Park Bench
Another Method of Factoring
Objective: To Factor Quadratics in the form of ax2 ± bx ± c = 0, where a, b, and c are real numbers.
NOTE: The method demonstrated in this handout is most effective when the leading coefficient
of the quadratic equation is not equal to one. a ≠ 1
Example:
Factor the following quadratic equation into its linear factors: 6 x 2 − 17 x + 12
Step 1 Find the key by multiplying the values for a and c.
The general equation for a quadratic is ax 2 ± bx ± c ; therefore, in our example a = 6, b = -17 and
c = 12. The key is formed by multiplying a * c, or 6 * 12 = 72
Step 2 List the factors for the key value. If the key value is negative, one of the factors must be
negative. If the key is positive and the b value is negative, all factors of the key are negative.
In the example, the key value is equal to 72 and the value for b is negative (b = -17); therefore,
all the factors listed must be negative.
Factors of 72
-1 * -72
-2 * -36
-3 * -24
-4 * -18
-6 * -12
-8 * -9
Step 3 Determine which factors of the key will give a total value equal to the value for b when
added.
Using the list of factors, determine what the sum of the factors would be. Look for the factors
that give you a sum equal to the value of b (b = -17).
Factors of 72
-1 * -72
-2 * -36
-3 * -24
-4 * -18
-6 * -12
-8 * -9
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Sum of the Factors
-73
-38
-27
-22
-18
-17
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factor )(ax + factor )
, where a is the value of the leading
a
coefficient and the factors are those identified in step 3.
Step 4 Use the following formula,
(ax +
In the example, a = 6 and the factors are -8 and -9.
(ax +
(6 x − 8)(6 x − 9)
factor )(ax + factor )
→→
a
6
Step 5 Factor out common multiples from each of the linear factors.
(6 x − 8)(6 x − 9)
6
→→
2(3x − 4) ∗ 3(2 x − 3)
6
Step 6 The common multiples that were factored out should cancel with the denominator leaving
only the linear factors.
2(3x − 4) ∗ 3(2 x − 3)
6 ∗ (3x − 4 )(2 x − 3)
6 ∗ (3x − 4 )(2 x − 3)
→→
→→
→ → (3 x − 4 )(2 x − 3)
6
6
6
Therefore, the linear factors of 6 x 2 − 17 x + 12 are (3 x − 4 )(2 x − 3) .
Listed below are other factoring techniques.
Special Factor Formulas:
x3 + a3 = (x + a) (x2 – ax + a2)
x3 – a3 = (x – a) (x2 + ax + a2)
x2 – a2 = (x – a) (x + a)
x4 – a4 = (x2 – a2) (x2 + a2)
x4 + a4 = (x2 +
2 ax + a2) (x2 –
2 ax + a2)
Factor by Grouping:
acx3 + adx2 + bcx + bd =
ax2 (cx + d) + b (cx + d) =
(ax2 + b) (cx + d)
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