4.3 Solve x2
+ bx + c = 0 by Factoring
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 ¹ 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) |
-l, _8_ |
l, _-8_ |
|
Sum of factors (m + n) |
_7_ |
_-7_ |
|
Factors of -8
(m, n) |
-2, _4_ |
2, _-4_ |
|
Sum of factors (m + n) |
_2_ |
_-2_ |
Notice that m =
_-1_
and n = _8_. So, x2 + 7x -
8 = ( x -
1 )( x + 8 ).
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. m2
-
22m + 121 |
Perfect square
trinomial |
|
=
m2 - 2(m)( 11 ) + _11_2 |
|
|
=
( m - 11 )2 |
|
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 |
Example 3
Find the roots of an
equation
Find the roots of the equation x2
- 2x - 15 = 0.
Solution
|
x2 - 2x - 15 = 0 |
Original
equation |
|
( x - 5 )( x + 3 )
= 0 |
Factor. |
|
_x - 5_ = 0 or _x
+ 3_ = 0 |
Zero
product property |
|
x = _5_ or x = _-3_ |
Solve
for x. |
|
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 |
Write
original equation. |
|
= ( x + 6 )( x - 1 ) |
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).
