Download Solving Quadratic Equations by Factoring

5.2
Solving Quadratic Equations
by Factoring
Goals p Factor quadratic expressions and solve quadratic
equations by factoring.
p Find zeros of quadratic functions.
Your Notes
VOCABULARY
Binomial An expression with two terms, such as x ! 1
Trinomial An expression with three terms, such as
x2 ! x ! 1
Factoring A process used to write a polynomial as a
product of other polynomials having equal or lesser
degree
Monomial An expression with one term, such as 3x
Quadratic equation in one variable An equation that can
be written in the form ax2 ! bx ! c " 0 where a # 0
Zero of a function A number k is a zero of a function f if
f(k) " 0.
Example 1
Factoring a Trinomial of the Form x2 ! bx ! c
Factor x2 $ 9x ! 14.
Solution
You want x2 $ 9x ! 14 " (x ! m)(x ! n) where mn " 14
and m ! n " $9 .
Factors of 14
(m, n)
Sum of factors
(m ! n)
1,
14
15
$1, $14
$15
2,
7
9
$2, $7
$9
The table shows that m " $2 and n " $7 . So,
x2 $ 9x ! 14 " ( x $ 2 )( x $ 7 ).
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Your Notes
Example 2
Factoring a Trinomial of the Form ax2 ! bx ! c
Factor 2x2 ! 13x ! 6.
Solution
You want 2x2 ! 13x ! 6 " (kx ! m)(lx ! n) where k and l
are factors of 2 and m and n are ( positive ) factors of 6 .
Check possible factorizations by multiplying.
(2x ! 3)(x ! 2) " 2x 2 ! 7x ! 6
(2x ! 2)(x ! 3) " 2x 2 ! 8x ! 6
(2x ! 6)(x ! 1) " 2x 2 ! 8x ! 6
(2x ! 1)(x ! 6) " 2x 2 ! 13x ! 6
The correct factorization is
2x2 ! 13x ! 6 " (2x ! 1)(x ! 6) .
SPECIAL FACTORING PATTERNS
Difference of Two Squares
a2 $ b2 " (a ! b)(a $ b)
Example
x2 $ 9 " ( x ! 3 )( x $ 3 )
Perfect Square Trinomial
a2 ! 2ab ! b2 " (a ! b)2
Example
x2 ! 12x ! 36 " ( x ! 6 )2
a2 $ 2ab ! b2 " (a $ b)2
x2 $ 8x ! 16 " ( x $ 4 )2
Example 3
Factoring with Special Patterns
Factor the quadratic expression.
a. 9x2 $ 16 " ( 3x )2 $ 4
2
" ( 3x ! 4 )( 3x $ 4 )
b. 16y2 ! 40y ! 25
" ( 4y )2 ! 2( 4y )( 5 ) ! 5
Difference of two
squares
Perfect square trinomial
2
" ( 4y ! 5 )2
c. 64x2 $ 32x ! 4
" ( 8x )2 $ 2( 8x )( 2 ) ! 2
" ( 8x $ 2 )2
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Algebra 2 Notetaking Guide • Chapter 5
Perfect square trinomial
2
Your Notes
Factoring Monomials First
Example 4
Factor the quadratic expression.
a. 12x2 $ 3 " 3 (4x 2 $ 1) " 3 (2x $ 1)(2x ! 1)
b. 3u2 $ 9u ! 6 " 3 (u2 $ 3u ! 2) " 3 (u $ 2)(u $ 1)
c. 7v2 $ 42v " 7v (v $ 6)
d. 2x2 ! 8x ! 2 " 2 (x 2 ! 4x ! 1)
Checkpoint Factor the expression.
1. 6c2 $ 48c $ 54
2. 81x2 $ 1
6(c ! 1)(c $ 9)
(9x ! 1)(9x $ 1)
3. 49h2 ! 42h ! 9
4. 16x2 $ 4
(7h ! 3)2
4(2x ! 1)(2x $ 1)
ZERO PRODUCT PROPERTY
Let A and B be real numbers or algebraic expressions. If
AB " 0, then A " 0 or B " 0 .
Example 5
Solving Quadratic Equations
Solve 4x2 ! 13x ! 11 " $3x $ 5.
Solution
4x2 ! 13x ! 11 " $3x $ 5
4x 2
Write original equation.
! 16x ! 16 " 0
Write in standard form.
x 2 ! 4x ! 4 " 0
Divide each side by 4 .
(x ! 2)2 " 0
x!2 "0
x " $2
Factor.
Use zero product property.
Solve for x.
The solution is $2 . Check this in the original equation.
Lesson 5.2 • Algebra 2 Notetaking Guide
99
Your Notes
Checkpoint Solve the quadratic equation.
5. x2 ! 15x ! 26 " 0
6. 2x2 ! x ! 3 "
$5x ! 19 ! x2
$8, 2
$13, $2
Example 6
Finding the Zeros of a Quadratic Function
Find the zeros of y " x2 ! 4x ! 3.
Use factoring to write the function in intercept form.
y " x2 ! 4x ! 3
" (x ! 3)(x ! 1)
The zeros of the function are $3
and $1 .
Check Graph y " x2 ! 4x ! 3. The
graph passes through ( $3 , 0) and
( $1 , 0), so the zeros are $3 and
$1 .
Checkpoint Complete the following exercise.
7. Find the zeros of y " 3x2 $ x $ 2.
Homework
2
$%%, 1
3
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Algebra 2 Notetaking Guide • Chapter 5