Factoring rewrites a quadratic like x² + 5x + 6 as a product of two simpler pieces, (x + 2)(x + 3). It is the reverse of multiplying out, and it is the fastest way to solve many quadratic equations.
Try it with your own problem ↓
The steps
- Make sure the quadratic is in order: x² + bx + c.
- Find two numbers that multiply to c and add to b. List the factor pairs of c to find them.
- Write the factors as (x + p)(x + q), where p and q are those two numbers.
- Check by multiplying out (FOIL).
- To solve = 0, set each factor equal to 0.
A worked example
Factor x² + 5x + 6.
- We need two numbers that multiply to 6 and add to 5.
- Pairs that multiply to 6: 1 × 6 (sum 7) and 2 × 3 (sum 5). The pair 2 and 3 works.
- Write (x + 2)(x + 3).
- Check: x · x = x², x · 3 + 2 · x = 5x, 2 · 3 = 6. So x² + 5x + 6. ✓
- If asked to solve x² + 5x + 6 = 0: x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.
x² + 5x + 6 = (x + 2)(x + 3).
Common mistakes
- Getting the sign of the numbers wrong. If c is positive, p and q have the same sign as b. If c is negative, they have opposite signs.
- Finding numbers that only add, or only multiply. You need both.
- Skipping the check. Multiply back to catch sign slips.
- Forgetting to take out a common factor first. In 2x² + 10x + 12, pull out the 2 before factoring.
Questions people ask
What if the first number isn't 1, like 6x² + 7x − 3?
Use the "ac method": find two numbers that multiply to 6 × (−3) = −18 and add to 7 (9 and −2), split the middle term, and factor by grouping. This gives (3x − 1)(2x + 3).
What if it won't factor?
Some quadratics don't factor with whole numbers. Use the quadratic formula instead.
What is a difference of squares?
a² − b² = (a − b)(a + b). So x² − 9 = (x − 3)(x + 3).
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