How to Factor Quadratics When a Is Not 1 (AC Method, Step by Step)

To factor a quadratic ax² + bx + c when a ≠ 1, use the AC method: multiply a·c, find two numbers that multiply to ac and add to b, split the middle term, and factor by grouping. For example, 2x² + 3x − 2 = (2x − 1)(x + 2).

This page shows the method in full, with two solved examples and the mistakes students most often make.

Worked Example 1: Factor 2x² + 3x − 2

Here a = 2, b = 3, c = −2.

1. Compute a·c. a·c = 2·(−2) = −4.
2. Find two numbers that multiply to −4 and add to 3. The pair is 4 and −1, because 4·(−1) = −4 and 4 + (−1) = 3.
3. Split the middle term using those numbers: 2x² + 4x − x − 2.
4. Group and factor: 2x(x + 2) − 1(x + 2).
5. Pull out the common binomial: (2x − 1)(x + 2).

Check by expanding: (2x − 1)(x + 2) = 2x² + 4x − x − 2 = 2x² + 3x − 2. ✓

So 2x² + 3x − 2 = (2x − 1)(x + 2), with zeros x = 1/2 and x = −2.

The Method in 5 Steps

1. Identify a, b, c. Write the quadratic in standard form ax² + bx + c. Example: in 6x² − 7x − 3, a = 6, b = −7, c = −3.
2. Multiply a·c. This is the key number the AC method is named after. For 6x² − 7x − 3, a·c = −18.
3. Find two integers that multiply to a·c and add to b. List factor pairs of a·c and pick the pair whose sum is b. For a·c = −18, b = −7: the pair is −9 and 2.
4. Split the middle term using those two integers. 6x² − 7x − 3 = 6x² − 9x + 2x − 3.
5. Factor by grouping. Group the first two and last two terms, factor each group, then extract the common binomial: 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3).

What if there's a common factor first?

Always check for a GCF before applying AC. For 4x² + 10x + 4, factor out 2 first: 2(2x² + 5x + 2) = 2(2x + 1)(x + 2).

Worked Example 2: Factor 6x² − 7x − 3

Here a = 6, b = −7, c = −3.

1. a·c = 6·(−3) = −18.
2. Two numbers that multiply to −18 and add to −7: −9 and 2. Check: (−9)(2) = −18 and −9 + 2 = −7. ✓
3. Split the middle term: 6x² − 9x + 2x − 3.
4. Group: (6x² − 9x) + (2x − 3) = 3x(2x − 3) + 1(2x − 3).
5. Common binomial factor: (3x + 1)(2x − 3).

Check: (3x + 1)(2x − 3) = 6x² − 9x + 2x − 3 = 6x² − 7x − 3. ✓

Zeros: x = −1/3 and x = 3/2.

Common Mistakes

• Forgetting to multiply a by c. Students often look for numbers that multiply to c alone (which works only when a = 1). When a ≠ 1, you must use a·c.
• Getting the signs wrong on the factor pair. If a·c is negative, the two numbers have opposite signs. If a·c is positive, they share the sign of b.
• Not pulling out a GCF first. Missing an initial common factor (like the 2 in 4x² + 10x + 4) leads to messy numbers and often an unfactorable-looking result.
• Grouping incorrectly. After splitting the middle term, the two grouped binomials must be identical. If they aren't, swap the order of the two middle terms and try again.
• Stopping too early. After grouping you must factor out the common binomial to get the final (px + q)(rx + s) form — writing 3x(2x − 3) + 1(2x − 3) is not the final answer.

Reading the AC method is one thing; catching your own sign error when you split the middle term is another. LernOS walks you through quadratics like these one question at a time, nudging you when you get stuck instead of showing the answer — sign up to try it on the next problem you're stuck on.