car (in miles per hour). A car has 78 feet to come to a complete stop. Find the maximum speed at which the car can travel. GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN

47.

MULTIPLE REPRESENTATIONS For the period 198522001, the average salary y (in thousands of dollars) per season of a Major League Baseball player can be modeled by y 5 7x2 2 4x 1 392 where x is the number of years since 1985.

a. Solving an Equation Write and solve an equation to find the year

when the average salary was $1,904,000. b. Drawing a Graph Use a graph to check your solution to part (a).

10.5 Solve Quadratic Equations by Completing the Square

667

48. MULTI-STEP PROBLEM You have 80 feet of fencing to make

a rectangular horse pasture that covers 750 square feet. A barn will be used as one side of the pasture as shown. a. Write equations for the perimeter and area of

the pasture. b. Use substitution to solve the system of equations

W

W

from part (a). What are the possible dimensions of the pasture? 49.



SHORT RESPONSE You purchase stock for $16 per share, and you sell the stock 30 days later for $23.50 per share. The price y (in dollars) of a share during the 30 day period can be modeled by y 5 20.025x2 1 x 1 16 where x is the number of days after the stock is purchased. Could you have sold the stock earlier for $23.50 per share? Explain.

50. SNOWBOARDING During a “big air” competition,

snowboarders launch themselves from a half pipe, perform tricks in the air, and land back in the half pipe.

Initial vertical velocity = 24 ft/sec

a. Model Use the vertical motion model to write

an equation that models the height h (in feet) of a snowboarder as a function of the time t (in seconds) she is in the air.

16.4 ft

b. Apply How long is the snowboarder in the air if

she lands 13.2 feet above the base of the half pipe? Round your answer to the nearest tenth of a second. "MHFCSB

Cross section of a half pipe

at classzone.com

51. CHALLENGE You are knitting a rectangular scarf. The pattern you have

created will result in a scarf that has a length of 60 inches and a width of 4 inches. However, you happen to have enough yarn to cover an area of 480 square inches. You decide to increase the dimensions of the scarf so that all of your yarn will be used. If the increase in the length is 10 times the increase in the width, what will the dimensions of the scarf be?

MIXED REVIEW PREVIEW

Evaluate the expression for the given value of x. (p. 74)

Prepare for Lesson 10.6 in Exs. 52–57.

52. 3 1 x 2 6; x 5 8

53. 11 2 (2x) 1 15; x 5 21

54. 2x 1 18 2 20; x 5 210

55. 32 2 x 2 5; x 5 5

56. x 1 14.7 2 16.2; x 5 2.3

57. 29.2 2 (211.4) 2 x; x 5 24.5

Solve the proportion. (p. 168) 4 8 58. } 5} m23

3

5 3 59. } a5}

a15

c12 6

2c 2 3 5

60. } 5 }

Solve the equation.

668

61. (x 2 4)(x 1 9) 5 0 (p. 575)

62. x2 2 15x 1 26 5 0 (p. 583)

63. 3x2 1 10x 1 7 5 0 (p. 593)

64. 4x2 2 20x 1 25 5 0 (p. 600)

EXTRA PRACTICE for Lesson 10.5, p. 947

ONLINE QUIZ at classzone.com

Extension Use after Lesson 10.5

Graph Quadratic Functions in Vertex Form GOAL Graph quadratic functions in vertex form.

Key Vocabulary • vertex form

In Lesson 10.2, you graphed quadratic functions in standard form. Quadratic functions can also be written in vertex form, y 5 a(x 2 h)2 1 k where a Þ 0. In this form, the vertex of the graph can be easily determined.

For Your Notebook

KEY CONCEPT Graph of Vertex Form y 5 a(x 2 h)2 1 k

The graph of y 5 a(x 2 h)2 1 k is the graph of y 5 ax2 translated h units horizontally and k units vertically. Characteristics of the graph of y 5 a(x 2 h)2 1 k:

y

y 5 a(x 2 h) 2 1 k

• The vertex is (h, k).

(h, k)

y 5 ax 2

• The axis of symmetry is x 5 h. • The graph opens up if a > 0, and

k (0, 0)

x

h

the graph opens down if a < 0.

EXAMPLE 1

Graph a quadratic function in vertex form

Graph y 5 2(x 1 2)2 1 3. Solution

STEP 1 Identify the values of a, h, and k: a 5 21, h 5 22, and k 5 3. Because a < 0, the parabola opens down.

STEP 2 Draw the axis of symmetry, x 5 22. STEP 3 Plot the vertex (h, k) 5 (22, 3).

y

(22, 3)

3

STEP 4 Plot four points. Evaluate the function for two x-values less than the x-coordinate of the vertex.

1 x

2

x 5 23: y 5 2(23 1 2) 1 3 5 2 x 5 25: y 5 2(25 1 2)2 1 3 5 26

x 5 22

Plot the points (23, 2) and (25, 26) and their reflections, (21, 2) and (1, 26), in the axis of symmetry.

STEP 5 Draw a parabola through the plotted points.

Extension: Graph Quadratic Functions in Vertex Form

669

EXAMPLE 2

Graph a quadratic function

Graph y = x 2 2 8x 1 11. Solution

STEP 1 Write the function in vertex form by completing the square. y 5 x2 2 8x 1 11 y 1 ■ 5 (x2 2 8x 1 ■ ) 1 11

Write original function.

y 1 16 5 (x2 2 8x 1 16) 1 11

Add }

y 1 16 5 (x 2 4)2 1 11

Write x2 2 8x 1 16 as a square of a binomial.

y 5 (x 2 4)2 2 5

Prepare to complete the square. 2

1 28 2 2

5 (24) 2 5 16 to each side.

Subtract 16 from each side.

STEP 2 Identify the values of a, h, and k: a 5 1, h 5 4, and k 5 25. Because a > 0, the parabola opens up.

STEP 3 Draw the axis of symmetry, x 5 4.

y

STEP 4 Plot the vertex (h, k) 5 (4, 25).

x54

STEP 5 Plot four more points. Evaluate the

1

function for two x-values less than the x-coordinate of the vertex.

1

x

2

x 5 3: y 5 (3 2 4) 2 5 5 24 x 5 1: y 5 (1 2 4)2 2 5 5 4 Plot the points (3, 24) and (1, 4) and their reflections, (5, 24) and (7, 4), in the axis of symmetry.

(4, 25)

STEP 6 Draw a parabola through the plotted points.

PRACTICE EXAMPLE 1 on p. 669 for Exs. 1–6

EXAMPLE 2 on p. 670 for Exs. 7–12

Graph the quadratic function. Label the vertex and axis of symmetry. 1. y 5 (x 1 2)2 2 5

2. y 5 2(x 2 4)2 1 1

3. y 5 x2 1 3

4. y 5 3(x 2 1)2 2 2

5. y 5 22(x 1 5)2 2 2

1 6. y 5 2 } (x 1 4)2 1 4 2

Write the function in vertex form, then graph the function. Label the vertex and axis of symmetry. 7. y 5 x2 2 12x 1 36

8. y 5 x2 1 8x 1 15

9. y 5 2x2 1 10x 2 21

10. y 5 2x2 2 12x 1 19

11. y 5 23x2 2 6x 2 1

12. y 5 2}x2 2 6x 2 21

1 2

13. Write an equation in vertex form of the

parabola shown. Use the coordinates of the vertex and the coordinates of a point on the graph to write the equation.

y

(210, 5)

(22, 5)

1

(26, 1) 21

670

Chapter 10 Quadratic Equations and Functions

x