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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*
• Trinomial –
• Binomial –
• Monomial –
Consisting of three terms (Ex: 5x3 – 9x2 + 3)
Consisting of 2 terms (Ex: 2x6 + 2x)
Consisting of one term (Ex: x2)
Specific Expressions
An expression in x that can be written in the standard form:
ax2 + bx +c
Where a, b, and c are any number except a ≠ 0.
Quadratic Expression
Factoring
The process of rewriting a mathematical expression involving a sum to a product. It is the opposite of distributing.
Example:
2 10 24 2 12x x x x SUM PRODUCT
Factor
If x2 + 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)
10x
4x2 -6x
-15
Finding the Dimensions of a Generic Rectangle
Mr. Wells’ Way to find the product for a generic rectangle:
First, find the POSITIVE Greatest Common
Factor of two terms in the bottom row.
2x
Second, find missing WHOLE
NUMBER dimensions
on the individual
boxes.2x -3
5
Make sure to Check
Lastly, write the answer as a Product:
2 5 2 3x x
7x
(2x2)(6) 12x2
3x
4x
+6
2x2
ax2 is always in the bottom
left corner
Factoring with the Box and Diamond22 7 6x x
___ 4x 3x
Because of our pattern, the missing
boxes need to multiply to:
The missing boxes also have to add up
to bx in the sum
2x
x 2
3
( 2x + 3 )( x + 2 )
GCF
Factor: c is always in the top right
corner
Dia
mo
nd P
robl
em
Fill in the results from the diamond and find the dimensions of the box:
Write the expression as a product:
(3x2)(-10)
-30x2-1015x
-2x 3x2ax2c
ax2
Factoring Example 1 23 13 10x x
___ bx
13x
-2x 15x
c Product
Sum
x
3x -2
5
( x + 5 )( 3x – 2 )
GCF
Factor:
33x
-35x
(15x2)(-77)
-1155x2-77
15x2ax2c
ax2
Factoring Example 222 15 77x x
___ bx
-2x
-35x 33x
c Product
Sum
5x
3x -7
11
( 5x + 11 )( 3x – 7 )
GCF
Factor:215 2 77x x
Rewrite in Standard Form: ax2 + bx + c
3x
3x
(x2)(9)
9x29
x2ax2c
ax2
Factoring Example 32 6 9x x
___ bx
6x
3x 3x
c Product
Sum
x
x 3
3
( x + 3 )2
GCF
Factor:
( x + 3 )( x + 3 )
6x
-6x
(9x2)(-4)
-36x2-4
9x2ax2c
ax2
Factoring Example 429 4x
___ bx
0
-6x 6x
c Product
Sum
3x
3x -2
2
( 3x + 2 )( 3x – 2 )
GCF
Factor:29 0 4x x
Factoring: Which Expression is correct?
210 25 15x x Factor:
If you use the box and diamond, the following products are possible:
3 10 5
5 15 2 1
x x
and
x x
Which is the best possible answer?
Notice that every term is divisible by 5
÷5x5
Factoring Completely Example 1210 25 15x x
210 25 15x x 5
22x 35( x + 3 )( 2x – 1 ) 25 2 5 3x x
Factor:
5x
6x
-x
(2x2)(-3)
-6x2-3
2x2ax2c
ax2___ bx
5x
-x 6x
c Product
Sum
x
2x -1
3
GCF
Reverse Box to factor out the GCF Ignore the GCF and factor the quadratic
Don’t forget the GCF
Factoring Completely Example 23 23 6 45x x x
3 23 6 45x x x 3x
2x 153x(x + 3)(x – 5) 23 2 15x x x
Factor:
2x
3x
-5x
(x2)(-15)
-15x2-15
x2ax2c
ax2___ bx
-2x
-5x 3x
c Product
Sum
x
x -5
3
GCF
Reverse Box to factor out the GCF Ignore the GCF and factor the quadratic
Don’t forget the GCF
Factoring Completely Example 3
24 20x x
4 5x x
24 20x x4xx 5
Factor:
Reverse Box to factor out the GCF
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 42 13 42x x
2 13 42x x 1
2x 42-( x + 6 )( x + 7 ) 2 13 42x x
Factor:
13x
6x
7x
(x2)(42)
42x242
x2ax2c
ax2___ bx
13x
6x 7x
c Product
Sum
x
x 7
6
GCF
Reverse Box to factor out the negative Ignore the GCF and factor the quadratic
Don’t forget the GCF
When the x2 term is negative, it is difficult to factor.