AC method factoring guide

Factoring a trinomial like x² + 7x + 12 is easy: you hunt for two numbers that multiply to 12 and add to 7. But the moment the x² has a coefficient, like 6x² + 7x + 2, guessing stops working. The AC method is the systematic way through it. It works on every factorable trinomial of the form ax² + bx + c, and our free AC method calculator runs the whole thing for you, showing each step so you can copy the method yourself.

The four steps

For a trinomial ax² + bx + c:

Step 1: Multiply a times c. This is the product you will be hunting pairs for.

Step 2: Find two numbers that multiply to ac and add to b. Same skill you already use for simple trinomials, just with the product from step 1.

Step 3: Split the middle term. Rewrite bx as two terms using the pair you found.

Step 4: Factor by grouping. Pull the GCF out of each pair of terms, then factor out the common binomial. Expand to check.

That is the entire method. Steps 1 through 3 just set up step 4.

Worked example: 6x² + 7x + 2

Step 1. a = 6, c = 2, so ac = 12.

Step 2. Numbers that multiply to 12 and add to 7: 3 and 4. (3 × 4 = 12, 3 + 4 = 7.)

Step 3. Split 7x: 6x² + 3x + 4x + 2.

Step 4. Group: 3x(2x + 1) + 2(2x + 1). Both halves share (2x + 1), so pull it out: (3x + 2)(2x + 1).

Check: (3x + 2)(2x + 1) = 6x² + 3x + 4x + 2 = 6x² + 7x + 2. It works.

Worked example with negatives: 4x² − 5x − 6

Step 1. a = 4, c = −6, so ac = −24.

Step 2. Numbers that multiply to −24 and add to −5. List the pairs of 24: 1×24, 2×12, 3×8, 4×6. With signs, 3 and −8 work: 3 × (−8) = −24, 3 + (−8) = −5.

Step 3. Split −5x: 4x² + 3x − 8x − 6.

Step 4. Group: x(4x + 3) − 2(4x + 3) = (x − 2)(4x + 3).

Check: (x − 2)(4x + 3) = 4x² + 3x − 8x − 6 = 4x² − 5x − 6. Done.

Before you start: check the GCF

12x² + 18x + 6 looks like an AC method problem, but every term is divisible by 6. Pull it out first: 6(2x² + 3x + 1). Then factor 2x² + 3x + 1 the normal way (ac = 2, pair 1 and 2, → (2x + 1)(x + 1)), giving 6(2x + 1)(x + 1). Factoring completely always starts with the GCF, which is why our GCF calculator is the first stop for messy trinomials.

Three ways this goes wrong

  • Wrong signs in step 2. When ac is negative, one number must be negative. Write the sign into the pair from the start instead of fixing it later.
  • Splitting b with numbers that add correctly but multiply wrong. Both conditions have to hold. If the grouping in step 4 does not produce a common binomial, the pair was wrong; go back to step 2.
  • Trying it on something it cannot factor. x² + x + 1 has ac = 1 and no integer pair multiplies to 1 and adds to 1. Some quadratics just do not factor over the integers; the quadratic formula (see our quadratic formula calculator) will tell you that instead.

Related tools

The AC method sits between two tools you probably already use: the general trinomial calculator for ax² + bx + c and the simple trinomial calculator for x² + bx + c. If your expression has four terms instead of three, try the factoring by grouping calculator (step 4 above IS factoring by grouping). Our step-by-step factoring calculator tries every method automatically, and the how to factor polynomials guide explains when to reach for which one. Working with plain numbers instead of algebra? The factor tree calculator breaks whole numbers into prime factors the same systematic way. Browse every tool on the calculators page.

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