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Factoring a Quadratic
Expression
Expression v. Equation
Expression: Numbers, symbols, and
operators grouped together that show
the value of something. For example:
3x – 5
Equation: A Statement formed when a
equality symbol is placed between two
equal expressions. For example:
y = 3x – 5
*Make sure to leave the right form on any assessment*
Specific Expressions
• Trinomial – Consisting of three terms
3
2
(Ex: 5x – 9x + 3)
• Binomial – Consisting of 2 terms
6
(Ex: 2x + 2x)
• Monomial – Consisting of one term
2
(Ex: x )
Quadratic Expression
An expression in x that can be written in the
standard form:
2
ax + bx +c
Where a, b, and c are any number except
a ≠ 0.
Factoring
The process of rewriting a mathematical
expression involving a sum to a product.
It is the opposite of distributing.
Example:
x  10 x  24   x  2  x  12 
2
SUM
PRODUCT
Factor
2
If x + 8x + 15 = ( x + 3 )( x + 5 )
then
x + 3 and x + 5 are
called factors of x2 + 8x + 15
(Remember that 3 and 4 are factors of 12 since 3.4=12)
Finding the Dimensions of a Generic
Rectangle
Mr. Wells’ Way to find the product for a generic
rectangle:
Make sure
to Check
First, find the
POSITIVE
Greatest
Common
Factor of two
terms in the
bottom row.
5
10x
-15
2x
2
-6x
2x
-3
4x
Lastly, write the answer
as a Product:
Second, find
missing
WHOLE
NUMBER
dimensions
on the
individual
boxes.
 2 x  5 2 x  3
Factoring with the Box and Diamond
2x  7 x  6
Fill in the results from the diamond
and find the dimensions of the box:
3
GCF 2x
___
ax2 is always
in the bottom
left corner
3x
+6
2x2
4x
x
2
c is
always in
the top
right
corner
Write the expression as a product:
( 2x + 3 )( x + 2 )
Because of our
pattern, the missing
boxes need to
multiply to:
2
Diamond Problem
Factor:
2
(2x )(6)
12x2
4x
3x
7x
The missing boxes
also have to add up
to bx in the sum
Factoring Example 1
3x  13x  10
2
Factor:
Product
c
5
15x
(3x2)(-10)
-10
-30x2
ax2c
x
GCF ___
3x2
-2x
-2x
ax2
3x
-2
( x + 5 )( 3x – 2 )
bx
13x
Sum
15x
Factoring Example 2
Factor: 2 x  15x  77
2
 15x  2 x  77
2
Rewrite in Standard Form: ax2 + bx + c
Product
c
11
33x
-77
(15x2)(-77)
-1155x2
ax2c
2
15x
GCF 5x
___
-35x
-35x
ax2
3x
-7
( 5x + 11 )( 3x – 7 )
bx
-2x
Sum
33x
Factoring Example 3
Factor:
x  6x  9
2
Product
c
3
3x
(x2)(9)
9
9x2
ax2c
x
GCF ___
x2
3x
3x
ax2
x
bx
6x
3
Sum
( x + 3 )( x + 3 )
( x + 3 )2
3x
Factoring Example 4
Factor:
9x  4  9x  0x  4
2
2
Product
c
2
6x
(9x2)(-4)
-4
-36x2
ax2c
GCF 3x
___
9x2
-6x
-6x
ax2
3x
-2
( 3x + 2 )( 3x – 2 )
bx
0
Sum
6x
Factoring: Which Expression is correct?
Factor:
10 x  25 x  15
2
Notice that every term is
divisible by 5
If you use the box and diamond, the following products are
possible:
 x  310 x  5
x5
and
÷5
 5 x  15 2 x  1
Which is the best possible answer?
Factoring Completely Example 1
Factor:
10 x  25x 15
2
Ignore the GCF and factor
the quadratic
Reverse Box to factor out the GCF
2x 2 5x 3
5 10 x2  25x 15
5  2 x  5 x  3  5( x + 3 )( 2x – 1 )
2
Product
c
3
6x
(2x2)(-3)
-3
Don’t forget
the GCF
-6x2
ax2c
x
GCF ___
2x2
-x
ax2
2x
-1
-x
bx
5x
Sum
6x
Factoring Completely Example 2
Factor:
3x  6 x  45x
3
2
Ignore the GCF and factor
the quadratic
Reverse Box to factor out the GCF
x 2 2x 15
3x 3x3  6 x2  45x
3x  x  2 x  15  3x(x + 3)(x – 5)
2
Product
c
3
3x
-15
(x2)(-15)
Don’t forget
the GCF
-15x2
ax2c
x
GCF ___
x2
-5x
ax2
x
-5
-5x
bx
-2x
Sum
3x
Factoring Completely Example 3
Factor:
4 x  20 x
2
Reverse Box to factor out the GCF
x 5
2
4x 4 x  20 x
4 x  x  5
There is no longer a
quadratic, it is not
possible to factor
anymore. There is not
always more factoring
after the GCF.
Factoring Completely Example 4
Factor:
 x  13x  42
2
Reverse Box to factor out the negative
When the x2 term is negative, it is
difficult to factor.
Ignore the GCF and factor
the quadratic
x 2 13x 42
2
2
1  x  13x  42   x  13x  42   -( x + 6 )( x + 7 )
Product
c
6
6x
(x2)(42)
42
Don’t forget
the GCF
42x2
ax2c
x
GCF ___
x2
7x
ax2
x
7
6x
bx
13x
Sum
7x