Factoring Difference of Squares Formula

The Formula

a^2 - b^2 = (a + b)(a - b)

When to use: When you multiply (a + b)(a - b), the middle terms cancel: a^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

x^2 - 25 = (x + 5)(x - 5)
4x^2 - 9 = (2x + 3)(2x - 3)

Notation

a^2 and b^2 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 a^2 - b^2 into the product (a + b)(a - b).

When you multiply (a + b)(a - b), the middle terms cancel: a^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

\forall a, b \in \mathbb{R}:\; a^2 - b^2 = (a + b)(a - b). This is an identity in \mathbb{R}[a, b]. Note: a^2 + b^2 is irreducible over \mathbb{R} (factors only over \mathbb{C} as (a + bi)(a - bi)).

Worked Examples

Example 1

easy
Factor x^2 - 49.

Solution

  1. 1
    Step 1: Recognize the form a^2 - b^2 where a = x and b = 7.
  2. 2
    Step 2: Apply the formula: (a+b)(a-b) = (x+7)(x-7).
  3. 3
    Step 3: Verify: (x+7)(x-7) = x^2 - 7x + 7x - 49 = x^2 - 49 ✓

Answer

(x + 7)(x - 7)
The difference of squares pattern 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 16x^2 - 25y^2.

Common Mistakes

  • Trying to factor a SUM of squares: x^2 + 9 \neq (x + 3)(x - 3)
  • Not recognizing disguised forms like 16x^4 - 1 = (4x^2 + 1)(4x^2 - 1), which can be factored further
  • Forgetting to identify both a and b correctly: in 4x^2 - 9, a = 2x and b = 3

Why This Formula Matters

One of the most commonly tested factoring patterns. Appears throughout algebra, calculus (rationalizing), and number theory.

Frequently Asked Questions

What is the Factoring Difference of Squares formula?

Recognizing and factoring expressions of the form a^2 - b^2 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: a^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?

a^2 and b^2 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?

One of the most commonly tested factoring patterns. Appears throughout algebra, calculus (rationalizing), and number theory.

What do students get wrong about Factoring Difference of Squares?

A SUM of squares a^2 + b^2 does NOT factor over the real numbers. This pattern only works for differences.

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?

This formula is covered in depth in our complete guide:

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