AC Method — Definition, Formula & Examples

The AC method is a systematic way to factor trinomials of the form

ax^2 + bx + c

by finding two numbers that multiply to

ac

and add to

b

, then using grouping to complete the factorization.

Given a trinomial $ax^2 + bx + c$ with integer coefficients, the AC method requires identifying integers $m$ and $n$ such that $mn = ac$ and $m + n = b$ . The middle term $bx$ is then rewritten as $mx + nx$ , and the resulting four-term polynomial is factored by grouping.

How It Works

Start by computing the product

ac

. Find two integers

m

and

n

whose product equals

ac

and whose sum equals

b

. Rewrite the trinomial as

ax^2 + mx + nx + c

. Group the first two terms and the last two terms, then factor each group. A common binomial factor will appear, which you factor out to get the final product of two binomials.

Worked Example

Problem: Factor

6x^2 + 11x + 4

using the AC method.

Compute ac: Multiply the leading coefficient by the constant term.

ac = 6 \times 4 = 24

Find m and n: Find two integers that multiply to 24 and add to 11. The pair 3 and 8 works because

3 \times 8 = 24

and

3 + 8 = 11

m = 3, \quad n = 8

Rewrite and group: Split the middle term using m and n, then factor by grouping.

6x^2 + 3x + 8x + 4 = 3x(2x + 1) + 4(2x + 1)

Factor out common binomial: Both groups share the factor

(2x + 1)

(2x + 1)(3x + 4)

Answer:

6x^2 + 11x + 4 = (2x + 1)(3x + 4)

Why It Matters

The AC method is especially valuable when the leading coefficient

a \neq 1

, where simple guess-and-check becomes tedious. It is a standard technique in Algebra 1 and Algebra 2 courses and a prerequisite for solving quadratic equations by factoring.

Common Mistakes

Mistake: Finding two numbers that multiply to

c

instead of

ac

Correction: You must multiply the leading coefficient

a

by the constant

c

first. The two numbers need to multiply to the full product

ac

, not just

c