4.3 Solve x2 + bx + c = 0 by Factoring
Goal  Solve quadratic equations.
Your Notes
VOCABULARY
Monomial
An expression that is either a number, a variable, or the product of a number and one or
more variables
Binomial
The sum of two monomials
Trinomial
The sum of three monomials
Quadratic equation
An equation in one variable that can be written in the form ax2 + bx + c = 0 where a  0
Root of an equation
A solution of a quadratic function
Zero of a function
The numbers p and q of a function in intercept form are also called the zeros of the
function.
Example 1
Factor trinomials of the form x2 + bx + c
Factor the expression x2 + 7x  8.
Solution
You want x2 + 7x  8 = (x + m)(x + n) where mn = _8_ and m + n = _7_.
Factors of 8 (m, n)
Sum of factors (m + n)
l, _8_
_7_
l, _8_
_7_
Factors of 8 (m, n)
Sum of factors (m + n)
2, _4_
_2_
2, _4_
_2_
Notice that m = _1_ and n = _8_. So, x2 + 7x  8 = ( x  1 )( x + 8 ).
Your Notes
SPECIAL FACTORING PATTERNS
Pattern Name
Difference of
Two Squares
a2  b2 = ( a + b )( a  b )
x2  4 = (x + 2)(x  2)
Perfect Square
Trinomial
a2 + 2ab + b2 = ( a + b )2
x2 + 6x + 9 = (x + 3)2
Perfect Square
Trinomial
a2  2ab + b2 = ( a  b )2
x2  4x + 4 = (x  2)2
Example 2
Factor with special patterns
Factor the expression.
a. x2  25 = x2  _52_
Difference of two squares
= ( x + 5 )( x  5 )
b. m  22m + 121
2
Perfect square trinomial
= m2  2(m)( 11 ) + _11_2
= ( m  11 )2
Checkpoint Factor the expression. If it cannot be factored, say so.
1.
x2 + 7x + 12
(x + 4)(x + 3)
2.
x2  81
(x  9)(x + 9)
ZERO PRODUCT PROPERTY
Words
If the _product_ of two expressions is zero, then _one_ or _both_ of the
expressions equals zero.
Algebra
If A and B are expressions and AB = _0_ , then A = _0_ or B = _0_ .
Example
If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = _5_ or
x = _2_ .
Your Notes
Example 3
Find the roots of an equation
Find the roots of the equation x2  2x  15 = 0.
Solution
x2  2x  15 = 0
( x  5 )( x + 3 ) = 0
Original equation
Factor.
_x  5_ = 0 or _x + 3_ = 0
Zero product property
x = _5_ or
Solve for x.
x = _3_
The roots are _5_ and _3_ .
Example 4
Find the zeros of a quadratic function
Find the zeros of the function y = x2 + 5x  6 by rewriting the function in intercept
form.
Solution
y = x2 + 5x  6
= ( x + 6 )( x  1 )
Write original equation.
Factor.
The zeros of the function are _6_ and _1_.
CHECK Graph y = x2 + 5x  6. The graph passes through (_6_ , 0) and (_1_ , 0).
Checkpoint Complete the following exercises.
3.
Find the roots of the equation x2  3x + 2 = 0.
1,2
Find the zeros of the function y = x2 + 3x  40 by rewriting the function in intercept
form.
8,5
Homework
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4.
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