Factoring Difference of Squares Formula

Factoring difference of squares is recognizing and factoring expressions of the form a^2 - b^2 into the product (a + b)(a - b).

The Formula

a2−b2=(a+b)(a−b)

When to use: When you multiply (a+b)(a−b), the middle terms cancel: a2−ab+ab−b2=a2−b2. So any time you see a perfect square minus a perfect square, you can instantly factor it. Think of it as a rectangle whose area is the difference of two square areas.

Quick Example

x2−25=(x+5)(x−5)
4x2−9=(2x+3)(2x−3)

Notation

a2 and b2 are perfect squares. The minus sign between them is required. a and b can be any expression (e.g., a=2x, b=3).

What This Formula Means

Recognizing and factoring expressions of the form a2−b2 into the product (a+b)(a−b).

When you multiply (a+b)(a−b), the middle terms cancel: a2−ab+ab−b2=a2−b2. So any time you see a perfect square minus a perfect square, you can instantly factor it. Think of it as a rectangle whose area is the difference of two square areas.

Formal View

∀a,b∈R:  a2−b2=(a+b)(a−b). This is an identity in R[a,b]. Note: a2+b2 is irreducible over R (factors only over C as (a+bi)(a−bi)).

Worked Examples

Example 1

easy
Factor x2−49.

Answer

(x+7)(x−7)

First step

1
Step 1: Recognize the form a2−b2 where a=x and b=7.

Full solution

  1. 2
    Step 2: Apply the formula: (a+b)(a−b)=(x+7)(x−7).
  2. 3
    Step 3: Verify: (x+7)(x−7)=x2−7x+7x−49=x2−49 ✓
The difference of squares pattern a2−b2=(a+b)(a−b) works because the middle terms cancel. Both terms must be perfect squares separated by subtraction.

Example 2

medium
Factor 16x2−25y2.

Example 3

medium
Factor 49x2−100y2.

Common Mistakes

  • Trying to factor a2+b2 the same way — a sum of squares does not factor over the reals; only the difference does.
  • Stopping after one step on x4−16 — refactor the new difference of squares: (x2−4)(x2+4)=(x−2)(x+2)(x2+4).
  • Forgetting a coefficient is a square too — 4x2−9 has a=2x, b=3, giving (2x+3)(2x−3), not (2x+9)(2x−1).

Why This Formula Matters

It is the fastest factoring pattern in algebra and the engine behind rationalizing binomial denominators and simplifying rational expressions; missing it forces students into slow trinomial methods on a problem that should take one line. Recognizing it by "Are both terms perfect squares with a minus sign between them and nothing in the middle?" — rather than by familiar numbers — is what lets a student tell it apart from factoring trinomials and perfect-square trinomial and sum of squares in a mixed problem set.

Frequently Asked Questions

What is the Factoring Difference of Squares formula?

Recognizing and factoring expressions of the form a2−b2 into the product (a+b)(a−b).

How do you use the Factoring Difference of Squares formula?

When you multiply (a+b)(a−b), the middle terms cancel: a2−ab+ab−b2=a2−b2. So any time you see a perfect square minus a perfect square, you can instantly factor it. Think of it as a rectangle whose area is the difference of two square areas.

What do the symbols mean in the Factoring Difference of Squares formula?

a2 and b2 are perfect squares. The minus sign between them is required. a and b can be any expression (e.g., a=2x, b=3).

Why is the Factoring Difference of Squares formula important in Math?

It is the fastest factoring pattern in algebra and the engine behind rationalizing binomial denominators and simplifying rational expressions; missing it forces students into slow trinomial methods on a problem that should take one line. Recognizing it by "Are both terms perfect squares with a minus sign between them and nothing in the middle?" — rather than by familiar numbers — is what lets a student tell it apart from factoring trinomials and perfect-square trinomial and sum of squares in a mixed problem set.

What do students get wrong about Factoring Difference of Squares?

The procedure for factoring difference of squares is the easy part; the trap is trying to factor a2+b2 the same way. Asking "Are both terms perfect squares with a minus sign between them and nothing in the middle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Factoring Difference of Squares formula?

Before studying the Factoring Difference of Squares formula, you should understand: factoring, polynomials.

Want the Full Guide?

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Factoring Polynomials: All Methods Explained with Step-by-Step Examples →
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