Factoring Difference of Squares Formula

Factoring difference of squares are 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)a^2 - b^2 = (a + b)(a - b)

When to use: When you multiply (a+b)(a−b)(a + b)(a - b), the middle terms cancel: a2−ab+ab−b2=a2−b2a^2 - ab + ab - b^2 = a^2 - b^2. 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)x^2 - 25 = (x + 5)(x - 5)
4x2−9=(2x+3)(2x−3)4x^2 - 9 = (2x + 3)(2x - 3)

Notation

a2a^2 and b2b^2 are perfect squares. The minus sign between them is required. aa and bb can be any expression (e.g., a=2xa = 2x, b=3b = 3).

What This Formula Means

Recognizing and factoring expressions of the form a2−b2a^2 - b^2 into the product (a+b)(a−b)(a + b)(a - b).

When you multiply (a+b)(a−b)(a + b)(a - b), the middle terms cancel: a2−ab+ab−b2=a2−b2a^2 - ab + ab - b^2 = a^2 - b^2. 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)\forall a, b \in \mathbb{R}:\; a^2 - b^2 = (a + b)(a - b). This is an identity in R[a,b]\mathbb{R}[a, b]. Note: a2+b2a^2 + b^2 is irreducible over R\mathbb{R} (factors only over C\mathbb{C} as (a+bi)(a−bi)(a + bi)(a - bi)).

Worked Examples

Example 1

easy
Factor x2−49x^2 - 49.

Answer

(x+7)(x−7)(x + 7)(x - 7)

First step

1
Step 1: Recognize the form a2−b2a^2 - b^2 where a=xa = x and b=7b = 7.

Full solution

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

Example 2

medium
Factor 16x2−25y216x^2 - 25y^2.

Example 3

medium
Factor 49x2−100y249x^2 - 100y^2.

Common Mistakes

  • Trying to factor a2+b2a^2+b^2 the same way — a sum of squares does not factor over the reals; only the difference does.
  • Stopping after one step on x4−16x^4-16 — refactor the new difference of squares: (x2−4)(x2+4)=(x−2)(x+2)(x2+4)(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4).
  • Forgetting a coefficient is a square too — 4x2−94x^2-9 has a=2x, b=3a=2x,\ b=3, giving (2x+3)(2x−3)(2x+3)(2x-3), not (2x+9)(2x−1)(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−b2a^2 - b^2 into the product (a+b)(a−b)(a + b)(a - b).

How do you use the Factoring Difference of Squares formula?

When you multiply (a+b)(a−b)(a + b)(a - b), the middle terms cancel: a2−ab+ab−b2=a2−b2a^2 - ab + ab - b^2 = a^2 - b^2. 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?

a2a^2 and b2b^2 are perfect squares. The minus sign between them is required. aa and bb can be any expression (e.g., a=2xa = 2x, b=3b = 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+b2a^2+b^2 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.

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