Difference of Squares

Learn to recognize and factor expressions in the form a squared minus b squared.

Intermediate25 minLesson
Printable worksheet

Definition

The difference of squares is a special factoring pattern:
a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)
This pattern works because when you multiply (a+b)(a−b)(a + b)(a - b), the middle terms cancel out:
(a+b)(a−b)=a2−ab+ab−b2=a2−b2(a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2
The two factors (a+b)(a + b) and (a−b)(a - b) are called conjugates - they have the same terms but opposite signs in the middle.

Try it now

Which expression is a difference of squares?

Worked Examples

Factor x2−16x^2 - 16
1

Check if it's a difference of squares

x2x^2 is a perfect square, 16=4216 = 4^2 is a perfect square, and there's a minus sign between them → Yes, it's a difference of squares
2

Identify aa and bb

a2=x2⇒a=xa^2 = x^2 \Rightarrow a = x, and b2=16⇒b=4b^2 = 16 \Rightarrow b = 4 → a=xa = x, b=4b = 4
3

Apply the formula

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b) → (x+4)(x−4)(x + 4)(x - 4)
4

Verify by expanding

(x+4)(x−4)=x2−4x+4x−16=x2−16(x + 4)(x - 4) = x^2 - 4x + 4x - 16 = x^2 - 16 \checkmark → Correct!

Common Mistakes

Trying to factor a2+b2a^2 + b^2 as (a+b)(a−b)(a + b)(a - b)

Why it's wrong: The pattern only works for SUBTRACTION. When you expand (a+b)(a−b)(a + b)(a - b), you get a2−b2a^2 - b^2, not a2+b2a^2 + b^2.

Correct: The sum of squares a2+b2a^2 + b^2 cannot be factored over the real numbers. Only the DIFFERENCE of squares can be factored.

Forgetting to check if each factor can be factored further

Why it's wrong: Some expressions like x4−16x^4 - 16 can be factored multiple times: first to (x2+4)(x2−4)(x^2 + 4)(x^2 - 4), then (x2−4)(x^2 - 4) factors again.

Correct: Always check: is the result still a difference of squares? If yes, factor again until no more factoring is possible.

Not recognizing perfect squares like 4949, 121121, or 4x24x^2

Why it's wrong: To use this pattern, you must identify that both terms are perfect squares.

Correct: Memorize perfect squares: 1,4,9,16,25,36,49,64,81,100,121,1441, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. For variables: (2x)2=4x2(2x)^2 = 4x^2, (3y)2=9y2(3y)^2 = 9y^2.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Factoring the Difference of Two Squares | Step-by-Step Examples

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Interactive Visual

Area model: difference of squares

Cut an a × a corner out of x²: what remains rearranges into an (x + a) × (x − a) rectangle.

(x−4)2(x - 4)^2
4(x−4)4(x - 4)
4(x−4)4(x - 4)
−16-16
x−4x - 4
44
x−4x - 4
44

Length × width = area

(x+4)(x−4)=x2−16(x + 4)(x - 4) = x^2 - 16

Remove the 4 × 4 corner (16) from x²: x² − 16 = (x + 4)(x − 4).

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which expression is a difference of squares?

Why It Matters

The difference of squares pattern is one of the most useful shortcuts in algebra:
  • Mental Math: Calculate 47×5347 \times 53 instantly as 502−32=2500−9=249150^2 - 3^2 = 2500 - 9 = 2491
  • Simplifying: Factor complex expressions quickly without trial and error
  • Problem Solving: Many geometry and physics problems involve this pattern
  • Foundation: This pattern appears throughout calculus and higher mathematics
Recognizing this pattern saves time and reduces errors in countless algebra problems!

Real World Applications

Mental Math Multiplication

Calculate products of numbers equidistant from a round number using the difference of squares pattern.

Example:

To calculate 23×2723 \times 27: Both numbers are 2 away from 25, so 23×27=(25−2)(25+2)=252−22=625−4=62123 \times 27 = (25-2)(25+2) = 25^2 - 2^2 = 625 - 4 = 621

1Try It Yourself

A store sells items for 48 dollars and 52 dollars. A customer buys one of each.

Use the difference of squares to find the total mentally.

Step 1: Write the mathematical expression

Write as (50−2)(50+2)(50-2)(50+2) and simplify:

Area and Geometry

The difference of squares appears when calculating the area between two squares.

Example:

A large square has side x+3x + 3 and a small square inside has side x−3x - 3. The area of the border is (x+3)2−(x−3)2(x+3)^2 - (x-3)^2. Using our pattern (with a=x+3a = x+3 and b=x−3b = x-3): =[(x+3)+(x−3)][(x+3)−(x−3)]=(2x)(6)=12x= [(x+3)+(x-3)][(x+3)-(x-3)] = (2x)(6) = 12x

2Try It Yourself

A picture frame is made by cutting a square hole (side 10 cm) from a larger square (side 14 cm).

Find the area of the frame using difference of squares.

Step 1: Write the mathematical expression

Calculate 142−10214^2 - 10^2 using the pattern:

Key Takeaways

  • 1The difference of squares formula is a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)
  • 2Both terms must be perfect squares with a minus sign between them
  • 3The factors (a+b)(a + b) and (a−b)(a - b) are called conjugates
  • 4Always check if the result can be factored again (like x4−16x^4 - 16)
  • 5The sum of squares a2+b2a^2 + b^2 cannot be factored over real numbers

Frequently Asked Questions

No, the sum of squares cannot be factored using real numbers. The difference of squares pattern ONLY works when there's a minus sign: a2−b2a^2 - b^2.
No, the sum of squares cannot be factored using real numbers. The difference of squares pattern ONLY works when there's a minus sign: a2−b2a^2 - b^2.
When you expand (a+b)(a−b)(a+b)(a-b), you get a2−ab+ab−b2a^2 - ab + ab - b^2. The −ab-ab and +ab+ab cancel each other out, leaving just a2−b2a^2 - b^2.
For numbers: check if it's in the list 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc. For variables: the exponent must be even (like x2x^2, x4x^4, y6y^6), and any coefficient must itself be a perfect square (like 4x2=(2x)24x^2 = (2x)^2).

Glossary

Difference of squares
An expression of the form a2−b2a^2 - b^2 that factors as (a+b)(a−b)(a+b)(a-b)
Perfect square
A number or expression that is the square of an integer or monomial (e.g., 16, x2x^2, 9y49y^4)
Conjugates
A pair of binomials that differ only in the sign between their terms: (a+b)(a+b) and (a−b)(a-b)
Factor completely
To break down an expression into factors that cannot be factored further

Formula Card

Difference of Squares

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Factor the difference of two perfect squares into conjugate binomials

Perfect Square Trinomial (Plus)

a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2

Factor a trinomial that is the square of a binomial sum

Perfect Square Trinomial (Minus)

a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2

Factor a trinomial that is the square of a binomial difference

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