Factoring quadratics

Find two numbers that multiply to c and add to b — the quadratic factor trick.

To factor x² + bx + c, find two numbers that multiply to c and add to b.

Walk through

Step 1 of 4

The target

Factor x² + 7x + 12. Goal: two numbers that **multiply to 12** and **add to 7**.

Reading the signs

c positive, b positive → both numbers positive: x² + 7x + 12 = (x + 3)(x + 4).
c positive, b negative → both negative: x² − 5x + 6 = (x − 2)(x − 3).
c negative → one of each sign; the bigger one matches b's sign: x² + x − 6 = (x + 3)(x − 2).
Difference of squares: x² − a² = (x − a)(x + a).

Your turn

Factor x² − 2x − 15.

Watch out

Check the sum as well as the product. For x² − 5x + 6, the pair 2 and 3 multiplies to 6 but adds to +5 — you need −2 and −3.

Try it

x² + 7x + 12

Need two numbers: ×12, +7. That's 3 and 4. So (x+3)(x+4).

Try it

x² − 5x + 6

Need ×6, −5. That's −2 and −3. So (x−2)(x−3).