MATH LESSON

FACTORING: DIFFERENCE OF TWO SQUARES

When you see two perfect squares separated by a minus sign, you can factor them into a sum and difference!

A difference of two squares is a binomial of the form a² − b², which always factors into (a + b)(a − b).

a² − b² = (a + b)(a − b)

a² − b²

=

(a + b)

×

(a − b)

How to Recognize It

A Difference of Two Squares (DOTS) must meet ALL three conditions:

1. It must be a binomial (exactly 2 terms)

2. Both terms must be perfect squares

3. The sign between them must be MINUS (−)

a² + b² is NOT a difference of two squares!

a² − b² YES, this is DOTS!

How to Identify a and b

For each perfect square term, find its square root. That gives you a and b:

Given: x² − 25

First term: x² = (x)² → a = x

Second term: 25 = (5)² → b = 5

Substitute into the pattern:

(a + b)(a − b) = (x + 5)(x − 5)

Order Does NOT Matter!

Even if the smaller square comes first, it is still DOTS. Just be careful with signs:

x² − 25 = (x + 5)(x − 5)

25 − x² = (5 + x)(5 − x)

Both are correct! The larger term determines which is a and which is b.

Common Mistakes!

Mistake 1: Factoring a sum of squares (a² + b²).

x² + 25 = (x + 5)(x + 5) × WRONG

x² + 25 = PRIME (cannot be factored using DOTS) ✓

Mistake 2: Using the same sign in both factors.

x² − 25 = (x + 5)(x + 5) × WRONG

x² − 25 = (x + 5)(x − 5) ✓ One PLUS, one MINUS!

Mistake 3: Forgetting to take the square root of coefficients.

4x² − 9 = (4x + 9)(4x − 9) × WRONG

4x² − 9 = (2x + 3)(2x − 3) ✓ √4 = 2, √9 = 3

Worked Examples

Example 1: Factor: x² − 36
a² = x² → a = x  |  b² = 36 → b = 6
→ (x + 6)(x − 6)

Example 2: Factor: 4x² − 25
a² = 4x² = (2x)² → a = 2x  |  b² = 25 → b = 5
→ (2x + 5)(2x − 5)

Example 3: Factor: 9x² − 16
a² = 9x² = (3x)² → a = 3x  |  b² = 16 → b = 4
→ (3x + 4)(3x − 4)

Example 4: Factor: 25x² − 1
a² = 25x² = (5x)² → a = 5x  |  b² = 1 = (1)² → b = 1
→ (5x + 1)(5x − 1)

Example 5: Factor: x&sup4; − 81
a² = x⁴ = (x²)² → a = x²  |  b² = 81 → b = 9
→ (x² + 9)(x² − 9) (x²−9 can be factored further!)

Shortcut: Find the square root of each term, then write (root + root)(root − root). Always one PLUS and one MINUS!

MATH SEATWORK

Seatwork Time!

Answer all 5 questions. Good luck!

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Made for students who want to master Factoring Difference of Two Squares