Factoring
ax
^{2} + bx + c
This section explains how to factor expressions of the
form
ax
^{2} + bx + c
, where
a
,
b
, and
c
are integers.
First, factor out all constants which evenly divide all three terms. If
a
is negative, factor out 1. This will leave an expression of the
form
d (ax
^{2} + bx + c)
, where
a
,
b
,
c
, and
d
are integers, and
a > 0
. We can now turn to factoring the inside expression.
Here is how to factor an expression
ax
^{2} + bx + c
, where
a > 0
:
 Write out all the pairs of numbers that, when multiplied, produce
a
.
 Write out all the pairs of numbers that, when multiplied, produce
c
.
 Pick one of the
a
pairs 
(a
_{1}, a
_{2})
 and one of the
c
pairs 
(c
_{1}, c
_{2})
.
 If
c > 0
: Compute
a
_{1}
c
_{1} + a
_{2}
c
_{2}
. If
 a
_{1}
c
_{1} + a
_{2}
c
_{2} = b
,
then the factored form of the quadratic is

(a
_{1}
x + c
_{2})(a
_{2}
x + c
_{1})
if
b > 0
.

(a
_{1}
x  c
_{2})(a
_{2}
x  c
_{1})
if
b < 0
.
 If
a
_{1}
c
_{1} + a
_{2}
c
_{2}≠b
, compute
a
_{1}
c
_{2} + a
_{2}
c
_{1}
. If
a
_{1}
c
_{2} + a
_{2}
c
_{1} = b
, then the factored form of the quadratic is
(a
_{1}
x + c
_{1})(a
_{2}
x + c
_{2})
or
(a
_{1}
x + c
_{1})(a
_{2}
x + c
_{2})
. If
a
_{1}
c
_{2} + a
_{2}
c
_{1}≠b
, pick another set of pairs.
 If
c < 0
: Compute
a
_{1}
c
_{1} a
_{2}
c
_{2}
. If
 a
_{1}
c
_{1}  a
_{2}
c
_{2} = b
, then
the factored form of the quadratic is:
(a
_{1}
x  c
_{2})(a
_{2}
x + c
_{1})
where
a
_{1}
c
_{1} > a
_{2}
c
_{2}
if
b > 0
and
a
_{1}
c
_{1} < a
_{2}
c
_{2}
if
b < 0
.
Using FOIL, the outside pair plus (or minus) the inside pair must equal
b
.
 Check.
Example 1: Factor
3x
^{2}  8x + 4
.
 Numbers that produce 3: (1, 3).
 Numbers that produce 4: (1, 4), (2, 2).

 (1, 3) and (1, 4):
1(1) + 3(4) = 11≠8
.
1(4) + 3(1) = 7≠ = 8
.
 (1, 3) and (2, 2):
1(2) + 3(2) = 8
.

(x  2)(3x  2)
.
 Check:
(x  2)(3x  2) = 3x
^{2} 2x  6x + 4 = 3x
^{2}  8x + 4
.
Example 2: Factor
12x
^{2} + 17x + 6
.
 Numbers that produce 12: (1, 12), (2, 6), (3, 4).
 Numbers that produce 6: (1, 6), (2, 3).

 (1, 12) and (1, 6):
1(1) + 12(6) = 72
.
1(6) + 12(1) = 18
.
 (1, 12) and (2, 3):
1(2) + 12(3) = 38
.
1(3) + 12(2) = 27
.
 (2, 6) and (1, 6):
2(1) + 6(6) = 38
.
2(6) + 6(1) = 18
.
 (2, 6) and (2, 3):
2(2) + 6(3) = 22
.
2(3) + 6(2) = 18
.
 (3, 4) and (1, 6):
3(1) + 4(6) = 27
.
3(6) + 4(1) = 22
.
 (3, 4) and (2, 3):
3(2) + 4(3) = 18
.
3(3) + 4(2) = 17
.
(3x + 2)(4x + 3)
.
 Check:
(3x + 2)(4x + 3) = 12x
^{2} +9x + 8x + 6 = 12x
^{2} + 17x + 6
.
Example 3: Factor
4x
^{2}  5x  21
.
 Numbers that produce 4: (1, 4), (2, 2).
 Numbers that produce 21: (1, 21), (3, 7).

 (1, 4) and (1, 21):
1(1) 4(21) =  83
.
1(21)  4(1) = 17
.
 (1, 4) and (3, 7):
1(3)  4(7) =  25
.
1(7)  4(3) =  5
.
(x  3)(4x + 7)
.
 Check:
(x  3)(4x + 7) = 4x
^{2} +7x  12x  21 = 4x
^{2}  5x  21
.
Any easy way to check your answer is to pick a value of
x
(
x = 10
is
a good value to pick) and evaluate the original expression and the
factored form. You should get the same result for both expressions.