652 Chapter 10 Quadratic Equations and Functions

Key Vocabulary• square root,

p. 110

• perfect square,p. 111

Before You solved a quadratic equation by graphing.

Now You will solve a quadratic equation by finding square roots.

Why? So you can solve a problem about a falling object, as in Example 5.

10.4

Solve the equation.

a. 2x2 5 8 b. m2 2 18 5 218 c. b2 1 12 5 5

Solution

a. 2x2 5 8 Write original equation.

x2 5 4 Divide each side by 2.

x 5 6Ï}4 5 62 Take square roots of each side. Simplify.

c The solutions are 22 and 2.

b. m2 2 18 5 218 Write original equation.

m2 5 0 Add 18 to each side.

m 5 0 The square root of 0 is 0.

c The solution is 0.

c. b2 1 12 5 5 Write original equation.

b2 5 27 Subtract 12 from each side.

c Negative real numbers do not have real square roots. So, there isno solution.

E X A M P L E 1 Solve quadratic equations

To use square roots to solve a quadratic equation of the form ax2 1 c 5 0, firstisolate x2 on one side to obtain x2 5 d. Then use the following informationabout the solutions of x2 5 d to solve the equation.

KEY CONCEPT For Your Notebook

Solving x2 5 d by Taking Square Roots

• If d . 0, then x2 5 d has two solutions:x 5 6Ï

}

d .

• If d 5 0, then x2 5 d has one solution: x 5 0.

• If d , 0, then x2 5 d has no solution.x

y

d > 0

d < 0

d 5 0

FPO

ANOTHER WAY

You can also usefactoring to solve2x2 2 8 5 0: 2x2 2 8 5 0 2(x2 2 4) 5 02(x 2 2)(x 1 2) 5 0 x 5 2 or x 5 22

Use Square Roots toSolve Quadratic Equations

READING

Recall that in thiscourse, solutionsrefers to real-numbersolutions.

10.4 Use Square Roots to Solve Quadratic Equations 653

SIMPLIFYING SQUARE ROOTS In cases where you need to take the squareroot of a fraction whose numerator and denominator are perfect squares,

the radical can be written as a fraction. For example, Î}16}25

can be written

as 4}5

because 14}5 225 16

}25

.

Solve 4z2 5 9.

Solution

4z2 5 9 Write original equation.

z2 5 9}4

Divide each side by 4.

z 5 6Î}9}4

Take square roots of each side.

z 5 63}2 Simplify.

c The solutions are 23}2

and 3}2

.

E X A M P L E 2 Take square roots of a fraction

Solve 3x2 2 11 5 7. Round the solutions to the nearest hundredth.

Solution

3x2 2 11 5 7 Write original equation.

3x2 5 18 Add 11 to each side.

x2 5 6 Divide each side by 3.

x 5 6 Ï}

6 Take square roots of each side.

x ø 6 2.45 Use a calculator. Round to the nearest hundredth.

c The solutions are about 22.45 and about 2.45.

E X A M P L E 3 Approximate solutions of a quadratic equation

✓ GUIDED PRACTICE for Examples 1, 2, and 3

Solve the equation.

1. c2 2 25 5 0 2. 5w2 1 12 5 28 3. 2x2 1 11 5 11

4. 25x2 5 16 5. 9m2 5 100 6. 49b2 1 64 5 0

Solve the equation. Round the solutions to the nearest hundredth.

7. x2 1 4 5 14 8. 3k2 2 1 5 0 9. 2p2 2 7 5 2

APPROXIMATING SQUARE ROOTS In cases where d in the equation x2 5 dis not a perfect square or a fraction whose numerator and denominator arenot perfect squares, you need to approximate the square root. A calculatorcan be used to find an approximation.

654 Chapter 10 Quadratic Equations and Functions

E X A M P L E 4 Solve a quadratic equation

Solve 6(x 2 4)2 5 42. Round the solutions to the nearest hundredth.

6(x 2 4)2 5 42 Write original equation.

(x 2 4)2 5 7 Divide each side by 6.

x 2 4 5 6 Ï}7 Take square roots of each side.

x 5 4 6 Ï}7 Add 4 to each side.

c The solutions are 4 1 Ï}7 ø 6.65 and 4 2 Ï

}7 ø 1.35.

CHECK To check the solutions, first writethe equation so that 0 is on oneside as follows: 6(x 2 4)2 2 42 5 0.Then graph the related functiony 5 6(x 2 4)2 2 42. The x-interceptsappear to be about 6.6 and about1.3. So, each solution checks.

1.3 6.6

SPORTS EVENT During an ice hockey game, aremote-controlled blimp flies above the crowdand drops a numbered table-tennis ball. Thenumber on the ball corresponds to a prize.Use the information in the diagram to find theamount of time that the ball is in the air.

Solution

STEP 1 Use the vertical motion model to writean equation for the height h (in feet)of the ball as a function of time t (inseconds).

h 5 216t2 1 vt 1 s Vertical motion model

h 5 216t2 1 0t 1 45 Substitute for v and s.

STEP 2 Find the amount of time the ball is in theair by substituting 17 for h and solving for t.

h 5 216t2 1 45 Write model.

17 5 216t2 1 45 Substitute 17 for h.

228 5 216t2 Subtract 45 from each side.

28}16

5 t2 Divide each side by 216.

Î}28}16

5 t Take positive square root.

1.32 ø t Use a calculator.

c The ball is in the air for about 1.32 seconds.

E X A M P L E 5 Solve a multi-step problem

INTERPRETSOLUTION

Because the timecannot be a negativenumber, ignore thenegative square root.

ANOTHER WAY

For alternative methodsfor solving the problemin Example 5, turnto page 659 for theProblem SolvingWorkshop.

17 ft

45 ft

Not drawn to scale

DETERMINEVELOCITY

When an object isdropped, it has aninitial vertical velocityof 0 feet per second.

10.4 Use Square Roots to Solve Quadratic Equations 655

1. VOCABULARY Copy and complete: If b2 5 a, then b is a(n) ? of a.

2. ★ WRITING Describe two methods for solving a quadratic equation of theform ax2 1 c 5 0.

SOLVING EQUATIONS Solve the equation.

3. 3x2 2 3 5 0 4. 2x2 2 32 5 0 5. 4x2 2 400 5 0

6. 2m2 2 42 5 8 7. 15d2 5 0 8. a2 1 8 5 3

9. 4g2 1 10 5 11 10. 2w2 1 13 5 11 11. 9q2 2 35 5 14

12. 25b2 1 11 5 15 13. 3z2 2 18 5 218 14. 5n2 2 17 5 219

15. ★ MULTIPLE CHOICE Which of the following is a solution of the equation61 2 3n2 5 214?

A 5 B 10 C 25 D 625

16. ★ MULTIPLE CHOICE Which of the following is a solution of the equation13 2 36x2 5 212?

A 26}5

B 1}6

C 5}6

D 5

APPROXIMATING SQUARE ROOTS Solve the equation. Round the solutions tothe nearest hundredth.

17. x2 1 6 5 13 18. x2 1 11 5 24 19. 14 2 x2 5 17

20. 2a2 2 9 5 11 21. 4 2 k2 5 4 22. 5 1 3p2 5 38

23. 53 5 8 1 9m2 24. 221 5 15 2 2z2 25. 7c2 5 100

26. 5d2 1 2 5 6 27. 4b2 2 5 5 2 28. 9n2 2 14 5 23

29. ★ MULTIPLE CHOICE The equation 17 21}4 x2 5 12 has a solution between

which two integers?

A 1 and 2 B 2 and 3 C 3 and 4 D 4 and 5

10.4 EXERCISES

✓ GUIDED PRACTICE for Examples 4 and 5

Solve the equation. Round the solutions to the nearest hundredth, if necessary.

10. 2(x 2 2)2 5 18 11. 4(q 2 3)2 5 28 12. 3(t 1 5)2 5 24

13. WHAT IF? In Example 5, suppose the table-tennis ball is released 58 feetabove the ground and is caught 12 feet above the ground. Find the amountof time that the ball is in the air. Round your answer to the nearesthundredth of a second.

EXAMPLE 3

on p. 653for Exs. 17–29

EXAMPLES1 and 2

on pp. 652–653for Exs. 3–16

HOMEWORKKEY

5 WORKED-OUT SOLUTIONSon p. WS1 for Exs. 25 and 59

★ 5 STANDARDIZED TEST PRACTICEExs. 2, 15, 16, 29, 51, 52, 57, and 60

5 MULTIPLE REPRESENTATIONSEx. 62

SKILL PRACTICE

656

ERROR ANALYSIS Describe and correct the error in solving the equation.

30. 2x2 2 54 5 18 31. 7d2 2 6 5 217

SOLVING EQUATIONS Solve the equation. Round the solutions to the nearesthundredth.

32. (x 2 7)2 5 6 33. 7(x 2 3)2 5 35 34. 6(x 1 4)2 5 18

35. 20 5 2(m 1 5)2 36. 5(a 2 2)2 5 70 37. 21 5 3(z 1 14)2

38. 1}2

(c 2 8)2 5 3 39. 3}2

(n 1 1)2 5 33 40. 4}3

(k 2 6)2 5 20

SOLVING EQUATIONS Solve the equation. Round the solutions to the nearesthundredth, if necessary.

41. 3x2 2 35 5 45 2 2x2 42. 42 5 3(x2 1 5) 43. 11x2 1 3 5 5(4x2 2 3)

44. 1 t 2 5}

3 225 49 45. 111w 2 7

}2 22 2 20 5 101 46. (4m2 2 6)2 5 81

GEOMETRY Use the given area A of the circle to find the radius r or thediameter d to the nearest hundredth.

47. A 5 144π in.2 48. A 5 21π m2 49. A 5 34π ft2

r r d

50. REASONING An equation of the graph shown is

y 5 1}2

(x 2 2)2 1 1. Two points on the parabola have

y-coordinates of 9. Find the x-coordinates of these points.

51. ★ SHORT RESPONSE Solve x2 5 1.44 without using a calculator. Explainyour reasoning.

52. ★ OPEN – ENDED Give values for a and c so that ax2 1 c 5 0 has(a) two solutions, (b) one solution, and (c) no solution.

CHALLENGE Solve the equation without graphing.

53. x2 2 12x 1 36 5 64 54. x2 1 14x 1 49 5 16 55. x2 1 18x 1 81 5 25

2x2 2 54 5 18

2x2 5 72

x2 5 36

x 5 Ï}36

x 5 6

The solution is 6.

7d2 2 6 5 217

7d2 5 211

d2 5 211}7

d ø 61.25

The solutions are about 21.25and about 1.25.

x

y

1

1

EXAMPLE 4

on p. 654for Exs. 32–40

★ 5 STANDARDIZEDTEST PRACTICE

5 WORKED-OUT SOLUTIONSon p. WS1

10.4 Use Square Roots to Solve Quadratic Equations 657

56. FALLING OBJECT Fenway Park is a Major League Baseball park in Boston,Massachusetts. The park offers seats on top of the left field wall. A personsitting in one of these seats accidentally drops his sunglasses on the field.The height h (in feet) of the sunglasses can be modeled by the functionh 5 216t2 1 38 where t is the time (in seconds) since the sunglasseswere dropped. Find the time it takes for the sunglasses to reach the field.Round your answer to the nearest hundredth of a second.

57. ★ MULTIPLE CHOICE Which equation can be used to find the time ittakes for an object to hit the ground after it was dropped from a heightof 68 feet?

A 216t2 5 0 B 216t2 2 68 5 0 C 216t2 1 68 5 0 D 216t2 5 68

58. INTERNET USAGE For the period 1995–2001, the number y (inthousands) of Internet users worldwide can be modeled by thefunction y 5 12,697x2 1 55,722 where x is the number of years since1995. Between which two years did the number of Internet usersworldwide reach 100,000,000?

59. GEMOLOGY To find the weight w (in carats) of round faceted gems,gemologists use the formula w 5 0.0018D2ds where D is the diameter (inmillimeters) of the gem, d is the depth (in millimeters) of the gem, ands is the specific gravity of the gem. Find the diameter to the nearest tenthof a millimeter of each round faceted gem in the table.

a.

b.

c.

60. ★ SHORT RESPONSE In deep water, the speed s (in meters per second)of a series of waves and the wavelength L (in meters) of the waves arerelated by the equation 2πs2 5 9.8L.

a. Find the speed to the nearest hundredth of a meter per second of aseries of waves with the following wavelengths: 6 meters, 10 meters,and 25 meters. (Use 3.14 for π.)

b. Does the speed of a series of waves increase or decrease as thewavelength of the waves increases? Explain.

PROBLEM SOLVING

EXAMPLE 5

on p. 654for Exs. 56–57

Gem Weight(carats)

Depth(mm)

Specificgravity

Diameter(mm)

Amethyst 1 4.5 2.65 ?

Diamond 1 4.5 3.52 ?

Ruby 1 4.5 4.00 ?

The wavelength L is the distance between one crest and the next.Crest Crest

L

658

61. MULTI-STEP PROBLEM The Doyle log rule is a formulaused to estimate the amount of lumber that can besawn from logs of various sizes. The amount of lumber

V (in board feet) is given by V 5L(D 2 4)2

}16

where L is

the length (in feet) of a log and D is the small-enddiameter (in inches) of the log.

a. Solve the formula for D.

b. Use the rewritten formula to find the diameters, to the nearest tenthof a foot, of logs that will yield 50 board feet and have the followinglengths: 16 feet, 18 feet, 20 feet, and 22 feet.

62. MULTIPLE REPRESENTATIONS A ride at an amusementpark lifts seated riders 250 feet above the ground. Thenthe riders are dropped. They experience free fall until thebrakes are activated at 105 feet above the ground.

a. Writing an Equation Use the vertical motion model towrite an equation for the height h (in feet) of the ridersas a function of the time t (in seconds) into the free fall.

b. Making a Table Make a table that shows the height of theriders after 0, 1, 2, 3, and 4 seconds. Use the tableto estimate the amount of time the riders experiencefree fall.

c. Solving an Equation Use the equation to find the amountof time, to the nearest tenth of a second, that the ridersexperience free fall.

63. CHALLENGE The height h (in feet) of a dropped object on any planet

can be modeled by h 5 2g}2

t2 1 s where g is the acceleration (in feet per

second per second) due to the planet’s gravity, t is the time (in seconds)after the object is dropped, and s is the initial height (in feet) of theobject. Suppose the same object is dropped from the same height onEarth and Mars. Given that g is 32 feet per second per second on Earthand 12 feet per second per second on Mars, on which planet will theobject hit the ground first? Explain.

EXTRA PRACTICE for Lesson 10.4, p. 947 ONLINE QUIZ at classzone.com

MIXED REVIEW

Evaluate the power. (p. 2)

64. 15}2 2

265. 19

}5 22 66. 13

}4 22 67. 17

}2 2

2

Write an equation of the line with the given slope and y-intercept. (p. 283)

68. slope: 29 69. slope: 7 70. slope: 3y-intercept: 11 y-intercept: 27 y-intercept: 22

Write an equation of the line that passes through the given point and isperpendicular to the given line. (p. 319)

71. (1, 21), y 5 2x 72. (0, 8), y 5 4x 1 1 73. (29, 24), y 5 23x 1 6

PREVIEW

Prepare forLesson 10.5 inExs. 64–67.

Diameter

Boards

Using Alternative Methods 659

PRO B L E M

Using Factoring One alternative approach is to use factoring.

STEP 1 Write an equation for the height h (in feet) of the ball as a function oftime t (in seconds) after it is dropped using the vertical motion model.

h 5 216t2 1 vt 1 s Vertical motion model

h 5 216t2 1 0t 1 45 Substitute 0 for v and 45 for s.

STEP 2 Substitute 17 for h to find the time it takes the ball to reach aheight of 17 feet. Then write the equation so that 0 is on one side.

17 5 216t2 1 45 Substitute 17 for h.

0 5 216t2 1 28 Subtract 17 from each side.

STEP 3 Solve the equation by factoring. Replace 28 with the closestperfect square, 25, so that the right side of the equation isfactorable as a difference of two squares.

0 5 216t2 1 25 Use 25 as an approximation for 28.

0 5 2(16t2 2 25) Factor out 21.

0 5 2(4t 2 5)(4t 1 5) Difference of two squares pattern

4t 2 5 5 0 or 4t 1 5 5 0 Zero-product property

t 5 5}4

or t 5 2 5}4

Solve for t.

c The ball is in the air about 5}4

, or 1.25, seconds.

Another Way to Solve Example 5, page 654

MULTIPLE REPRESENTATIONS In Example 5 on page 654, you saw how tosolve a problem about a dropped table-tennis ball by using a square root.You can also solve the problem by using factoring or by using a table.

M E T H O D 1

LESSON 10.4

SPORTS EVENT During an ice hockeygame, a remote-controlled blimp fliesabove the crowd and drops a numberedtable-tennis ball. The number on theball corresponds to a prize. Use theinformation in the diagram to find theamount of time that the ball is in the air.

17 ft

45 ft

Not drawn to scale

USE ANAPPROXIMATION

By replacing 28 with25, you will obtainan answer that is anapproximation of theamount of time that theball is in the air.

ALTERNATIVE METHODSALTERNATIVE METHODSUsingUsing

660 Chapter 10 Quadratic Equations and Functions

Using a Table Another approach is to make and use a table.

STEP 1 Make a table that shows the heighth (in feet) of the ball by substitutingvalues for time t (in seconds) in the functionh 5 216t2 1 45. Use increments of 1 second.

STEP 2 Identify the time interval in which theheight of the ball is 17 feet. This happensbetween 1 and 2 seconds.

STEP 3 Make a second table using increments of0.1 second to get a closer approximation.

c The ball is in the air about 1.3 seconds.

M E T H O D 2

PR AC T I C E

1. WHAT IF? In the problem on page 659,suppose the ball is caught at a height of10 feet. For how many seconds is the ballin the air? Solve this problem using twodifferent methods.

2. OPEN-ENDED Describe a problem about adropped object. Then solve the problem andexplain what your solution means in thissituation.

3. GEOMETRY The box below is arectangular prism with the dimensionsshown.

x in.

5 in.

5x in.

a. Write an equation that gives the volumeV (in cubic inches) of the box as afunction of x.

b. The volume of the box is 83 cubic inches.Find the dimensions of the box. Usefactoring to solve the problem.

c. Make a table to check your answer frompart (b).

4. TRAPEZE You are learning how to performon a trapeze. While hanging from a stilltrapeze bar, your shoe comes loose and fallsto a safety net that is 6 feet off the ground. Ifyour shoe falls from a height of 54 feet, howlong does it take your shoe to hit the net?Choose any method for solving the problem.Show your steps.

5. ERROR ANALYSIS A student solved theproblem in Exercise 4 as shown below.Describe and correct the error.

Time t(seconds)

Height h(feet)

0 45

1 29

2 219

Time t(seconds)

Height h(feet)

1.0 29.00

1.1 25.64

1.2 21.96

1.3 17.96

1.4 13.64

Let t be the time (in seconds) that theshoe is in the air.

6 5 216t2 1 54

0 5 216t2 1 60

Replace 60 with the closest perfectsquare, 64.

0 5 216t2 1 64

0 5 216(t 2 2)(t 1 2)

t 5 2 or t 5 22

It takes about 2 seconds.

Mixed Review of Problem Solving 661

Lessons 10.1–10.4

STATE TEST PRACTICEclasszone.com

2x ft

(14 2 x) ft

1. MULTI-STEP PROBLEM A company’s yearlyprofits from 1996 to 2006 can be modeled bythe function y 5 x2 2 8x 1 80 where y is theprofit (in thousands of dollars) and x is thenumber of years since 1996.

a. In what year did the company experienceits lowest yearly profit?

b. What was the lowest yearly profit?

2. MULTI-STEP PROBLEM Use the rectanglebelow.

a. Find the value of x that gives the greatestpossible area of the rectangle.

b. What is the greatest possible area of therectangle?

3. EXTENDED RESPONSE You throw a lacrosseball twice using a lacrosse stick.

a. For your first throw, the ball is released8 feet above the ground with an initialvertical velocity of 35 feet per second.Use the vertical motion model to write anequation for the height h (in feet) of theball as a function of time t (in seconds).

b. For your second throw, the ball is released7 feet above the ground with an initialvertical velocity of 45 feet per second.Use the vertical motion model to write anequation for the height h (in feet) of theball as a function of time t (in seconds).

c. If no one catches either throw, for whichthrow is the ball in the air longer? Explain.

4. OPEN-ENDED Describe a real-world situationof an object being dropped. Then write anequation that models the height of the objectas a function of time. Use the equation todetermine the time it takes the object to hitthe ground.

5. SHORT RESPONSE A football player isattempting a field goal. The path of thekicked football can be modeled by thegraph of y 5 20.03x2 1 1.8x where x is thehorizontal distance (in yards) traveled by thefootball and y is the corresponding height(in feet) of the football. Will the football passover the goal post that is 10 feet above theground and 45 yards away? Explain.

6. GRIDDED ANSWER The force F (in newtons)a rider feels while a train goes around a

curve is given by F 5 mv2}r where m is the

mass (in kilograms) of the rider, v is thevelocity (in meters per second) of thetrain, and r is the radius (in meters) of thecurve. A rider with a mass of 75 kilogramsexperiences a force of 18,150 newtons, whilegoing around a curve that has a radius of8 meters. Find the velocity (in meters persecond) the train travels around the curve.

7. SHORT RESPONSE The opening of thetunnel shown can be modeled by the graphof the equation y 5 20.18x2 1 4.4x 2 12where x and y are measured in feet.

a. Find the maximum height of the tunnel.

b. A semi trailer is 7.5 feet wide, and the topof the trailer is 10.5 feet above the ground.Given that traffic travels one way on onelane through the center of the tunnel, willthe semi trailer fit through the opening ofthe tunnel? Explain.

MIXED REVIEW of Problem SolvingMIXED REVIEW of Problem Solving

662 Chapter 10 Quadratic Equations and Functions

D R A W C O N C L U S I O N S Use your observations to complete these exercises

1. Copy and complete the table using algebra tiles.

2. In the statement x2 1 bx 1 c 5 (x 1 d)2, how are b and d related?How are c and d related?

3. Use your answer to Exercise 2 to predict the number of 1-tilesyou would need to add to complete the square for theexpression x2 1 18x.

Expression Number of 1-tiles neededto complete the square

Expression writtenas a square

x2 1 4x 4 x2 1 4x 1 4 5 (x 1 2)2

x2 1 6x ? ?

x2 1 8x ? ?

x2 1 10x ? ?

Q U E S T I O N How can you use algebra tiles to complete the square?

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

E X P L O R E Complete the square

Find the value of c that makes x2 1 4x 1 c a perfect square trinomial.

STEP 1 Model expression

Use algebra tiles to model theexpression x2 1 4x. You willneed one x2-tile and fourx-tiles for this expression.

STEP 2 Rearrange tiles

Arrange the tiles to form asquare. The arrangement willbe incomplete in one of thecorners.

STEP 3 Complete the square

Determine the number of1-tiles needed to complete thesquare. The number of 1-tilesis the value of c. So, the perfectsquare trinomial is x2 1 4x 1 4or (x 1 2)2.

Algebraclasszone.com

Use before Lesson 10.5

10.5 Completing the Square Using Algebra TilesMATERIALS • algebra tiles

ACTIVITYACTIVITYInvestigating Algebra

InvestigatingAlgebr

ggarr