Practice with ExamplesFor use with pages 256–263

5.2LESSON

CONTINUED

Finding the Zeros of a Quadratic Function

Find the zeros of

SOLUTION

First, check to see if the terms have a commonmonomial factor.

The terms have an x in common.

The zeros of the function are 0 and 6.

Check by graphing The graph passesthrough and

Exercises for Example 4

Find the zeros of the function.

13.

14.

15. y � x2 � 3x � 10

y � x2 � 1

y � x2 � x � 20

�6, 0�.�0, 0�y � x2 � 6x.

x2 � 6x � x�x � 6�

y � x2 � 6x.

84 Algebra 2Practice Workbook with Examples

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EXAMPLE 4

1

2

Solving Quadratic Equations byFinding Square RootsGoals p Solve quadratic equations.

p Use quadratic equations to solve real-life problems.

5.3

VOCABULARY

Square root

Radical sign 3

Radicand

Radical

Rationalizing the denominator

Your Notes

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

Product Property: �ab� � p

Quotient Property: ��ab

�� �

Lesson 5.3 • Algebra 2 Notetaking Guide 101

3

Your Notes

Checkpoint Simplify the expression.

1. �5� p �8� 2. ��35

��

Simplify the expression.

a. �27� � p �

b. �5� p �15� � � p �

c. ��356�� � �

d. ��133�� � p �

Example 1 Using Properties of Square Roots

Solve �12

�(x � 2)2 � 8.

Solution

�12

�(x � 2)2 � 8 Write original equation.

� Multiply each side by .

� Take square roots of each side.

x � Add to each side.

The solutions are .

Check Check the solutions either by substituting them into

the original equation or by graphing y �

and observing the x-intercepts.

Example 2 Solving a Quadratic Equation

102 Algebra 2 Notetaking Guide • Chapter 5

4

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5.3 Practice with ExamplesFor use with pages 264–270

LESSONNAME _________________________________________________________ DATE ____________

Ch

apter 5GOAL

EXAMPLE 2

Solve quadratic equations by finding square roots and use quadratic

equations to solve real-life problems

Using Properties of Square Roots

Simplify the expression.

a. b.

c. d.

Exercises for Example 1

Simplify the expression.

1. 2. 3.

Solving a Quadratic Equation

Solve

SOLUTION

Write original equation.

Add 4 to each side.

Multiply both sides by 6.

Take square roots of both sides.

Simplify.

The solutions are and �2�21.2�21

x � ± 2�21

x � ±�84

x2 � 84

x2

6� 14

x2

6� 4 � 10

x2

6� 4 � 10.

� 81121

�2 � �18�60

�365

��36�5

�6�5

��5�5

�6�5

5� 325

��3�25

��35

�6 � �8 � �48 � �16 � �3 � 4�3�99 � �9 � �11 � 3�11

EXAMPLE 1

VOCABULARY

If then b is a square root of a. A positive number a has twosquare roots, and The symbol is a radical sign, a is theradicand, and is a radical.

Rationalizing the denominator is the process of eliminating squareroots in the denominator of a fraction.

�a� ��a.�a

b2 � a,

5

Practice with ExamplesFor use with pages 264–270

5.3LESSON

CONTINUED

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EXAMPLE 3

Exercises for Example 2

Solve the equation.

4. 5. 6.

Solving a Quadratic Equation

Solve

Write original equation.

Divide both sides by 5.

Take the square roots of both sides.

Simplify.

Add 7 to both sides.

The solutions are and

Exercises for Example 3

Solve the equation.

7. 8. 9.

10. 11. 12.13 (z � 3�2 � 55�x � 3�2 � 50�r � 8�2 � 50

�2�x � 3�2 � �12�w � 1�2 � 196�y � 3�2 � 9

7 � 3�3.7 � 3�3

x � 7 ± 3�3

x � 7 � ± 3�3

x � 7 � ±�27

�x � 7�2 � 27

5�x � 7�2 � 135

5�x � 7�2 � 135.

p2

4� 3 � 3312 � 2y2 � 44x2 � 5 � �1

6

5.4 Practice with ExamplesFor use with pages 272–280

LESSON

88 Algebra 2Practice Workbook with Examples

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NAME _________________________________________________________ DATE ____________

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apte

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GOAL Solve quadratic equations with complex solutions and perform operations

with complex numbers

Solving a Quadratic Equation

Solve

SOLUTION

Write original equation.

Add 12 to each side.

Divide each side by 2.

Take square roots of each side.

Write in terms of i.

Exercises for Example 1

Solve the equation.

1. 2. 3.

4. 5. 6. y2 � 25 � 03x2 � 1 � �35�2t2 � 8 � 46

r2 � 4 � �85y2 � �40x2 � �16

x � ± i�14

x � ±��14

x2 � �14

2x2 � �28

2x2 � 12 � �40

2x2 � 12 � �40

EXAMPLE 1

VOCABULARY

The imaginary unit i is defined as

A complex number written in standard form is a number where a and b are real numbers.

If then is an imaginary number.

If and then is a pure imaginary number.

Sum of complex numbers:

Difference of complex numbers:

In the complex plane, the horizontal axis is called the real axis and thevertical axis is called the imaginary axis.

The expressions and are called complex conjugates. Theproduct of complex conjugates is always a real number.

a � bia � bi

�a � bi� � �c � di� � �a � c� � �b � d�i

�a � bi� � �c � di� � �a � c� � �b � d�i

a � bib � 0,a � 0

a � bib � 0,

a � bi,

i � ��1.

7

What you need to know about “i”?

What is √-1 ?

What is √-4 ?

What is √-7 ?

What is i?

What is i2?

What is i3?

What is i4?

8

Practice with ExamplesFor use with pages 272–280

5.4LESSO4

CONTINUED

Algebra 2 89Practice Workbook with Examples

Copyright © McDougal Littell Inc. All rights reserved.

NAME _________________________________________________________ DATE ____________C

hap

ter 5EXAMPLE 2 Adding and Subtracting Complex Numbers

Write as a complex number in standard form.

Definition of complex addition

Simplify and use standard form.

Exercises for Example 2

Write the expression as a complex number in standard form.

7. 8. 9.

10. 11. 12.

Multiplying Complex Numbers

Write as a complex number in standard form.

SOLUTION

Use FOIL.

Simplify and use

Standard form

Exercises for Example 3

Write as a complex number in standard form.

13. 14. 15.

16. 17. 18. �8 � i��2 � i��5 � 3i�2�4 � i���2 � 6i�

�2 � 3i��2 � 3i�4i�3 � 5i)�2i�5 � i�

� �5 � 31i

i2 � �1. � 7 � 31i � 12��1� �7 � 3i��1 � 4i� � 7 � 28i � 3i � 12i2

�7 � 3i��1 � 4i�

12 � �8 � 10i��6 � 7i� � ��3 � i��12 � 8i� � �6 � 6i�

i � �5 � 6i���6 � 3i� � �5 � i��5 � 4i� � �7 � 2i�

� 10 � 2i

�6 � 3i� � ��4 � 2i� � 7i � �6 � 4� � �3 � 2 � 7�i

�6 � 3i� � ��4 � 2i� � 7i

EXAMPLE 3

9

10

The Quadratic Formula andthe DiscriminantGoals p Solve equations using the quadratic formula.

p Use the quadratic formula in real-life situations.

5.6

VOCABULARY

Discriminant of a quadratic equation

Your Notes

THE QUADRATIC FORMULA

Let a, b, and c be real numbers such that a ≠ 0. Thesolutions of the quadratic equation ax2 � bx � c � 0 are:

x �� � 4

2

�

Solve 3x2 � 3x � 5 � 0.

3x2 � 3x � 5 � 0 Original equation

x � Quadratic formula

x � a � , b � ,c �

x � Simplify.

The solutions are

x � and x � .

Example 1 Quadratic Equation with Two Real Solutions

� � 4

2

� � 4

2

�

�

112 Algebra 2 Notetaking Guide • Chapter 5

11

Your Notes

Solve x2 � 4x � 11 � 7.

x2 � 4x � 11 � 7 Write original equation.

x2 � 4x � 4 � 0 a � , b � , c �

x � Quadratic formula

x � Simplify.

x � Simplify.

The solution is .

Check Substitute for x in the original equation.

( )2 � 4( ) � 11 � 7

� 7

Example 2 Quadratic Equation with One Real Solution

Solve x2 � 4x � �8.

x2 � 4x � �8 Write original equation.

x2 � 4x � 8 � 0 a � , b � , c �

x � Quadratic formula

x � Simplify.

x � Write using the imaginary unit i.

x � Simplify.

The solutions are .

Check Substitute an imaginary solution into theoriginal equation.

( )2 � 4( ) � �8

� �8

� �8

Example 3 Quadratic Equation with Two Imaginary Solutions

Lesson 5.6 • Algebra 2 Notetaking Guide 113

12

Your Notes Checkpoint Solve the quadratic equation.

1. x2 � 3x � 10 2. x2 � 7 � 8x � 9

3. x2 � 6x � 3 � �7

NUMBER AND TYPE OF SOLUTIONS OF AQUADRATIC EQUATION

Consider the quadratic equation ax2 � bx � c � 0.

• If b2 � 4ac > 0, then the equation has .

• If b2 � 4ac � 0, then the equation has .

• If b2 � 4ac < 0, then the equation has .

Find the discriminant of the quadratic equation and give thenumber and type of solutions of the equation.

a. x2 � 2x � 3 � 0 b. x2 � 2x � 1 � 0

c. x2 � 2x � 5 � 0

SolutionDiscriminant Solution(s)

b2 � 4ac x ���b � �

2ba

2 � 4�ac��

a.

b.

c.

Example 4 Using the Discriminant

114 Algebra 2 Notetaking Guide • Chapter 5

13

14

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5.5 Practice with ExamplesFor use with pages 282–289

LESSONNAME _________________________________________________________ DATE ____________

Ch

apter 5GOAL Solve quadratic equations by completing the square, and use completing

the square to write quadratic functions in vertex form

Completing the Square

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

SOLUTION

Use the form and that

Perfect square trinomial

Square of a binomial:

Exercises for Example 1

Find the value of c that makes the expression a perfect square

trinomial. Then write the expression as the square of a binomial.

1. 2.

3. x2 � 2x � c

x2 � 12x � cx2 � 6x � c

�x �b2�2

� �x � 0.8�2

x2 � 1.6x � c � x2 � 1.6x � 0.64

c � �b2�

2

� �1.62 �

2

� �0.8�2 � 0.64

b � 1.6.x2 � bx � �b2�2

x2 � 1.6x � c

EXAMPLE 1

VOCABULARY

Completing the square is the process of rewriting an expression of theform as the square of the binomial. To complete the square for

you need to add This leads to the rule

x2 � bx � �b2�2

� �x �b2�2

.

�b2�2

.x2 � bx,x2 � bx

15

Practice with ExamplesFor use with pages 282–289

5.5LESSON

CONTINUED

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EXAMPLE 2 Solving a Quadratic Equation if the Coefficient of is 1

Solve by completing the square.

SOLUTION

Write original equation.

To isolate the terms containing x, add 2 to each side.

Add to each side.

Write the left side as a binomial squared.

Take sqare roots of each side.

To solve for x, add 1 to each side.

The solutions are and

Exercises for Example 2

Solve the equation by completing the square.

4. 5.

6.

Solving a Quadratic Equation if the Coefficient of is Not 1

Solve by completing the square.

SOLUTION

Write original equation.

Divide by 4 to make the coefficient of be 1.

Write the left side in the form

Add to each side.

Write the left side as the square of a binomial.

Take square roots of each side.

To solve for x, add to each side.

The solutions are and 34 ��54 .3

4 ��54

34 x �

34 ± �5

4

x �34 � ± �5

4

�x �34�2

�5

16

��34 �2

�916 x2 �

32 x � ��3

4�2� �

14 �

916

x2 � bx. x2 �32 x � �

14

x2 x2 �32 x �

14 � 0

4x2 � 6x � 1 � 0

4x2 � 6x � 1 � 0

x 2

x2 � 4x � 13

x2 � 6x � 5 � 0x2 � x � 1

1 � �3.1 � �3

x � 1±�3

x � 1 � ±�3

�x � 1�2 � 3

��22 �2

� ��1�2 � 1 x2 � 2x � ��1�2 � 2 � 1

x2 � 2x � 2

x2 � 2x � 2 � 0

x2 � 2x � 2 � 0

x 2

EXAMPLE 3

16

Copyright © McDougal Littell/Houghton Mifflin Company. Lesson 5.2 • Algebra 2 Notetaking Guide 97

Solving Quadratic Equations by FactoringGoals p Factor quadratic expressions and solve quadratic

equations by factoring.p Find zeros of quadratic functions.

5.2

VOCABULARY

Binomial

Trinomial

Factoring

Monomial

Quadratic equation in one variable

Zero of a function

Your Notes

Factor x2 � 9x � 14.

SolutionYou want x2 � 9x � 14 � (x � m)(x � n) where mn �and m � n � .

The table shows that m � and n � . So,x2 � 9x � 14 � ( )( ).

Example 1 Factoring a Trinomial of the Form x2 � bx � c

Factors of 14 (m, n) 1, �1, 2, �2,

Sum of factors (m � n)

17

Factor the quadratic expression.

a. 9x2 � 16 � ( )2 � 2 Difference of two

� ( )( ) squares

b. 16y2 � 40y � 25 Perfect square trinomial

� ( )2 � 2( )( ) � 2

� ( )2

c. 64x2 � 32x � 4 Perfect square trinomial

� ( )2 � 2( )( ) � 2

� ( )2

Example 3 Factoring with Special Patterns

Copyright © McDougal Littell/Houghton Mifflin Company. Lesson 5.2 • Algebra 2 Notetaking Guide 98

Your Notes

Factor 2x2 � 13x � 6.

SolutionYou want 2x2 � 13x � 6 � (kx � m)(lx � n) where k and lare factors of and m and n are ( ) factors of .Check possible factorizations by multiplying.

(2x � 3)(x � 2) �

(2x � 2)(x � 3) �

(2x � 6)(x � 1) �

(2x � 1)(x � 6) �

The correct factorization is 2x2 � 13x � 6 � .

Example 2 Factoring a Trinomial of the Form ax2 � bx � c

SPECIAL FACTORING PATTERNS

Difference of Two Squares Examplea2 � b2 � (a � b)(a � b) x2 � 9 � ( )( )

Perfect Square Trinomial Examplea2 � 2ab � b2 � (a � b)2 x2 � 12x � 36 � ( )2

a2 � 2ab � b2 � (a � b)2 x2 � 8x � 16 � ( )2

18

Copyright © McDougal Littell/Houghton Mifflin Company. Lesson 5.2 • Algebra 2 Notetaking Guide 99

Your Notes

Factor the quadratic expression.

a. 12x2 � 3 � 3 � 3

b. 3u2 � 9u � 6 � 3 � 3

c. 7v2 � 42v � 7v

d. 2x2 � 8x � 2 � 2

Example 4 Factoring Monomials First

Checkpoint Factor the expression.

1. 6c2 � 48c � 54 2. 81x2 � 1

3. 49h2 � 42h � 9 4. 16x2 � 4

ZERO PRODUCT PROPERTY

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

Solve 4x2 � 13x � 11 � �3x � 5.

Solution4x2 � 13x � 11 � �3x � 5 Write original equation.

� 0 Write in standard form.

� 0 Divide each side by .

� 0 Factor.

� 0 Use zero product property.

x � Solve for x.

The solution is . Check this in the original equation.

Example 5 Solving Quadratic Equations

19

Copyright © McDougal Littell/Houghton Mifflin Company. Lesson 5.2 • Algebra 2 Notetaking Guide 100

Your Notes Checkpoint Solve the quadratic equation.

5. x2 � 15x � 26 � 0 6. 2x2 � x � 3 ��5x � 19 � x2

Find the zeros of y � x2 � 4x � 3.

Use factoring to write the function in intercept form.

y � x2 � 4x � 3

�

The zeros of the function areand .

Check Graph y � x2 � 4x � 3. Thegraph passes through ( , 0) and ( , 0), so the zeros are and

.

Example 6 Finding the Zeros of a Quadratic Function

Checkpoint Complete the following exercise.

7. Find the zeros of y � 3x2 � x � 2.

Homework

20