example 4 finding the zeros of a quadratic function
TRANSCRIPT
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
Copyright © McDougal Littell Inc. All rights reserved.
NAME _________________________________________________________ DATE ____________
Ch
apte
r 5
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
Algebra 2 85Practice Workbook with Examples
Copyright © McDougal Littell Inc. All rights reserved.
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
86 Algebra 2Practice Workbook with Examples
Copyright © McDougal Littell Inc. All rights reserved.
NAME _________________________________________________________ DATE ____________
Ch
apte
r 5
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
Copyright © McDougal Littell Inc. All rights reserved.
NAME _________________________________________________________ DATE ____________
Ch
apte
r 5
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
Algebra 2 91Practice Workbook with Examples
Copyright © McDougal Littell Inc. All rights reserved.
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
92 Algebra 2Practice Workbook with Examples
Copyright © McDougal Littell Inc. All rights reserved.
NAME _________________________________________________________ DATE ____________
Ch
apte
r 5
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