Factoring a Quadratic
AC Method
Purpose:
Factor the quadratic ax^2 + bx + c into this form: ( )( ) of two terms multiplied.
Example:
2x^2 + 11x + 12
2x^2 + 11x + 12
Method:
Find factors of a times c that when added equal b
Find factors of a times c that when added equal b
Steps:
given: a=2, b=11, c=12
multiply a times c: (2)(12)=24
Factors of 24 Factors added
1*24 25
2*12 14
3*8 11
Separate the example equation into the a times c factors that add to equal b:
2x^2 + 11x + 12 = 2x^2 + 3x + 8x + 12
Factor the first two terms and the last two terms
2x^2 + 3x = x(2x +3) and 8x + 12 = 4(2x+3)
x(2x+3) + 4(2x+3)
Factor out the common term
(2x + 3)(x + 4)
The quadratic is now factored!
If you cannot find two factors of a times c that add to equal b, the equation cannot be factored in
this way.
this way.
An example of when c is negative:
2x^2 + 5x - 12
a=2, b=5, c=-12
(a)(c) = (2)(-12)= -24
Factors of 24 that subtract to equal b:
1*24 23
2*12 10
3*8 5
2x^2 - 3x + 8x - 12
x(2x - 3) + 4(2x - 3)
(2x - 3)(x + 4) the answer
An example of when b is negative:
3x^2 -4x -4
a=3, b=-4, c=-4
(a)(c) = (3)(-4) = -12
Factors of 12 that subtract to equal b:
1*12 11
2*6 4
3x^2 + 2x - 6x -4
x(3x + 2) - 2(3x + 2)
(3x + 2)(x - 2) the answer
An example of when it doesn't work:
3x^2 + 11x + 1
a=3, b=11, c=1
(a)(c) = (3)(1) = 3
Factors of 3 are:
1*3 added equals 4, since b does not equal 4, this equation cannot be factored this way.
