QUADRATIC FUNCTIONS - PBworksalgebra2text.pbworks.com/f/Chapter+5.pdf250 Chapter 5 Quadratic Functions Graphing a Quadratic Function Graph y = 2x2 º 8x + 6.SOLUTION Note that the

QUADRATICFUNCTIONSQUADRATICFUNCTIONS

� How can you model the heightof lava from an erupting volcano?

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http://www.classzone.com

APPLICATION: Volcanoes

Volcanic eruptions can eject lavahundreds of feet into the air, creat-ing spectacular but dangerous"lava fountains." As the lava coolsand hardens, it may accumulate toform the cone shape of a volcano.

Think & DiscussThe highest recorded lava fountain occurred during a1959 eruption at Kilauea Iki Crater in Hawaii. Thegraph models the height of a typical lava fragment inthe fountain while the fragment was in the air.

1. Estimate the lava fragment’s maximum heightabove the ground.

2. For how long was the lava fragment in the air?How did you use the graph to get your answer?

Learn More About ItYou will work with an equation modeling the heightof lava in Exercise 80 on p. 297.

APPLICATION LINK Visit www.mcdougallittell.comfor more information about volcanoes.

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Time in air (sec)4

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0 12 16 20

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248 Chapter 5

What’s the chapter about?Chapter 5 is about quadratic functions, equations, and inequalities. Many real-lifesituations can be modeled by quadratic functions. In Chapter 5 you’ll learn

• four ways to solve quadratic equations.• how to graph quadratic functions and inequalities.

CHAPTER

5Study Guide

PREVIEW

Are you ready for the chapter?SKILL REVIEW Do these exercises to review key skills that you’ll apply in thischapter. See the given reference page if there is something you don’t understand.

Solve the equation. (Review Examples 1–3, pp. 19 and 20)

1. 3x º 5 = 0 2. 4(x + 6) = 12 3. 2x + 1 = ºx + 7

Graph the inequality. (Review Example 3, p. 109)

4. x + y > 5 5. 3x º 2y ≤ 12 6. y ≥ º2x

Graph the function and label the vertex. (Review Example 1, p. 123)

7. y = |x| + 2 8. y = |x º 3| 9. y = º2|x + 1| º 4

PREPARE

Here’s a study strategy!STUDYSTRATEGY

� Review

• linear equation, p. 19• linear inequality, pp. 41, 108• absolute value, p. 50• linear function, p. 69• x-intercept, p. 84• best-fitting line, p. 101• vertex, p. 122

� New

• quadratic function, p. 249• parabola, p. 249• factoring, p. 256• quadratic equation, p. 257• zero of a function, p. 259• square root, p. 264• complex number, p. 272

• completing the square, p. 282

• quadratic formula, p. 291• discriminant, p. 293• quadratic inequality,

pp. 299, 301

• best-fitting quadratic model, p. 308

KEY VOCABULARY

STUDENT HELP

Study Tip“Student Help” boxesthroughout the chaptergive you study tips andtell you where to look forextra help in this bookand on the Internet.

Troubleshoot

After you complete each lesson, look back andidentify your trouble spots, such as concepts youdidn’t understand or homework problems you haddifficulty solving. Review the material given in thelesson and try to solve any difficult problemsagain. If you’re still having trouble, seek the helpof another student or your teacher.


5.1 Graphing Quadratic Functions 249

Graphing Quadratic FunctionsGRAPHING A QUADRATIC FUNCTION

A has the form y = ax2 + bx + cwhere a ≠ 0. The graph of a quadratic function is U-shaped and is called a

For instance, the graphs of y = x2 and y = ºx2 areshown at the right. The origin is the lowest point on the graph of y = x2 and the highest point on the graph ofy = ºx2. The lowest or highest point on the graph of aquadratic function is called the

The graphs of y = x2 and y = ºx2 are symmetric aboutthe y-axis, called the axis of symmetry. In general, the

for the graph of a quadratic functionis the vertical line through the vertex.

In the activity you examined the graph of the simple quadratic function y = ax2. The graph of the more general function y = ax2 + bx + c is described below.

axis of symmetry

vertex.

parabola.

quadratic function

GOAL 1

Graph quadraticfunctions.

Use quadraticfunctions to solve real-lifeproblems, such as findingcomfortable temperatures in Example 5.

� To model real-life objects,such as the cables of theGolden Gate Bridge inExample 6.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

5.1RE

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x

y � x 2

y � �x 2

vertex axis ofsymmetry

y

2

2

Investigating ParabolasUse a graphing calculator to graph each of these functions in the same

viewing window: y = �12�x2, y = x2, y = 2x2, and y = 3x2.

Repeat Step 1 for these functions: y = º�12�x2, y = ºx2, y = º2x2, and

y = º3x2.

What are the vertex and axis of symmetry of the graph of y = ax2?

Describe the effect of a on the graph of y = ax2.4

3

2

1

DevelopingConcepts

ACTIVITY

The graph of y = ax2 + bx + c is a parabola with these characteristics:

• The parabola opens up if a > 0 and opens down if a < 0. The parabola iswider than the graph of y = x2 if |a| < 1 and narrower than the graph of y = x2 if |a| > 1.

• The x-coordinate of the vertex is º�2ba�.

• The axis of symmetry is the vertical line x = º�2ba�.

THE GRAPH OF A QUADRATIC FUNCTIONCONCEPTSUMMARY


250 Chapter 5 Quadratic Functions

Graphing a Quadratic Function

Graph y = 2x2 º 8x + 6.

SOLUTIONNote that the coefficients for this function are a = 2,b = º8, and c = 6. Since a > 0, the parabola opens up.

Find and plot the vertex. The x-coordinate is:

x = º�2ba� = º�2

º(28)� = 2

The y-coordinate is:

y = 2(2)2 º 8(2) + 6 = º2

So, the vertex is (2, º2).

Draw the axis of symmetry x = 2.

Plot two points on one side of the axis of symmetry, such as (1, 0) and (0, 6). Usesymmetry to plot two more points, such as (3, 0) and (4, 6).

Draw a parabola through the plotted points.

. . . . . . . . . .

The quadratic function y = ax2 + bx + c is written in Two otheruseful forms for quadratic functions are given below.

Graphing a Quadratic Function in Vertex Form

Graph y = º�12�(x + 3)2 + 4.

SOLUTIONThe function is in vertex form y = a(x º h)2 + k

where a = º�12�, h = º3, and k = 4. Since a < 0,

the parabola opens down. To graph the function, first plotthe vertex (h, k) = (º3, 4). Draw the axis of symmetry x = º3 and plot two points on one side of it, such as(º1, 2) and (1, º4). Use symmetry to complete the graph.

E X A M P L E 2

standard form.

E X A M P L E 1

x

(0, 6) (4, 6)

(1, 0)(3, 0)

(2, �2)

y

1

FORM OF QUADRATIC FUNCTION CHARACTERISTICS OF GRAPH

y = a(x º h)2 + k The vertex is (h, k).

The axis of symmetry is x = h.

y = a(x º p)(x º q) The x-intercepts are p and q.

The axis of symmetry is halfwaybetween (p, 0) and (q, 0).

For both forms, the graph opens up if a > 0 and opens down if a < 0.

Intercept form:

Vertex form:

VERTEX AND INTERCEPT FORMS OF A QUADRATIC FUNCTION

y

x

(1, �4)(�7, �4)

(�5, 2)

(�3, 4)(�1, 2)

4

1

Skills Review For help with symmetry,see p. 919.

STUDENT HELP

Look Back For help with graphingfunctions, see p. 123.

STUDENT HELP



Graphing a Quadratic Function in Intercept Form

Graph y = º(x + 2)(x º 4).

SOLUTIONThe quadratic function is in intercept form y = a(x º p)(x º q) where a = º1, p = º2, and q = 4. The x-intercepts occur at (º2, 0) and (4, 0). The axis of symmetry lies halfway between these points, at x = 1. So, the x-coordinate of the vertex is x = 1 and the y-coordinate of the vertex is:

y = º(1 + 2)(1 º 4) = 9

The graph of the function is shown.

. . . . . . . . . .

You can change quadratic functions from intercept form or vertex form to standardform by multiplying algebraic expressions. One method for multiplying expressionscontaining two terms is FOIL. Using this method, you add the products of the Firstterms, the Outer terms, the Inner terms, and the Last terms. Here is an example:

F O I L

(x + 3)(x + 5) = x2 + 5x + 3x + 15 = x2 + 8x + 15

Methods for changing from standard form to intercept form or vertex form will bediscussed in Lessons 5.2 and 5.5.

Writing Quadratic Functions in Standard Form

Write the quadratic function in standard form.

a. y = º(x + 4)(x º 9) b. y = 3(x º 1)2 + 8

SOLUTIONa. y = º(x + 4)(x º 9) Write original function.

= º(x2 º 9x + 4x º 36) Multiply using FOIL.

= º(x2 º 5x º 36) Combine like terms.

= ºx2 + 5x + 36 Use distributive property.

b. y = 3(x º 1)2 + 8 Write original function.

= 3(x º 1)(x º 1) + 8 Rewrite (x º 1)2.

= 3(x2 º x º x + 1) + 8 Multiply using FOIL.

= 3(x2 º 2x + 1) + 8 Combine like terms.

= 3x2 º 6x + 3 + 8 Use distributive property.

= 3x2 º 6x + 11 Combine like terms.

E X A M P L E 4

E X A M P L E 3

x

y (1, 9)

1

24�2

Skills Review For help with multiplyingalgebraic expressions,see p. 937.

STUDENT HELP



USING QUADRATIC FUNCTIONS IN REAL LIFE

Using a Quadratic Model in Standard Form

Researchers conducted an experiment to determine temperatures at which people feelcomfortable. The percent y of test subjects who felt comfortable at temperature x(in degrees Fahrenheit) can be modeled by:

y = º3.678x2 + 527.3x º 18,807

What temperature made the greatest percent of test subjects comfortable? At thattemperature, what percent felt comfortable? � Source: Design with Climate

SOLUTIONSince a = º3.678 is negative, the graph of the quadratic function opens down and the function has a maximum value. The maximum value occurs at:

x = º�2ba� = º�2(º

5237.6.378)� ≈ 72

The corresponding value of y is:

y = º3.678(72)2 + 527.3(72) º 18,807 ≈ 92

� The temperature that made the greatest percent of test subjects comfortable wasabout 72°F. At that temperature about 92% of the subjects felt comfortable.

Using a Quadratic Model in Vertex Form

CIVIL ENGINEERING The Golden Gate Bridge in San Francisco has two towers thatrise 500 feet above the road and are connected by suspension cables as shown. Eachcable forms a parabola with equation

y = �89160� (x º 2100)

2 + 8

where x and y are measured in feet.� Source: Golden Gate Bridge, Highway and

Transportation District

a. What is the distance d betweenthe two towers?

b. What is the height ¬ above theroad of a cable at its lowest point?

SOLUTIONa. The vertex of the parabola is (2100, 8), so a cable’s lowest point is 2100 feet

from the left tower shown above. Since the heights of the two towers are thesame, the symmetry of the parabola implies that the vertex is also 2100 feetfrom the right tower. Therefore, the towers are d = 2(2100) = 4200 feet apart.

b. The height ¬ above the road of a cable at its lowest point is the y-coordinate ofthe vertex. Since the vertex is (2100, 8), this height is ¬ = 8 feet.

E X A M P L E 6

E X A M P L E 5

GOAL 2

REAL LIFE

REAL LIFE

Temperature

X=71.691489 Y=92.217379

CIVIL ENGINEERCivil engineers

design bridges, roads, build-ings, and other structures. In 1996 civil engineers heldabout 196,000 jobs in theUnited States.

CAREER LINKwww.mcdougallittell.com

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1. Complete this statement: The graph of a quadratic function is called a(n) ��? .

2. Does the graph of y = 3x2 º x º 2 open up or down? Explain.

3. Is y = º2(x º 5)(x º 8) in standard form, vertex form, or intercept form?

Graph the quadratic function. Label the vertex and axis of symmetry.

4. y = x2 º 4x + 7 5. y = 2(x + 1)2 º 4 6. y = º(x + 2)(x º 1)

7. y = º�13�x2 º 2x º 3 8. y = º�35�(x º 4)

2 + 6 9. y = �52�x(x º 3)

Write the quadratic function in standard form.

10. y = (x + 1)(x + 2) 11. y = º2(x + 4)(x º 3) 12. y = 4(x º 1)2 + 5

13. y = º(x + 2)2 º 7 14. y = º�12�(x º 6)(x º 8) 15. y = �23�(x º 9)

2 º 4

16. The equation given in Example 5 is based on temperaturepreferences of both male and female test subjects. Researchers also analyzed datafor males and females separately and obtained the equations below.

Males: y = º4.290x2 + 612.6x º 21,773

Females: y = º6.224x2 + 908.9x º 33,092

What was the most comfortable temperature for the males? for the females?

MATCHING GRAPHS Match the quadratic function with its graph.

17. y = (x + 2)(x º 3) 18. y = º(x º 3)2 + 2 19. y = x2 º 6x + 11

A. B. C.

GRAPHING WITH STANDARD FORM Graph the quadratic function. Label thevertex and axis of symmetry.

20. y = x2 º 2x º 1 21. y = 2x2 º 12x + 19 22. y = ºx2 + 4x º 2

23. y = º3x2 + 5 24. y = �12�x2 + 4x + 5 25. y = º�16�x

2 º x º 3

GRAPHING WITH VERTEX FORM Graph the quadratic function. Label thevertex and axis of symmetry.

26. y = (x º 1)2 + 2 27. y = º(x º 2)2 º 1 28. y = º2(x + 3)2 º 4

29. y = 3(x + 4)2 + 5 30. y = º�13�(x + 1)2 + 3 31. y = �54�(x º 3)

2

y

x1

1y

x

1

1

y

x

1

3

PRACTICE AND APPLICATIONS

SCIENCE CONNECTION

GUIDED PRACTICEVocabulary Check ✓

Concept Check ✓

Skill Check ✓

STUDENT HELP

Extra Practice to help you masterskills is on p. 945.

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 17–25Example 2: Exs. 17–19,

26–31Example 3: Exs. 17–19,

32–37Example 4: Exs. 38–49Examples 5, 6: Exs. 51–54



GRAPHING WITH INTERCEPT FORM Graph the quadratic function. Label thevertex, axis of symmetry, and x-intercepts.

32. y = (x º 2)(x º 6) 33. y = 4(x + 1)(x º 1) 34. y = º(x + 3)(x + 5)

35. y = �13�(x + 4)(x + 1) 36. y = º�12�(x º 3)(x + 2) 37. y = º3x(x º 2)

WRITING IN STANDARD FORM Write the quadratic function in standard form.

38. y = (x + 5)(x + 2) 39. y = º(x + 3)(x º 4) 40. y = 2(x º 1)(x º 6)

41. y = º3(x º 7)(x + 4) 42. y = (5x + 8)(4x + 1) 43. y = (x + 3)2 + 2

44. y = º(x º 5)2 + 11 45. y = º6(x º 2)2 º 9 46. y = 8(x + 7)2 º 20

47. y = º(9x + 2)2 + 4x 48. y = º�73�(x + 6)(x + 3) 49. y = �12�(8x º 1)

2 º �23

�

50. VISUAL THINKING In parts (a) and (b), use a graphing calculator to examine how b and c affect the graph of y = ax2 + bx + c.

a. Graph y = x2 + c for c = º2, º1, 0, 1, and 2. Use the same viewing windowfor all the graphs. How do the graphs change as c increases?

b. Graph y = x2 + bx for b = º2, º1, 0, 1, and 2. Use the same viewingwindow for all the graphs. How do the graphs change as b increases?

51. AUTOMOBILES The engine torque y (in foot-pounds) of one modelof car is given by

y = º3.75x2 + 23.2x + 38.8

where x is the speed of the engine (in thousands of revolutions per minute). Find the engine speed that maximizes torque. What is the maximum torque?

52. SPORTS Although a football field appears to be flat, its surface is actuallyshaped like a parabola so that rain runs off to either side. The cross section of afield with synthetic turf can be modeled by

y = º0.000234(x º 80)2 + 1.5

where x and y are measured in feet. What is the field’s width? What is the maximum height of the field’s surface? � Source: Boston College

53. PHYSIOLOGY Scientists determined that the rate y (in calories per minute)at which you use energy while walking can be modeled by

y = 0.00849(x º 90.2)2 + 51.3, 50 ≤ x ≤ 150

where x is your walking speed (in meters per minute). Graph the function on thegiven domain. Describe how energy use changes as walking speed increases.What speed minimizes energy use? � Source: Bioenergetics and Growth

54. The woodland jumpingmouse can hop surprisingly long distances given itssmall size. A relatively long hop can be modeled by

y = º�29�x(x º 6)

where x and y are measured in feet. How far can awoodland jumping mouse hop? How high can it hop?� Source: University of Michigan Museum of Zoology

BIOLOGY CONNECTION

y

xNot drawn to scale

surface offootball field

y

xNot drawn to scale

TORQUE, the focusof Ex. 51, is the

“twisting force” produced by the crankshaft in a car’sengine. As torque increases,a car is able to acceleratemore quickly.

APPLICATION LINKwww.mcdougallittell.com

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HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor help with problemsolving in Ex. 54.

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55. MULTI-STEP PROBLEM A kernel of popcorn contains water that expandswhen the kernel is heated, causing it to pop. The equations below give the“popping volume” y (in cubic centimeters per gram) of popcorn with moisturecontent x (as a percent of the popcorn’s weight). � Source: Cereal Chemistry

Hot-air popping: y = º0.761x2 + 21.4x º 94.8

Hot-oil popping: y = º0.652x2 + 17.7x º 76.0

a. For hot-air popping, what moisture content maximizes popping volume? What is the maximum volume?

b. For hot-oil popping, what moisture content maximizes popping volume? What is the maximum volume?

c. The moisture content of popcorn typically ranges from 8% to 18%. Graph the equations for hot-air and hot-oil popping on the interval 8 ≤ x ≤ 18.

d. Writing Based on the graphs from part (c), what general statement can you make about the volume of popcorn produced from hot-air popping versushot-oil popping for any moisture content in the interval 8 ≤ x ≤ 18?

56. LOGICAL REASONING Write y = a(x º h)2 + k and y = a(x º p)(x º q) in

standard form. Knowing that the vertex of the graph of y = ax2 + bx + c occurs

at x = º�2ba�, show that the vertex for y = a(x º h)

2 + k occurs at x = h and that

the vertex for y = a(x º p)(x º q) occurs at x = �p +

2q

�.

SOLVING LINEAR EQUATIONS Solve the equation. (Review 1.3 for 5.2)

57. x º 2 = 0 58. 2x + 5 = 0 59. º4x º 7 = 21

60. 3x + 9 = ºx + 1 61. 6(x + 8) = 18 62. 5(4x º 1) = 2(x + 3)

63. 0.6x = 0.2x + 2.8 64. �78x� º �35

x� = �12

1� 65. �1

52x� + �14� = �6

x� º �2

1�

GRAPHING IN THREE DIMENSIONS Sketch the graph of the equation. Labelthe points where the graph crosses the x-, y-, and z-axes. (Review 3.5)

66. x + y + z = 4 67. x + y + 2z = 6 68. 3x + 4y + z = 12

69. 5x + 5y + 2z = 10 70. 2x + 7y + 3z = 42 71. x + 3y º 3z = 9

USING CRAMER’S RULE Use Cramer’s rule to solve the linear system. (Review 4.3)

72. x + y = 1 73. 2x + y = 5 74. 7x º 10y = º15º5x + y = 19 3x º 4y = 2 x + 2y = º9

75. 5x + 2y + 2z = 4 76. x + 3y + z = 5 77. 2x º 3y º 9z = 113x + y º 6z = º4 ºx + y + z = 7 6x + y º z = 45ºx º y º z = 1 2x º 7y + 5z = 28 9x º 2y + 4z = 56

78. WEATHER In January, 1996, rain and melting snow caused the depth of theSusquehanna River in Pennsylvania to rise from 7 feet to 22 feet in 14 hours.Find the average rate of change in the depth during that time. (Review 2.2)

MIXED REVIEW

TestPreparation

★★ Challenge

EXTRA CHALLENGE

www.mcdougallittell.com



Factor quadraticexpressions and solvequadratic equations byfactoring.

Find zeros ofquadratic functions, asapplied in Example 8.

� To solve real-lifeproblems, such as findingappropriate dimensions for amural in Ex. 97.


GOAL 2

GOAL 1


5.2 Solving Quadratic Equations byFactoringFACTORING QUADRATIC EXPRESSIONS

You know how to write (x + 3)(x + 5) as x2 + 8x + 15. The expressions x + 3 and x + 5 are because they have two terms. The expression x2 + 8x + 15 is a because it has three terms. You can use to write a trinomialas a product of binomials. To factor x2 + bx + c, find integers m and n such that:

x2 + bx + c = (x + m)(x + n)

= x2 + (m + n)x + mn

So, the sum of m and n must equal b and the product of m and n must equal c.

Factoring a Trinomial of the Form x 2 + bx + c

Factor x2 º 12x º 28.

SOLUTIONYou want x2 º 12x º 28 = (x + m)(x + n) where mn = º28 and m + n = º12.

� The table shows that m = 2 and n = º14. So, x2 º 12x º 28 = (x + 2)(x º 14).. . . . . . . . . .

To factor ax2 + bx + c when a ≠ 1, find integers k, l, m, and n such that:

ax2 + bx + c = (kx + m)(lx + n)

= klx2 + (kn + lm)x + mn

Therefore, k and l must be factors of a, and m and n must be factors of c.

Factoring a Trinomial of the Form ax2 + bx + c

Factor 3x2 º 17x + 10.

SOLUTIONYou want 3x2 º 17x + 10 = (kx + m)(lx + n) where k and l are factors of 3 and m and n are (negative) factors of 10. Check possible factorizations by multiplying.

(3x º 10)(x º 1) = 3x2 º 13x + 10 (3x º 1)(x º 10) = 3x2 º 31x + 10

(3x º 5)(x º 2) = 3x2 º 11x + 10 (3x º 2)(x º 5) = 3x2 º 17x + 10 ✓

� The correct factorization is 3x2 º 17x + 10 = (3x º 2)(x º 5).

E X A M P L E 2

E X A M P L E 1

factoringtrinomialbinomials

GOAL 1

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Factors of º28 (m, n) º1, 28 1, º28 º2, 14 2, º14 º4, 7 4, º7Sum of factors (m + n) 27 º27 12 º12 3 º3

Skills Review For help with factoring,see p. 938.

STUDENT HELP


5.2 Solving Quadratic Equations by Factoring 257

As in Example 2, factoring quadratic expressions often involves trial and error.However, some expressions are easy to factor because they follow special patterns.

Factoring with Special Patterns

Factor the quadratic expression.

a. 4x2 º 25 = (2x)2 º 52 Difference of two squares

= (2x + 5)(2x º 5)

b. 9y2 + 24y + 16 = (3y)2 + 2(3y)(4) + 42 Perfect square trinomial

= (3y + 4)2

c. 49r2 º 14r + 1 = (7r)2 º 2(7r)(1) + 12 Perfect square trinomial

= (7r º 1)2

. . . . . . . . . .

A is an expression that has only one term. As a first step to factoring, youshould check to see whether the terms have a common monomial factor.

Factoring Monomials First

Factor the quadratic expression.

a. 5x2 º 20 = 5(x2 º 4) b. 6p2 + 15p + 9 = 3(2p2 + 5p + 3)

= 5(x + 2)(x º 2) = 3(2p + 3)( p + 1)

c. 2u2 + 8u = 2u(u + 4) d. 4x2 + 4x º 4 = 4(x2 + x º 1)

. . . . . . . . . .

You can use factoring to solve certain quadratic equations. A in one variable can be written in the form ax2 + bx + c = 0 where a ≠ 0. This iscalled the of the equation. If the left side of ax2 + bx + c = 0 canbe factored, then the equation can be solved using the zero product property.

standard form

quadratic equation

E X A M P L E 4

monomial

E X A M P L E 3

Let A and B be real numbers or algebraic expressions. If AB = 0, then A = 0 or B = 0.

ZERO PRODUCT PROPERTY

PATTERN NAME PATTERN EXAMPLE

Difference of Two Squares a2 º b2 = (a + b)(a º b) x2 º 9 = (x + 3)(x º 3)

Perfect Square Trinomial a2 + 2ab + b2 = (a + b)2 x2 + 12x + 36 = (x + 6)2

a2 º 2ab + b2 = (a º b)2 x2 º 8x + 16 = (x º 4)2

SPECIAL FACTORING PATTERNS

STUDENT HELP

Study TipIt is not always possibleto factor a trinomialinto a product of twobinomials with integercoefficients. Forinstance, the trinomial x 2 + x º 1 in part (d) ofExample 4 cannot befactored. Such trinomialsare called irreducible.



Solving Quadratic Equations

Solve (a) x2 + 3x º 18 = 0 and (b) 2t2 º 17t + 45 = 3t º 5.

SOLUTIONa. x2 + 3x º 18 = 0 Write original equation.

(x + 6)(x º 3) = 0 Factor.

x + 6 = 0 or x º 3 = 0 Use zero product property.

x = º6 or x = 3 Solve for x.

� The solutions are º6 and 3. Check the solutions in the original equation.b. 2t2 º 17t + 45 = 3t º 5 Write original equation.

2t2 º 20t + 50 = 0 Write in standard form.

t2 º 10t + 25 = 0 Divide each side by 2.

(t º 5)2 = 0 Factor.

t º 5 = 0 Use zero product property.

t = 5 Solve for t.

� The solution is 5. Check the solution in the original equation.

Using a Quadratic Equation as a Model

You have made a rectangular stained glass window that is 2 feet by 4 feet. You have 7 square feet of clear glass to createa border of uniform width around the window. What shouldthe width of the border be?

SOLUTION

� Reject the negative value, º3.5. The border’s width should be 0.5 ft, or 6 in.

E X A M P L E 6

E X A M P L E 5

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Crafts

VERBALMODEL

ALGEBRAICMODEL

PROBLEMSOLVING

STRATEGY= º

Width of border = x (feet)

Area of border = 7 (square feet)

Area of border and window = (square feet)

Area of window = 2 • 4 = 8 (square feet)

7 = º 8 Write algebraic model.

0 = 4x2 + 12x º 7 Write in standard form.

0 = (2x + 7)(2x º 1) Factor.

2x + 7 = 0 or 2x º 1 = 0 Use zero product property.

x = º3.5 or x = 0.5 Solve for x.

(2 + 2x)(4 + 2x)

(2 + 2x)(4 + 2x)

Area ofwindow

Area of borderand window

Area ofborder

LABELS

4 � 2x

2 � 2x

x x

x x

x x

x x

2

4

Look Back For help with solving equations, see p. 19.

STUDENT HELP



FINDING ZEROS OF QUADRATIC FUNCTIONS

In Lesson 5.1 you learned that the x-intercepts of the graph of y = a(x º p)(x º q)are p and q. The numbers p and q are also called of the function because thefunction’s value is zero when x = p and when x = q. If a quadratic function is givenin standard form y = ax2 + bx + c, you may be able to find its zeros by usingfactoring to rewrite the function in intercept form.

Finding the Zeros of a Quadratic Function

Find the zeros of y = x2 º x º 6.

SOLUTIONUse factoring to write the function in intercept form.

y = x2 º x º 6

= (x + 2)(x º 3)

� The zeros of the function are º2 and 3.✓ CHECK Graph y = x2 º x º 6. The graph passes

through (º2, 0) and (3, 0), so the zeros are º2 and 3.

. . . . . . . . . .

From Lesson 5.1 you know that the vertex of the graph of y = a(x º p)(x º q) lieson the vertical line halfway between (p, 0) and (q, 0). In terms of zeros, the functionhas its maximum or minimum value when x equals the average of the zeros.

Using the Zeros of a Quadratic Model

BUSINESS You maintain a music-oriented Web site that allows subscribingcustomers to download audio and video clips of their favorite bands. When thesubscription price is $16 per year, you get 30,000 subscribers. For each $1 increasein price, you expect to lose 1000 subscribers. How much should you charge tomaximize your annual revenue? What is your maximum revenue?

SOLUTION

= •

Let R be your annual revenue and let x be the number of $1 price increases.

R = (30,000 º 1000x)(16 + x)

= (º1000x + 30,000)(x + 16)

= º1000(x º 30)(x + 16)

The zeros of the revenue function are 30 and º16. The value of x that maximizes R

is the average of the zeros, or x = = 7.

� To maximize revenue, charge $16 + $7 = $23 per year for a subscription.Your maximum revenue is R = º1000(7 º 30)(7 + 16) = $529,000.

30 + (º16)��2

Subscription priceNumber of subscribersRevenue

E X A M P L E 8

E X A M P L E 7

zeros

GOAL 2

STUDENT HELP

Study TipIn Example 7 note thatº2 and 3 are zeros of thefunction and x-interceptsof the graph. In general,functions have zeros andgraphs have x-intercepts.


www.mcdougallittell.comfor extra examples.

INTE

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STUDENT HELP


1. What is a zero of a function y = ƒ(x)?

2. In Example 2, how do you know that m and n must be negative factors of 10?

3. ERROR ANALYSIS A student solved x2 + 4x + 3 = 8 as shown. Explain the student’smistake. Then solve the equation correctly.

Factor the expression.

4. x2 º x º 2 5. 2x2 + x º 3 6. x2 º 16

7. y2 + 2y + 1 8. p2 º 4p + 4 9. q2 + q

Solve the equation.

10. (x + 3)(x º 1) = 0 11. x2 º 2x º 8 = 0 12. 3x2 + 10x + 3 = 0

13. 4u2 º 1 = 0 14. v2 º 14v = º49 15. 5w2 = 30w

Write the quadratic function in intercept form and give the function’s zeros.

16. y = x2 º 6x + 5 17. y = x2 + 6x + 8 18. y = x2 º 1

19. y = x2 + 10x + 25 20. y = 2x2 º 2x º 24 21. y = 3x2 º 8x + 4

22. URBAN PLANNING You have just planted arectangular flower bed of red roses in a park nearyour home. You want to plant a border of yellowroses around the flower bed as shown. Since youbought the same number of red and yellow roses,the areas of the border and inner flower bed will beequal. What should the width x of the border be?

FACTORING x2 + bx + c Factor the trinomial. If the trinomial cannot befactored, say so.

23. x2 + 5x + 4 24. x2 + 9x + 14 25. x2 + 13x + 40

26. x2 º 4x + 3 27. x2 º 8x + 12 28. x2 º 16x + 51

29. a2 + 3a º 10 30. b2 + 6b º 27 31. c2 + 2c º 80

32. p2 º 5p º 6 33. q2 º 7q º 10 34. r2 º 14r º 72

FACTORING ax2 + bx + c Factor the trinomial. If the trinomial cannot befactored, say so.

35. 2x2 + 7x + 3 36. 3x2 + 17x + 10 37. 8x2 + 18x + 9

38. 5x2 º 7x + 2 39. 6x2 º 9x + 5 40. 10x2 º 19x + 6

41. 3k2 + 32k º 11 42. 11m2 + 14m º 16 43. 18n2 + 9n º 14

44. 7u2 º 4u º 3 45. 12v2 º 25v º 7 46. 4w2 º 13w º 27


GUIDED PRACTICE


Vocabulary Check ✓Concept Check ✓

Skill Check ✓

STUDENT HELP


x2 + 4x + 3 = 8

(x + 3) (x + 1 ) = 8

x + 3 = 8 or x + 1 = 8

x = 5 or x = 7

xx

x

x

xx

x

x12 ft

8 ft



FACTORING WITH SPECIAL PATTERNS Factor the expression.

47. x2 º 25 48. x2 + 4x + 4 49. x2 º 6x + 9

50. 4r2 º 4r + 1 51. 9s2 + 12s + 4 52. 16t2 º 9

53. 49 º 100a2 54. 25b2 º 60b + 36 55. 81c2 + 198c + 121

FACTORING MONOMIALS FIRST Factor the expression.

56. 5x2 + 5x º 10 57. 18x2 º 2 58. 3x2 + 54x + 243

59. 8y2 º 28y º 60 60. 112a2 º 168a + 63 61. u2 + 7u

62. 6t2 º 36t 63. ºv2 + 2v º 1 64. 2d2 + 12d º 16

EQUATIONS IN STANDARD FORM Solve the equation.

65. x2 º 3x º 4 = 0 66. x2 + 19x + 88 = 0 67. 5x2 º 13x + 6 = 0

68. 8x2 º 6x º 5 = 0 69. k2 + 24k + 144 = 0 70. 9m2 º 30m + 25 = 0

71. 81n2 º 16 = 0 72. 40a2 + 4a = 0 73. º3b2 + 3b + 90 = 0

EQUATIONS NOT IN STANDARD FORM Solve the equation.

74. x2 + 9x = º20 75. 16x2 = 8x º 1

76. 5p2 º 25 = 4p2 + 24 77. 2y2 º 4y º 8 = ºy2 + y

78. 2q2 + 4q º 1 = 7q2 º 7q + 1 79. (w + 6)2 = 3(w + 12) º w2

FINDING ZEROS Write the quadratic function in intercept form and give thefunction’s zeros.

80. y = x2 º 3x + 2 81. y = x2 + 7x + 12 82. y = x2 + 2x º 35

83. y = x2 º 4 84. y = x2 + 20x + 100 85. y = x2 º 3x

86. y = 3x2 º 12x º 15 87. y = ºx2 + 16x º 64 88. y = 2x2 º 9x + 4

89. LOGICAL REASONING Is there a formula for factoring the sum of two squares?You will investigate this question in parts (a) and (b).

a. Consider the sum of squares x2 + 9. If this sum can be factored, then there areintegers m and n such that x2 + 9 = (x + m)(x + n). Write two equationsrelating the sum and the product of m and n to the coefficients in x2 + 9.

b. Show that there are no integers m and n that satisfy both equations you wrotein part (a). What can you conclude?

90. QUILTING You have made a quiltthat is 4 feet by 5 feet. You want to usethe remaining 10 square feet of fabric to add a decorative border of uniformwidth. What should the width of theborder be?

91. CONSTRUCTION A high schoolwants to double the size of its parkinglot by expanding the existing lot asshown. By what distance x should the lot be expanded?

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 23–34Example 2: Exs. 35–46Example 3: Exs. 47–55Example 4: Exs. 56–64Example 5: Exs. 65–79Example 6: Exs. 90, 91,

97, 98Example 7: Exs. 80–88Example 8: Exs. 99–101



Find the value of x.

92. Area of rectangle = 40 93. Area of rectangle = 105

94. Area of triangle = 22 95. Area of trapezoid = 114

96. VISUAL THINKING Use the diagram shown at the right.

a. Explain how the diagram models the factorization x2 + 5x + 6 = (x + 2)(x + 3).

b. Draw a diagram that models the factorization x2 + 7x + 12 = (x + 3)(x + 4).

97. As part of Black History Month in February, an artist iscreating a mural on the side of a building. A painting of Dr. Martin Luther King,Jr., will occupy the center of the mural and will be surrounded by a border ofuniform width showing other prominent African-Americans. The side of thebuilding is 50 feet wide by 30 feet high, and the artist wants to devote 25% of theavailable space to the border. What should the width of the border be?

98. ENVIRONMENT A student environmentalgroup wants to build an ecology garden asshown. The area of the garden should be 800 square feet to accommodate all thespecies of plants the group wants to grow. A construction company has donated 120 feetof iron fencing to enclose the garden. Whatshould the dimensions of the garden be?

99. ATHLETIC WEAR A shoe store sells about 200 pairs of a new basketballshoe each month when it charges $60 per pair. For each $1 increase in price,about 2 fewer pairs per month are sold. How much per pair should the storecharge to maximize monthly revenue? What is the maximum revenue?

100. HOME ELECTRONICS The manager of a home electronics store isconsidering repricing a new model of digital camera. At the current price of$680, the store sells about 70 cameras each month. Sales data from other storesindicate that for each $20 decrease in price, about 5 more cameras per monthwould be sold. How much should the manager charge for a camera to maximizemonthly revenue? What is the maximum revenue?

101. Big Bertha, a cannon used in World War I, couldfire shells incredibly long distances. The path of a shell could be modeled by y = º0.0196x2 + 1.37x where x was the horizontal distance traveled (in miles)and y was the height (in miles). How far could Big Bertha fire a shell? What wasthe shell’s maximum height? � Source: World War I: Trenches on the Web

HISTORY CONNECTION

ART CONNECTION

4x � 3

2x � 1

x

3x � 1

x

x

2x � 1

x

x � 3

GEOMETRY CONNECTION

x 1 1 1

11

x

ENVIRONMENT Ecology gardens

are often used to conductresearch with different plantspecies under a variety ofgrowing conditions.

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Skills Review For help with areas ofgeometric figures,see p. 914.

STUDENT HELP

x ft

(60 – x) ft



102. MULTIPLE CHOICE Suppose x2 + 4x + c = (x + m)(x + n) where c, m, and nare integers. Which of the following are not possible values of m and n?

¡A m = 2, n = 2 ¡B m = º1, n = 5¡C m = º2, n = º2 ¡D m = 1, n = 3

103. MULTIPLE CHOICE What are all solutions of 2x2 º 11x + 16 = x2 º 3x?

¡A 2, 6 ¡B º4 ¡C º4, 4 ¡D 4104. MULTIPLE CHOICE Given that 4 is a zero of y = 3x2 + bx º 8, what is the

value of b?

¡A º40 ¡B º10 ¡C º8 ¡D 2105. MULTICULTURAL MATHEMATICS The following problem is from the

Chiu chang suan shu, an ancient Chinese mathematics text. Solve the problem.(Hint: Use the Pythagorean theorem.)

A rod of unknown length is used to measure the dimensions of a rectangular door. The rod is 4 ch’ih longer than the width of the door, 2 ch’ih longer than the height of the door, and the same length as the door’s diagonal. What are the dimensions of the door? (Note: 1 ch’ih is slightly greater than 1 foot.)

ABSOLUTE VALUE Solve the equation or inequality. (Review 1.7)

106. |x| = 3 107. |x º 2| = 6 108. |4x º 9| = 2

109. |º5x + 4| = 14 110. |7 º 3x| = º8 111. |x + 1| < 3

112. |2x º 5| ≤ 1 113. |x º 4| > 7 114. |�13�x + 1| ≥ 2GRAPHING LINEAR EQUATIONS Graph the equation. (Review 2.3)

115. y = x + 1 116. y = º2x + 3 117. y = 3x º 5

118. y = º�52�x + 7 119. x + y = 4 120. 2x º y = 6

121. 3x + 4y = º12 122. º5x + 3y = 15 123. y = 2

124. y = º3 125. x = º1 126. x = 4

GRAPHING QUADRATIC FUNCTIONS Graph the function. (Review 5.1 for 5.3)

127. y = x2 º 2 128. y = 2x2 º 5 129. y = ºx2 + 3

130. y = (x + 1)2 º 4 131. y = º(x º 2)2 + 1 132. y = º3(x + 3)2 + 7

133. y = �14�x2 º 1 134. y = �12�(x º 4)

2 º 6 135. y = º�23�(x + 1)(x º 3)

136. COMMUTING You can take either the subway or the bus to your after-school job. A round trip from your home to where you work costs $2 on thesubway and $3 on the bus. You prefer to take the bus as often as possible but canafford to spend only $50 per month on transportation. If you work 22 days eachmonth, how many of these days can you take the bus? (Review 1.5)

MIXED REVIEW

TestPreparation

★★ Challenge

EXTRA CHALLENGE



Solving Quadratic Equations byFinding Square Roots

SOLVING QUADRATIC EQUATIONS

A number r is a of a number s if r2 = s. A positive number s has two square roots denoted by �s� and º�s�. The symbol �� is a the number s beneath the radical sign is the and the expression �s� is a

For example, since 32 = 9 and (º3)2 = 9, the two square roots of 9 are �9� = 3 and º�9� = º3. You can use a calculator to approximate �s� when s is not a perfectsquare. For instance, �2� ≈ 1.414.

In the activity you may have discovered the following properties of square roots.You can use these properties to simplify expressions containing square roots.

A square-root expression is considered simplified if (1) no radicand has a perfect-square factor other than 1, and (2) there is no radical in a denominator.

Using Properties of Square Roots

Simplify the expression.

a. �2�4� = �4� • �6� = 2�6� b. �6� • �1�5� = �9�0� = �9� • �1�0� = 3�1�0�

c. ��17�6�� = = d. ��72�� = • = �1�4��2�2��2�

�7��2�

�7��4

�7��1�6�

E X A M P L E 1

radical.radicand,radical sign,

square root

GOAL 1


Solve quadraticequations by finding squareroots.

Use quadraticequations to solve real-lifeproblems, such as findinghow long a falling stunt manis in the air in Example 4.

� To model real-lifequantities, such as the heightof a rock dropped off theLeaning Tower of Pisain Ex. 69.


GOAL 2

GOAL 1


5.3

Investigating Properties of Square RootsEvaluate the two expressions. What do you notice about the square root ofa product of two numbers?

a. �3�6�, �4� • �9� b. �8�, �4� • �2� c. �3�0�, �3� • �1�0�

Evaluate the two expressions. What do you notice about the square root ofa quotient of two numbers?

a. ��49��, b. ��22�5��, c. ��17�9��, �1�9��7��2�5��

�2��4��9�

2

1

DevelopingConcepts

ACTIVITY

Product Property: �a�b� = �a� • �b� Quotient Property: ��ba

�� = �a��b�

PROPERTIES OF SQUARE ROOTS (a > 0, b > 0)

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5.3 Solving Quadratic Equations by Finding Square Roots 265

In part (d) of Example 1, the square root in the denominator of was

eliminated by multiplying both the numerator and the denominator by �2�. This process is called

You can use square roots to solve some types of quadratic equations. For instance, if s > 0, then the quadratic equation x2 = s has two real-number solutions: x = �s�and x = º�s�. These solutions are often written in condensed form as x = ±�s�. The symbol ±�s� is read as “plus or minus the square root of s.”

Solving a Quadratic Equation

Solve 2x2 + 1 = 17.

SOLUTIONBegin by writing the equation in the form x2 = s.

2x2 + 1 = 17 Write original equation.

2x2 = 16 Subtract 1 from each side.

x2 = 8 Divide each side by 2.

x = ±�8� Take square roots of each side.

x = ±2�2� Simplify.

� The solutions are 2�2� and º2�2�.

✓ CHECK You can check the solutions algebraically by substituting them into the original equation. Sincethis equation is equivalent to 2x2 º 16 = 0, you canalso check the solutions by graphing y = 2x2 º 16and observing that the graph’s x-intercepts appear tobe about 2.8 ≈ 2�2� and º2.8 ≈ º2�2�.


Solve �13�(x + 5)2 = 7.

SOLUTION

�13�(x + 5)

2 = 7 Write original equation.

(x + 5)2 = 21 Multiply each side by 3.

x + 5 = ±�2�1� Take square roots of each side.

x = º5 ± �2�1� Subtract 5 from each side.

� The solutions are º5 + �2�1� and º5 º �2�1�.

✓ CHECK Check the solutions either by substituting them into the original equationor by graphing y = �13�(x + 5)

2 º 7 and observing the x-intercepts.

E X A M P L E 3

E X A M P L E 2

rationalizing the denominator.

�7��2�


www.mcdougallittell.comfor extra examples.

INTE

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STUDENT HELP


USING QUADRATIC MODELS IN REAL LIFE

When an object is dropped, its speed continually increases, and therefore its heightabove the ground decreases at a faster and faster rate. The height h (in feet) of theobject t seconds after it is dropped can be modeled by the function

h = º16t2 + h0

where h0 is the object’s initial height. This model assumes that the force of airresistance on the object is negligible. Also, the model works only on Earth. Forplanets with stronger or weaker gravity, different models are used (see Exercise 71).

Modeling a Falling Object’s Height with a Quadratic Function

A stunt man working on the set of a movie is to fall out of a window 100 feet abovethe ground. For the stunt man’s safety, an air cushion 26 feet wide by 30 feet long by9 feet high is positioned on the ground below the window.

a. For how many seconds will the stunt man fall before he reaches the cushion?

b. A movie camera operating at a speed of 24 frames per second records the stuntman’s fall. How many frames of film show the stunt man falling?

SOLUTIONa. The stunt man’s initial height is h0 = 100 feet, so his height as a function of time

is given by h = º16t2 + 100. Since the top of the cushion is 9 feet above theground, you can determine how long it takes the stunt man to reach the cushionby finding the value of t for which h = 9 . Here are two methods:

Method 1: Make a table of values.

� From the table you can see that h = 9 at a value of t between t = 2 and t = 3. It takes between 2 sec and 3 sec for the stunt man to reach the cushion.

Method 2: Solve a quadratic equation.

h = º16t2 + 100 Write height function.

9 = º16t2 + 100 Substitute 9 for h.

º91 = º16t2 Subtract 100 from each side.

�91

16� = t2 Divide each side by –16.

��91�16�� = t Take positive square root.2.4 ≈ t Use a calculator.

� It takes about 2.4 seconds for the stunt man to reach the cushion.

b. The number of frames of film that show the stunt man falling is given by theproduct (2.4 sec)(24 frames/sec), or about 57 frames.

E X A M P L E 4

GOAL 2


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Movies

t 0 1 2 3

h 100 84 36 º44



1. Explain what it means to “rationalize the denominator” of a quotient containingsquare roots.

2. State the product and quotient properties of square roots in words.

3. How many real-number solutions does the equation x2 = s have when s > 0?when s = 0? when s < 0?

Simplify the expression.

4. �4�9� 5. �1�2� 6. �4�5� 7. �3� • �2�7�

8. ��12�65�� 9. ��79�� 10. 11. ��25��Solve the equation.

12. x2 = 64 13. x2 º 9 = 16 14. 4x2 + 7 = 23

15. �x6

2� º 2 = 0 16. 5(x º 1)2 = 50 17. �12�(x + 8)

2 = 14

18. ENGINEERING At an engineering school, students are challenged to designa container that prevents an egg from breaking when dropped from a height of 50 feet. Write an equation giving a container’s height h (in feet) above theground after t seconds. How long does the container take to hit the ground?

USING THE PRODUCT PROPERTY Simplify the expression.

19. �1�8� 20. �4�8� 21. �2�7� 22. �5�2�

23. �7�2� 24. �1�7�5� 25. �9�8� 26. �6�0�5�

27. 2�7� • �7� 28. �8� • �2� 29. �3� • �1�2� 30. 3�2�0� • 6�5�

31. �1�2� • �2� 32. �6� • �1�0� 33. 4�3� • �2�1� 34. �8� • �6� • �3�

USING THE QUOTIENT PROPERTY Simplify the expression.

35. ��19�� 36. ��44�9�� 37. ��32�65�� 38. ��18�010��39. ��13�6�� 40. ��16�14�� 41. ��73�56�� 42. ��14�60�9��43. 44. 45. ��65�� 46. ��11�414��

47. ��78�� 48. ��11�83�� 49. ��43�52�� 50. ��17�5�� • ��43��SOLVING QUADRATIC EQUATIONS Solve the equation.

51. x2 = 121 52. x2 = 90 53. 3x2 = 108

54. 2x2 + 5 = 41 55. ºx2 º 12 = º87 56. 7 º 10u2 = 1

57. �2v5

2� º 1 = 11 58. 6 º �

p8

2

� = º4 59. �56q2� º �

q3

2

� = 72

5��1�7�

2��3�


1��3�


Skill Check ✓

Concept Check ✓

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 19–50Example 2: Exs. 51–59Example 3: Exs. 60–68Example 4: Exs. 69–73

Extra Practiceto help you masterskills is on p. 946.

STUDENT HELP



SOLVING QUADRATIC EQUATIONS Solve the equation.

60. 2(x º 3)2 = 8 61. 4(x + 1)2 = 100 62. º3(x + 2)2 = º18

63. 5(x º 7)2 = 135 64. 8(x + 4)2 = 9 65. 2(a º 6)2 º 45 = 53

66. �14�(b º 8)2 = 7 67. (2r º 5)2 = 81 68. �

(s +10

1)2� º �15

2� = �12

5�

69. According to legend, in 1589 the Italian scientist GalileoGalilei dropped two rocks of different weights from the top of the Leaning Towerof Pisa. He wanted to show that the rocks would hit the ground at the same time.Given that the tower’s height is about 177 feet, how long would it have taken forthe rocks to hit the ground?

70. ORNITHOLOGY Many birds drop shellfish ontorocks to break the shell and get to the food inside.

Crows along the west coast of Canada use this techniqueto eat whelks (a type of sea snail). Suppose a crow dropsa whelk from a height of 20 feet, as shown. � Source: Cambridge Encyclopedia of Ornithology

a. Write an equation giving the whelk’s height h(in feet) after t seconds.

b. Use the Table feature of a graphing calculator to findh when t = 0, 0.1, 0.2, 0.3, . . . , 1.4, 1.5. (You’ll needto scroll down the table to see all the values.) To thenearest tenth of a second, how long does it take forthe whelk to hit the ground? Check your answer bysolving a quadratic equation.

71. ASTRONOMY On any planet, the height h (in feet) of a falling object t seconds after it is dropped can be modeled by

h = º�g2�t

2 + h0

where h0 is the object’s initial height and g is the acceleration (in feet per secondsquared) due to the planet’s gravity. For each planet in the table, find the time ittakes for a rock dropped from a height of 200 feet to hit the ground.

72. OCEANOGRAPHY The equation h = 0.019s2 gives the height h (in feet) ofthe largest ocean waves when the wind speed is s knots. How fast is the windblowing if the largest waves are 15 feet high? � Source: Encyclopaedia Britannica

73. TELEVISION The aspect ratio of a TV screenis the ratio of the screen’s width to its height. Formost TVs, the aspect ratio is 4:3. What are thewidth and height of the screen for a 27 inch TV?(Hint: Use the Pythagorean theorem and the factthat TV sizes such as 27 inches refer to the lengthof the screen’s diagonal.)

HISTORY CONNECTION

27 3x

4x

ASTRONOMY Theacceleration due to

gravity on the moon is about5.3 ft/sec2. This means thatthe moon’s gravity is onlyabout one sixth as strong asEarth’s.


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Planet Earth Mars Jupiter Neptune Pluto

g (ft/sec2) 32 12 81 36 2.1

� Source: STARLab, Stanford University

Skills ReviewFor help with thePythagorean theorem,see p. 917.

STUDENT HELP

20 ft



74. MULTI-STEP PROBLEM Building codes often require that buildings be able to withstand a certain amount of wind pressure. The pressure P (in pounds persquare foot) from wind blowing at s miles per hour is given by P = 0.00256s2.� Source: The Complete How to Figure It

a. You are designing a two-story library. Buildings this tall are often required towithstand wind pressure of 20 lb/ft2. Under this requirement, how fast can thewind be blowing before it produces excessive stress on a building?

b. To be safe, you design your library so that it can withstand wind pressure of 40 lb/ft2. Does this mean that the library can survive wind blowing at twicethe speed you found in part (a)? Justify your answer mathematically.

c. Writing Use the pressure formula to explain why even a relatively smallincrease in wind speed could have potentially serious effects on a building.

75. For a bathtub with a rectangular base, Torricelli’s lawimplies that the height h of water in the tub t seconds after it begins draining isgiven by

h = ��h�0� º t�2 where l and w are the tub’s length and width, d is the diameter of the drain, andh0 is the water’s initial height. (All measurements are in inches.) Suppose youcompletely fill a tub with water. The tub is 60 inches long by 30 inches wide by25 inches high and has a drain with a 2 inch diameter.

a. Find the time it takes for the tub to go from being full to half-full.

b. Find the time it takes for the tub to go from being half-full to empty.

c. CRITICAL THINKING Based on your results, what general statement can youmake about the speed at which water drains?

SOLVING SYSTEMS Solve the linear system by graphing. (Review 3.1)

76. x + y = 5 77. x º y = º1 78. º3x + y = 7ºx + 2y = 4 3x + y = 5 2x + y = 2

79. 2x º 3y = 9 80. x + 4y = 4 81. 2x + 3y = 64x º 3y = 3 3x º 2y = 12 x º 6y = 18

MATRIX OPERATIONS Perform the indicated operation(s). (Review 4.1)

82. � + � 83. � º � 84. º4 � 85. º2 � + 7 � WRITING IN STANDARD FORM Write the quadratic function in standard form. (Review 5.1 for 5.4)

86. y = (x + 5)(x º 2) 87. y = (x º 1)(x º 8) 88. y = (2x + 7)(x + 4)

89. y = (4x + 9)(4x º 9) 90. y = (x º 3)2 + 1 91. y = 5(x + 6)2 º 12

11º7

150

10º9

1220

º18

5º4

º3 4

4º1

º69

30

7 º2

º4º2

º510

º12

68

MIXED REVIEW

2πd2�3��lw

SCIENCE CONNECTION

TestPreparation

★★ Challenge

EXTRA CHALLENGE




Graph the function. (Lesson 5.1)

1. y = x2 º 2x º 3 2. y = 2(x + 2)2 + 1 3. y = º�13�(x + 5)(x º 1)

Solve the equation. (Lesson 5.2)

4. x2 º 6x º 27 = 0 5. 4x2 + 21x + 20 = 0 6. 7t2 º 4t = 3t2 º 1

Simplify the expression. (Lesson 5.3)

7. �5�4� 8. 7�2� • �1�0� 9. ��35�6�� 10.

11. SWIMMING The drag force F (in pounds) of water on a swimmer can bemodeled by F = 1.35s2 where s is the swimmer’s speed (in miles per hour). Howfast must you swim to generate a drag force of 10 pounds? (Lesson 5.3)

4��1�2�

QUIZ 1 Self-Test for Lessons 5.1–5.3

Telescopes

THENTHEN

NOWNOW

THE FIRST TELESCOPE is thought to have been made in 1608 by Hans Lippershey,a Dutch optician. Lippershey’s telescope, called a refracting telescope, used lensesto magnify objects. Another type of telescope is a reflecting telescope. Reflectingtelescopes magnify objects with parabolic mirrors, traditionally made from glass.

RECENTLY “liquid mirrors” for telescopes have been made by spinning reflective liquids, such as mercury. A cross section of the surface of a spinning liquid is a parabola with equation

y = x2 º

where ƒ is the spinning frequency (in revolutions per second) and R is the radius (in feet) of the container.

1. Write an equation for the surface of a liquid before it is spun. What does theequation tell you about the location of the x-axis relative to the liquid?

2. Suppose mercury is spun with a frequency of 0.5 revolution/sec in a containerwith radius 2 feet. Write and graph an equation for the mercury’s surface.

3. Find the x-intercepts of the graph of y = x2 º . Does changing the

spinning frequency affect the x-intercepts? Explain.

π2ƒ2R2�32

π2ƒ2�16

π2ƒ2R2�32

π2ƒ2�16


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y

R

ƒ

Liquid mirrors are first usedto do astronomical research.

Isaac Newton builds firstreflecting telescope.

Maria Mitchell is firstto use a telescope todiscover a comet.

1668

18471987

Galileo first uses arefracting telescope forastronomical purposes.

1609


5.3 Technology Activity 271

Solving Quadratic EquationsYou can use a graphing calculator to solve quadratic equations having real-number solutions.

� EXAMPLESolve 2(x º 3)2 = 5.

� SOLUTIONWrite the equation in the form ƒ(x) = 0.

2(x º 3)2 = 5

2(x º 3)2 º 5 = 0

Therefore, the solutions of the original equation are the zeros of the function y = 2(x º 3)2 º 5, or equivalently, the x-intercepts of this function’s graph.

Graph the function you entered in Step 2. Use your calculator’s Zero or Rootfeature to find the x-intercepts of the graph. (Root is another word for a solutionof an equation, in this case 2(x º 3)2 º 5 = 0.)

� The solutions are about 1.42 and about 4.58.

� EXERCISES

Use a graphing calculator to solve the equation.

1. 3x2 º 7 = 0 2. º2x2 + 9 = 3

3. 5x2 + 2 = 6x2 º 4 4. 1.2x2 º 5.6 = 0.8x2 º 2.3

5. (x + 1)2 º 3 = 0 6. º�13�(x º 4)2 = º8

7. x2 + 2x º 6 = 0 8. 2x2 + 8x + 3 = 4x2 + 5x º 1

9. MANUFACTURING A company sells ground coffee in cans having a radiusof 2 inches and a height of 6 inches. The company wants to manufacture a largercan that has the same height but holds twice as much coffee. Write an equationyou can use to find the larger can’s radius. (Hint: Use the formula V = πr2h forthe volume of a cylinder.) Solve the equation with a graphing calculator.

ZeroX=4.5811388 Y=0

ZeroX=1.4188612 Y=0

3

1

Using Technology

Graphing Calculator Activity for use with Lesson 5.3ACTIVITY 5.3

STUDENT HELP

KEYSTROKE HELP

See keystrokes for several models ofcalculators atwww.mcdougallittell.com

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Y1=2(X-3)2-5Y2=Y3=Y4=Y5=Y6=Y7=

Enter y = 2(x º 3)2 º 5 intoyour graphing calculator.

2



Complex NumbersOPERATIONS WITH COMPLEX NUMBERS

Not all quadratic equations have real-number solutions. For instance, x2 = º1 has no real-number solutions because the square of any real number x is never negative.To overcome this problem, mathematicians created an expanded system of numbersusing the i, defined as i = �º�1�. Note that i2 = º1. Theimaginary unit i can be used to write the square root of any negative number.


Solve 3x2 + 10 = º26.

SOLUTION3x2 + 10 = º26 Write original equation.

3x2 = º36 Subtract 10 from each side.

x2 = º12 Divide each side by 3.

x = ±�º�1�2� Take square roots of each side.

x = ±i�1�2� Write in terms of i.

x = ±2i�3� Simplify the radical.

� The solutions are 2i�3� and º2i�3�.. . . . . . . . . .

A written in is a number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part. If b ≠ 0, then a + bi is an If a = 0 and b ≠ 0, then a + bi is a The diagram shows how different types of complex numbers are related.

pure imaginary number.imaginary number.

standard formcomplex number

E X A M P L E 1

imaginary unit

GOAL 1

Solve quadraticequations with complexsolutions and performoperations with complexnumbers.

Apply complexnumbers to fractal geometry.

� To solve problems, suchas determining whether acomplex number belongs tothe Mandelbrot set in Example 7.


GOAL 2

GOAL 1


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PROPERTY EXAMPLE

1. If r is a positive real number, �º�5� = i �5�then �º�r� = i �r�.

2. By Property (1), it follows that (i �5�)2 = i2 • 5 = º5(i �r�)2 = ºr.

THE SQUARE ROOT OF A NEGATIVE NUMBER

Complex Numbers (a � bi )

RealNumbers(a � 0i )

ImaginaryNumbers

(a � bi, b � 0)

PureImaginaryNumbers

(0 � bi, b � 0)

�1

3

52

2 � 3i 5 � 5i

�4i 6i

�2π


5.4 Complex Numbers 273

Just as every real number corresponds to a point on the real number line, everycomplex number corresponds to a point in the As shown in the nextexample, the complex plane has a horizontal axis called the real axis and a verticalaxis called the imaginary axis.

Plotting Complex Numbers

Plot the complex numbers in the complex plane.

a. 2 º 3i b. º3 + 2i c. 4i

SOLUTIONa. To plot 2 º 3i, start at the origin, move 2 units to the

right, and then move 3 units down.

b. To plot º3 + 2i, start at the origin, move 3 units to theleft, and then move 2 units up.

c. To plot 4i, start at the origin and move 4 units up.

. . . . . . . . . .

Two complex numbers a + bi and c + di are equal if and only if a = c and b = d.For instance, if x + yi = 8 º i, then x = 8 and y = º1.

To add (or subtract) two complex numbers, add (or subtract) their real parts and theirimaginary parts separately.

Sum of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i

Difference of complex numbers: (a + bi) º (c + di) = (a º c) + (b º d)i

Adding and Subtracting Complex Numbers

Write the expression as a complex number in standard form.

a. (4 º i) + (3 + 2i) b. (7 º 5i) º (1 º 5i) c. 6 º (º2 + 9i) + (º8 + 4i)

SOLUTIONa. (4 º i) + (3 + 2i) = (4 + 3) + (º1 + 2)i Definition of complex addition

= 7 + i Standard form

b. (7 º 5i) º (1 º 5i) = (7 º 1) + (º5 + 5)i Definition of complex subtraction

= 6 + 0i Simplify.

= 6 Standard form

c. 6 º (º2 + 9i) + (º8 + 4i) = [(6 + 2) º 9i] + (º8 + 4i) Subtract.

= (8 º 9i) + (º8 + 4i) Simplify.

= (8 º 8) + (º9 + 4)i Add.

= 0 º 5i Simplify.

= º5i Standard form

E X A M P L E 3

E X A M P L E 2

complex plane.

imaginary

i

1 real

�3 � 2i

2 � 3i

4i



To multiply two complex numbers, use the distributive property or the FOIL methodjust as you do when multiplying real numbers or algebraic expressions. Otherproperties of real numbers that also apply to complex numbers include the associativeand commutative properties of addition and multiplication.

Multiplying Complex Numbers


a. 5i(º2 + i) b. (7 º 4i)(º1 + 2i) c. (6 + 3i)(6 º 3i)

SOLUTIONa. 5i(º2 + i) = º10i + 5i2 Distributive property

= º10i + 5(º1) Use i 2 = º1.

= º5 º 10i Standard form

b. (7 º 4i)(º1 + 2i) = º7 + 14i + 4i º 8i2 Use FOIL.

= º7 + 18i º 8(º1) Simplify and use i 2 = º1.

= 1 + 18i Standard form

c. (6 + 3i)(6 º 3i) = 36 º 18i + 18i º 9i2 Use FOIL.

= 36 º 9(º1) Simplify and use i 2 = º1.

= 45 Standard form

. . . . . . . . . .

In part (c) of Example 4, notice that the two factors 6 + 3i and 6 º 3i have the forma + bi and a º bi. Such numbers are called The product ofcomplex conjugates is always a real number. You can use complex conjugates towrite the quotient of two complex numbers in standard form.

Dividing Complex Numbers

Write the quotient �51+º

32

ii� in standard form.

SOLUTIONThe key step here is to multiply the numerator and the denominator by the complexconjugate of the denominator.

�51

+º

32

ii� = �

51

+º

32

ii� • �

11

++

22

ii� Multiply by 1 + 2i, the conjugate of 1 º 2i.

= Use FOIL.

= �º1 +513i

� Simplify.

= º�15� + �153�i Standard form

5 + 10i + 3i + 6i2��1 + 2i º 2i º 4i2

E X A M P L E 5

complex conjugates.

E X A M P L E 4



USING COMPLEX NUMBERS IN FRACTAL GEOMETRY

In the hands of a person who understands fractal geometry, the complex plane canbecome an easel on which stunning pictures called fractals are drawn. One veryfamous fractal is the Mandelbrot set, named after mathematician Benoit Mandelbrot.The Mandelbrot set is the black region in the complex plane below. (The points in the colored regions are not part of the Mandelbrot set.)

To understand how the Mandelbrot set is constructed, you need to know how theabsolute value of a complex number is defined.

Finding Absolute Values of Complex Numbers

Find the absolute value of each complex number. Which number is farthest from theorigin in the complex plane?

a. 3 + 4i b. º2i c. º1 + 5i

SOLUTIONa. |3 + 4i| = �3�2�+� 4�2� = �2�5� = 5

b. |º2i| = |0 + (º2i)| = �0�2�+� (�º�2�)2� = 2

c. |º1 + 5i| = �(º�1�)2� +� 5�2� = �2�6� ≈ 5.10Since º1 + 5i has the greatest absolute value, it isfarthest from the origin in the complex plane.

E X A M P L E 6

GOAL 2

The of a complex number z = a + bi, denoted |z|, is anonnegative real number defined as follows:

|z| = �a�2�+� b�2�Geometrically, the absolute value of a complex number is the number’s distance from the origin in the complex plane.

absolute value

ABSOLUTE VALUE OF A COMPLEX NUMBER

z � �1 � 5iimaginary

real

3i|z|� 5

|z|� 2z � �2i

z � 3 � 4i

4

|z|� �26

BENOITMANDELBROT

was born in Poland in 1924,came to the United States in1958, and is now a professorat Yale University. He pio-neered the study of fractalgeometry in the 1970s.

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FOCUS ONPEOPLE

–i

1

–1

–2

–3

–4

–i

1

–1

–2

–3

–4

–i

1

–1

–2

–3

–4

–i

1

–1

–2

–3

–4

–i

1

–1

–2

–3

–4

–i

1

–1

–2

–3

–4

–i

1

–1

–2

–3

–4



The following result shows how absolute value can be used to tell whether a givencomplex number belongs to the Mandelbrot set.

Determining if a Complex Number Is in the Mandelbrot Set

Tell whether the complex number c belongs to the Mandelbrot set.

a. c = i b. c = 1 + i c. c = º2

SOLUTIONa. Let ƒ(z) = z2 + i.

z0 = 0 |z0| = 0

z1 = ƒ(0) = 02 + i = i |z1| = 1

z2 = ƒ(i) = i2 + i = º1 + i |z2| = �2� ≈ 1.41

z3 = ƒ(º1 + i) = (º1 + i)2 + i = ºi |z3| = 1

z4 = ƒ(ºi) = (ºi)2 + i = º1 + i |z4| = �2� ≈ 1.41

At this point the absolute values alternate between 1 and �2�, and so all the absolute values are less than N = 2. Therefore, c = i belongs to theMandelbrot set.

b. Let ƒ(z) = z2 + (1 + i).

z0 = 0 |z0| = 0

z1 = ƒ(0) = 02 + (1 + i) = 1 + i |z1| ≈ 1.41

z2 = ƒ(1 + i) = (1 + i)2 + (1 + i) = 1 + 3i |z2| ≈ 3.16

z3 = ƒ(1 + 3i) = (1 + 3i)2 + (1 + i) = º7 + 7i |z3| ≈ 9.90

z4 = ƒ(º7 + 7i) = (º7 + 7i)2 + (1 + i) = 1 º 97i |z4| ≈ 97.0

The next few absolute values in the list are (approximately) 9409, 8.85 ª 107,and 7.84 ª 1015. Since the absolute values are becoming infinitely large, c = 1 + i does not belong to the Mandelbrot set.

c. Let ƒ(z) = z2 + (º2), or ƒ(z) = z2 º 2. You can show that z0 = 0, z1 = º2, andzn = 2 for n > 1. Therefore, the absolute values of z0, z1, z2, z3, . . . are all lessthan N = 3, and so c = º2 belongs to the Mandelbrot set.

E X A M P L E 7

To determine whether a complex number c belongs to the Mandelbrot set,consider the function ƒ(z) = z2 + c and this infinite list of complex numbers:

z0 = 0, z1 = ƒ(z0), z2 = ƒ(z1), z3 = ƒ(z2), . . .

• If the absolute values |z0|, |z1|, |z2|, |z3|, . . . are all less than some fixednumber N, then c belongs to the Mandelbrot set.

• If the absolute values |z0|, |z1|, |z2|, |z3|, . . . become infinitely large, then c does not belong to the Mandelbrot set.

COMPLEX NUMBERS IN THE MANDELBROT SET



1. Complete this statement: For the complex number 3 º 7i, the real part is ��? and the imaginary part is ��? .

2. ERROR ANALYSIS A student thinks that the complex conjugate of º5 + 2iis 5 º 2i. Explain the student’s mistake, and give the correct complex conjugate of º5 + 2i.

3. Geometrically, what does the absolute value of a complex number represent?

Solve the equation.

4. x2 = º9 5. 2x2 + 3 = º13 6. (x º 1)2 = º7


7. (1 + 5i) + (6 º 2i) 8. (4 + 3i) º (º2 + 4i)

9. (1 º i)(7 + 2i) 10. �31º+

4ii

�

Find the absolute value of the complex number.

11. 1 + i 12. 3i 13. º2 + 3i 14. 5 º 5i

15. Plot the numbers in Exercises 11–14 in the same complex plane.

16. FRACTAL GEOMETRY Tell whether c = 1 º i belongs to the Mandelbrot set.Use absolute value to justify your answer.

SOLVING QUADRATIC EQUATIONS Solve the equation.

17. x2 = º4 18. x2 = º11 19. 3x2 = º81

20. 2x2 + 9 = º41 21. 5x2 + 18 = 3 22. ºx2 º 4 = 14

23. 8r2 + 7 = 5r2 + 4 24. 3s2 º 1 = 7s2 25. (t º 2)2 = º16

26. º6(u + 5)2 = 120 27. º�18�(v + 3)2 = 7 28. 9(w º 4)2 + 1 = 0

PLOTTING COMPLEX NUMBERS Plot the numbers in the same complex plane.

29. 4 + 2i 30. º1 + i 31. º4i 32. 3

33. º2 º i 34. 1 + 5i 35. 6 º 3i 36. º5 + 4i

ADDING AND SUBTRACTING Write the expression as a complex number instandard form.

37. (2 + 3i) + (7 + i) 38. (6 + 2i) + (5 º i)

39. (º4 + 7i) + (º4 º 7i) 40. (º1 º i) + (9 º 3i)

41. (8 + 5i) º (1 + 2i) 42. (2 º 6i) º (º10 + 4i)

43. (º0.4 + 0.9i) º (º0.6 + i) 44. (25 + 15i) º (25 º 6i)

45. ºi + (8 º 2i) º (5 º 9i) 46. (30 º i) º (18 + 6i) + 30i



Concept Check ✓

Skill Check ✓

STUDENT HELP


STUDENT HELP

HOMEWORK HELPExample 1: Exs. 17–28Example 2: Exs. 29–36Example 3: Exs. 37–46Example 4: Exs. 47–55Example 5: Exs. 56–63Example 6: Exs. 64–71Example 7: Exs. 72–79



MULTIPLYING Write the expression as a complex number in standard form.

47. i(3 + i) 48. 4i(6 º i) 49. º10i(4 + 7i)

50. (5 + i)(8 + i) 51. (º1 + 2i)(11 º i) 52. (2 º 9i)(9 º 6i)

53. (7 + 5i)(7 º 5i) 54. (3 + 10i)2 55. (15 º 8i)2

DIVIDING Write the expression as a complex number in standard form.

56. �1 +8

i� 57. �12ºi

i� 58. �º5

4ºi

3i� 59. �3

3º+

ii

�

60. �25++

52

ii� 61. �

º97º+

46ii

� 62. 63.

ABSOLUTE VALUE Find the absolute value of the complex number.

64. 3 º 4i 65. 5 + 12i 66. º2 º i 67. º7 + i

68. 2 + 5i 69. 4 º 8i 70. º9 + 6i 71. �1�1� + i�5�

MANDELBROT SET Tell whether the complex number c belongs to theMandelbrot set. Use absolute value to justify your answer.

72. c = 1 73. c = º1 74. c = ºi 75. c = º1 º i

76. c = 2 77. c = º1 + i 78. c = º0.5 79. c = 0.5i

LOGICAL REASONING In Exercises 80–85, tell whether the statement is trueor false. If the statement is false, give a counterexample.

80. Every complex number is an imaginary number.

81. Every irrational number is a complex number.

82. All real numbers lie on a single line in the complex plane.

83. The sum of two imaginary numbers is always an imaginary number.

84. Every real number equals its complex conjugate.

85. The absolute values of a complex number and its complex conjugate are always equal.

86. VISUAL THINKING The graph shows howyou can geometrically add two complexnumbers (in this case, 3 + 2i and 1 + 4i) tofind their sum (in this case, 4 + 6i). Find eachof the following sums by drawing a graph.

a. (2 + i) + (3 + 5i)

b. (º1 + 6i) + (7 º 4i)

COMPARING REAL AND COMPLEX NUMBERS Tell whether the property istrue for (a) the set of real numbers and (b) the set of complex numbers.

87. If r, s, and t are numbers in the set, then (r + s) + t = r + (s + t).

88. If r is a number in the set and |r| = k, then r = k or r = ºk.

89. If r and s are numbers in the set, then r º s = s º r.

90. If r, s, and t are numbers in the set, then r(s + t) = rs + rt.

91. If r and s are numbers in the set, then |r + s| = |r| + |s|.

6 º i�2��6 + i�2�

�1�0��1�0� º i

4

4 � 6iimaginary

real

i 2i

1

4i

3

STUDENT HELP

Skills ReviewFor help withdisproving statementsby counterexample,see p. 927.


92. CRITICAL THINKING Evaluate �º�4� • �º�9� and �3�6�. Does the rule �a� • �b� = �a�b� on page 264 hold when a and b are negative numbers?

93. Writing Give both an algebraic argument and a geometric argumentexplaining why the definitions of absolute value on pages 50 and 275 areconsistent when applied to real numbers.

94. EXTENSION: ADDITIVE AND MULTIPLICATIVE INVERSES The additiveinverse of a complex number z is a complex number za such that z + za = 0. The multiplicative inverse of z is a complex number zm such that z • zm = 1. Find the additive and multiplicative inverses of each complex number.

a. z = 1 + i b. z = 3 º i c. z = º2 + 8i

ELECTRICITY In Exercises 95 and 96, use the following information.Electrical circuits may contain several types of components such as resistors, inductors, and capacitors. The resistance of each component to the flow of electrical current is the component’s impedance, denoted by Z. The value of Z is a real number R for a resistor of R ohms (�), a pure imaginary number Lifor an inductor of L ohms, and a pure imaginary number ºCi for a capacitor of C ohms. Examples are given in the table.

95. SERIES CIRCUITS A series circuit is a type of circuit found in switches,fuses, and circuit breakers. In a series circuit, there is only one pathway throughwhich current can flow. To find the total impedance of a series circuit, add theimpedances of the components in the circuit. What is the impedance of eachseries circuit shown below? (Note: The symbol denotes an alternating currentsource and does not affect the calculation of impedance.)

a. b. c.

96. PARALLEL CIRCUITS Parallel circuits are used in household lighting andappliances. In a parallel circuit, there is more than one pathway through whichcurrent can flow. To find the impedance Z of a parallel circuit with two pathways,first calculate the impedances Z1 and Z2 of the pathways separately by treatingeach pathway as a series circuit. Then apply this formula:

Z =

What is the impedance of each parallel circuit shown below?

a. b. c.4�

2�Z1 Z2

7�

5�5�

3�Z1 Z2

9�

8�3�

4�Z1 Z2

2�

6�

Z1Z2�Z1 + Z2

8�

4�

6�

2�12�

15�

8�

2�

5�

7�


ELECTRICIANAn electrician

installs, maintains, andrepairs electrical systems.This often involves workingwith the types of circuitsdescribed in Exs. 95 and 96.

CAREER LINKwww.mcdougallittell.com

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Component Symbol Z

Resistor 3

Inductor 5i

Capacitor º6i


www.mcdougallittell.comfor help with problemsolving in Exs. 95 and 96.

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3�

5�

6�


QUANTITATIVE COMPARISON In Exercises 97–99, choose the statement thatis true about the given quantities.

¡A The quantity in column A is greater.¡B The quantity in column B is greater.¡C The two quantities are equal.¡D The relationship cannot be determined from the given information.

97.

98.

99.

100. POWERS OF i In this exercise you will investigate a pattern that appears whenthe imaginary unit i is raised to successively higher powers.

a. Copy and complete the table.

b. Writing Describe the pattern you observe in the table. Verify that thepattern continues by evaluating the next four powers of i.

c. Use the pattern you described in part (b) to evaluate i26 and i83.

EVALUATING FUNCTIONS Evaluate ƒ(x) for the given value of x. (Review 2.1)

101. ƒ(x) = 4x º 1 when x = 3 102. ƒ(x) = x2 º 5x + 8 when x = º4

103. ƒ(x) = |ºx + 6| when x = 9 104. ƒ(x) = 2 when x = º30

SOLVING SYSTEMS Use an inverse matrix to solve the system. (Review 4.5)

105. 3x + y = 5 106. x + y = 2 107. x º 2y = 105x + 2y = 9 7x + 8y = 21 3x + 4y = 0

SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.3 for 5.5)

108. (x + 4)2 = 1 109. (x + 2)2 = 36 110. (x º 11)2 = 25

111. º(x º 5)2 = º10 112. 2(x + 7)2 = 24 113. 3(x º 6)2 º 8 = 13

114. The table shows the cumulative number N(in thousands) of DVD players sold in the United States from the end ofFebruary, 1997, to time t (in months). Make a scatter plot of the data.Approximate the equation of the best-fitting line. (Review 2.5)

STATISTICS CONNECTION

MIXED REVIEW


TestPreparation

★★ Challenge

t 1 2 3 4 5 6 7 8 9 10 11 12

N 34 69 96 125 144 178 213 269 307 347 383 416

Column A Column B

|5 + 4i| |3 º 6i|

|º6 + 8i| |º10i|

|2 + bi| where b < º1 |�3� + ci| where 0 < c < 1

Power of i i1 i2 i3 i4 i5 i6 i7 i8

Simplified form i º1 ºi ? ? ? ? ?

EXTRA CHALLENGE


DATA UPDATE of DVD Insider data at www.mcdougallittell.comINT

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Using Algebra Tilesto Complete the Square� QUESTION Given b, what is the value of c that makes x2 + bx + c

a perfect square trinomial?

� EXPLORING THE CONCEPTUse algebra tiles to model the expression x2 + 6x.

Arrange the tiles in a square. Your arrangement will be incomplete in one corner.

Determine the number of 1-tiles needed to complete the square.

� DRAWING CONCLUSIONS1. Copy and complete the table at the left by

following the steps above.

2. Look for patterns in the last column ofyour table. Consider the general statementx2 + bx + c = (x + d)2.

a. How is d related to b in each case?

b. How is c related to d in each case?

c. How can you obtain the numbers in thesecond column of the table directly from the coefficients of x in the expressions from the first column?

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Developing Concepts

ACTIVITY 5.5 Group Activity for use with Lesson 5.5

GROUP ACTIVITYWork with a partner.

MATERIALSalgebra tiles

5.5 Concept Activity 281

By adding nine 1-tiles,you can see that x2 + 6x + 9 = (x + 3)2.

You want the length and widthof your “square” to be equal.

You will need one x2-tileand six x-tiles.

Completing the Square

Number of 1-tiles Expression written

Expression needed to complete as a square

the square

x2 � 2x � ��??

?

x2 � 4x � ��??

?

x2 � 6x � ��?9 x

2 � 6x � 9 � (x � 3)2

x2 � 8x � ��??

?

x2 � 10x � ��??

?



Completing the SquareSOLVING QUADRATIC EQUATIONS BYCOMPLETING THE SQUARE

is a process that allows you to write an expression of theform x2 + bx as the square of a binomial. This process can be illustrated using anarea model, as shown below.

You can see that to complete the square for x2 + bx, you need to add ��b2��2, the areaof the incomplete corner of the square in the second diagram. This diagram modelsthe following rule:

x2 + bx + ��b2��2 = �x + �b2��2

Completing the Square

Find the value of c that makes x2 º 7x + c a perfect square trinomial. Then write theexpression as the square of a binomial.

SOLUTIONIn the expression x2 º 7x + c, note that b = º7. Therefore:

c = ��b2��2 = ��º27��2 = �449�Use this value of c to write x2 º 7x + c as a perfect square trinomial, and then as thesquare of a binomial.

x2 º 7x + c = x2 º 7x + �449� Perfect square trinomial

= �x º �72��2 Square of a binomial: �x + }b2}�2. . . . . . . . . .

In Lesson 5.2 you learned how to solve quadratic equations by factoring. However,many quadratic equations, such as x2 + 10x º 3 = 0, contain expressions that cannotbe factored. Completing the square is a method that lets you solve any quadraticequation, as the next example illustrates.

E X A M P L E 1

b

bxx 2

x

x x 2

x

x

b2

b2

b2

b2 ( )

2b2( )x

( )x

Completing the square

GOAL 1

Solve quadraticequations by completing thesquare.

Use completing thesquare to write quadraticfunctions in vertex form, asapplied in Example 7.

� To solve real-lifeproblems, such as findingwhere to position a fire hose in Ex. 91.


GOAL 2

GOAL 1


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Study Tip In Example 2 note thatyou must add 25 to bothsides of the equation x 2 + 10x = 3 whencompleting the square.

5.5 Completing the Square 283

Solving a Quadratic Equation if the Coefficient of x2 Is 1

Solve x2 + 10x º 3 = 0 by completing the square.

SOLUTIONx2 + 10x º 3 = 0 Write original equation.

x2 + 10x = 3 Write the left side in the form x 2 + bx.

x2 + 10x + 52 = 3 + 25 Add �}120}�2 = 52 = 25 to each side.(x + 5)2 = 28 Write the left side as a binomial squared.

x + 5 = ±�2�8� Take square roots of each side.

x = º5 ± �2�8� Solve for x.

x = º5 ± 2�7� Simplify.

� The solutions are º5 + 2�7� and º5 º 2�7�.✓ CHECK You can check the solutions by

substituting them back into the originalequation. Alternatively, you can graph y = x2 + 10x º 3 and observe that the x-intercepts are about 0.29 ≈ º5 + 2�7�and º10.29 ≈ º5 º 2�7�.

. . . . . . . . . .

If the coefficient of x2 in a quadratic equation is not 1, you should divide each side ofthe equation by this coefficient before completing the square.

Solving a Quadratic Equation if the Coefficient of x 2 Is Not 1

Solve 3x2 º 6x + 12 = 0 by completing the square.

SOLUTION3x2 º 6x + 12 = 0 Write original equation.

x2 º 2x + 4 = 0 Divide each side by the coefficient of x 2.

x2 º 2x = º4 Write the left side in the form x 2 + bx.

x2 º 2x + (º1)2 = º4 + 1 Add �}º22}�2 = (º1)2 = 1 to each side.(x º 1)2 = º3 Write the left side as a binomial squared.

x º 1 = ±�º�3� Take square roots of each side.

x = 1 ± �º�3� Solve for x.

x = 1 ± i �3� Write in terms of the imaginary unit i.

� The solutions are 1 + i�3� and 1 º i�3�.✓ CHECK Because the solutions are imaginary, you cannot check them graphically.

However, you can check the solutions algebraically by substituting them back intothe original equation.

E X A M P L E 3

ZeroX=-10.2915 Y=0

E X A M P L E 2



Using a Quadratic Equation to Model Distance

On dry asphalt the distance d (in feet) needed for a car to stop is given by

d = 0.05s2 + 1.1s

where s is the car’s speed (in miles per hour). What speed limit should be posted on aroad where drivers round a corner and have 80 feet to come to a stop?

SOLUTIONd = 0.05s2 + 1.1s Write original equation.

80 = 0.05s2 + 1.1s Substitute 80 for d.

1600 = s2 + 22s Divide each side by the coefficient of s 2.

1600 + 121 = s2 + 22s + 112 Add �}222}�2 = 112 = 121 to each side.1721 = (s + 11)2 Write the right side as a binomial squared.

±�1�7�2�1� = s + 11 Take square roots of each side.

º11 ± �1�7�2�1� = s Solve for s.

s ≈ 30 or s ≈ º52 Use a calculator.

� Reject the solution º52 because a car’s speed cannot be negative. The postedspeed limit should be at most 30 miles per hour.

Using a Quadratic Equation to Model Area

You want to plant a rectangular garden along part of a 40 foot side of your house. Tokeep out animals, you will enclose the garden with wire mesh along its three opensides. You will also cover the garden with mulch. If you have 50 feet of mesh andenough mulch to cover 100 square feet, what should the garden’s dimensions be?

SOLUTIONDraw a diagram. Let x be the length of the sides of the garden perpendicular to thehouse. Then 50 º 2x is the length of the third fenced side of the garden.

x(50 º 2x) = 100 Length ª Width = Area

50x º 2x2 = 100 Distributive property

º2x2 + 50x = 100 Write the x 2-term first.

x2 º 25x = º50 Divide each side by º2.

x2 º 25x + (º12.5)2 = º50 + 156.25 Complete the square.

(x º 12.5)2 = 106.25 Write as a binomial squared.

x º 12.5 = ±�1�0�6�.2�5� Take square roots of each side.

x = 12.5 ± �1�0�6�.2�5� Solve for x.

x ≈ 22.8 or x ≈ 2.2 Use a calculator.

� Reject x = 2.2 since 50 º 2x = 45.6 is greater than the house’s length. If x = 22.8, then 50 º 2x = 4.4. The garden should be about 22.8 feet by 4.4 feet.

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house

40 ft

x50 – 2x

x

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5.5 Completing the Square 285

WRITING QUADRATIC FUNCTIONS IN VERTEX FORM

Given a quadratic function in standard form, y = ax2 + bx + c, you can usecompleting the square to write the function in vertex form, y = a(x º h)2 + k.

Writing a Quadratic Function in Vertex Form

Write the quadratic function y = x2 º 8x + 11 in vertex form. What is the vertex ofthe function’s graph?

SOLUTIONy = x2 º 8x + 11 Write original function.

y + ��? = (x2 º 8x + ��? ) + 11 Prepare to complete the square for x 2 º 8x.

y + 16 = (x2 º 8x + 16) + 11 Add � �2 = (º4)2 = 16 to each side.y + 16 = (x º 4)2 + 11 Write x 2 º 8x + 16 as a binomial squared.

y = (x º 4)2 º 5 Solve for y.

� The vertex form of the function is y = (x º 4)2 º 5. The vertex is (4, º5).

Finding the Maximum Value of a Quadratic Function

The amoun

Documents

QUADRATIC FUNCTIONS - PBworksalgebra2text.pbworks.com/f/Chapter+5.pdf250 Chapter 5 Quadratic Functions Graphing a Quadratic Function Graph y = 2x2 º 8x + 6.SOLUTION Note that the