Difference of Squares

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

Intermediate25 minLesson

Definition

The difference of squares is a special factoring pattern:

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

This pattern works because when you multiply

(a + b)(a - b)

, the middle terms cancel out:

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

The two factors

(a + b)

and

(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 $x^2 - 16$

Check if it's a difference of squares

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

Identify $a$ and $b$

$a^2 = x^2 \Rightarrow a = x$ , and $b^2 = 16 \Rightarrow b = 4$ → $a = x$ , $b = 4$

Apply the formula

$a^2 - b^2 = (a + b)(a - b)$ → $(x + 4)(x - 4)$

Verify by expanding

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

Answer: $x^2 - 16 = (x + 4)(x - 4)$

Common Mistakes

Trying to factor $a^2 + b^2$ as $(a + b)(a - b)$

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

Correct: The sum of squares $a^2 + b^2$ 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 $x^4 - 16$ can be factored multiple times: first to $(x^2 + 4)(x^2 - 4)$ , then $(x^2 - 4)$ 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 $49$ , $121$ , or $4x^2$

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

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

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.

Length × width = area

$(x + 3)(x - 3) = x^2 - 9$

Remove the 3 × 3 corner (9) from x²: x² − 9 = (x + 3)(x − 3).

Interactive Sandbox

Expression Calculator

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History

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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 $47 \times 53$ instantly as $50^2 - 3^2 = 2500 - 9 = 2491$
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 $23 \times 27$ : Both numbers are 2 away from 25, so $23 \times 27 = (25-2)(25+2) = 25^2 - 2^2 = 625 - 4 = 621$

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 $(50-2)(50+2)$ and simplify:

Area and Geometry

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

Example:

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

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 $14^2 - 10^2$ using the pattern:

Key Takeaways

1The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$
2Both terms must be perfect squares with a minus sign between them
3The factors $(a + b)$ and $(a - b)$ are called conjugates
4Always check if the result can be factored again (like $x^4 - 16$ )
5The sum of squares $a^2 + b^2$ 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:

a^2 - b^2

Glossary

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

Formula Card

Difference of Squares

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

Factor the difference of two perfect squares into conjugate binomials

Perfect Square Trinomial (Plus)

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

Factor a trinomial that is the square of a binomial sum

Perfect Square Trinomial (Minus)

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

Factor a trinomial that is the square of a binomial difference

Difference of Squares

Definition

Worked Examples

Common Mistakes

Interactive Visual

Area model: difference of squares

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Mental Math Multiplication

Area and Geometry

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Difference of Squares

Perfect Square Trinomial (Plus)

Perfect Square Trinomial (Minus)

More in This Topic

Introduction to Factoring

Factoring Out the GCF

Factoring by Grouping

Factoring Trinomials (a=1)