How to Factor Quadratics: A Step-by-Step Guide

Factoring a quadratic means rewriting it as a product of two binomials. For example, x² + 5x + 6 becomes (x + 2)(x + 3). This is useful for solving equations and simplifying expressions. Here's how to do it.

Form: x² + bx + c (leading coefficient = 1)

Find two numbers that multiply to c and add to b. Those numbers go in the binomials.

Example: Factor x² + 7x + 12

We need two numbers that multiply to 12 and add to 7.

• 3 × 4 = 12 and 3 + 4 = 7 ✓

Check: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓

Example: Factor x² − 5x + 6

Two numbers that multiply to 6 and add to −5. Both must be negative.

• (−2)(−3) = 6 and (−2) + (−3) = −5 ✓

Example: Factor x² + 2x − 15

Multiply to −15, add to 2. One positive, one negative.

• 5 × (−3) = −15 and 5 + (−3) = 2 ✓

Form: ax² + bx + c (leading coefficient ≠ 1)

Use the AC method (or "split the middle term"):

1. Multiply a and c.
2. Find two numbers that multiply to ac and add to b.
3. Rewrite bx as the sum of those two terms.
4. Factor by grouping.

Example: Factor 2x² + 7x + 3

a = 2, b = 7, c = 3. ac = 6. Need two numbers that multiply to 6 and add to 7 → 1 and 6.

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

GCF First

Always check for a common factor before using other methods.

Difference of Squares

a² − b² = (a + b)(a − b). Use when you have a perfect square minus another perfect square.

Solving Quadratic Equations by Factoring

Once factored, set each factor equal to zero and solve.

Solutions: x = −3 or x = −4

For quadratics that don't factor nicely, use the quadratic formula.

Common mistakes

Wrong signs: In x² − 5x + 6, both factors must be negative to get +6 and −5: (x − 2)(x − 3). Check by expanding.

Forgetting the leading coefficient: When a ≠ 1 (e.g., 2x² + 7x + 3), you can't just find factors of c. Use the AC method: multiply a × c, find factor pairs of that product, then split the middle term.

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