Back to Lesson

Teacher Guide: Difference of Squares

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

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Factoring. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Recognize expressions that fit the difference of squares pattern
  • Factor difference of squares expressions using the formula a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
  • Identify when an expression can be factored multiple times
  • Apply the difference of squares to mental math calculations
  • Distinguish between difference of squares (factorable) and sum of squares (not factorable)
Prerequisites
  • • Understanding of perfect squares (numbers and variables)
  • • Basic polynomial multiplication (FOIL method)
  • • Familiarity with factoring concepts
  • • Knowledge of exponent rules
Discussion Starters
  • 1. Why do you think the sum of squares cannot be factored but the difference of squares can?
  • 2. How could a cashier use the difference of squares pattern to calculate prices quickly?
  • 3. If x4−16x^4 - 16 can be factored twice, can x8−256x^8 - 256 be factored even more times?
  • 4. What happens if you multiply three consecutive odd numbers like 7, 9, 11? Is there a pattern?
Common Misconceptions

Thinking x2+4x^2 + 4 factors as (x+2)(x+2)(x+2)(x+2)

Remediation: Have students expand (x+2)(x+2)=x2+4x+4(x+2)(x+2) = x^2 + 4x + 4, not x2+4x^2 + 4. Emphasize that sum of squares has no middle term to cancel.

Only looking for numerical coefficients as perfect squares

Remediation: Show examples like x4=(x2)2x^4 = (x^2)^2 and 16y6=(4y3)216y^6 = (4y^3)^2 to demonstrate that variable expressions can also be perfect squares.

Differentiation Ideas

For Struggling Students:

  • • Start with numerical examples only: 36−25=(6+5)(6−5)=11×1=1136 - 25 = (6+5)(6-5) = 11 \times 1 = 11
  • • Use area model drawings to visualize the factorization
  • • Provide a list of perfect squares for reference

For On-Level Students:

  • • Factor expressions with coefficients like 4x2−494x^2 - 49
  • • Apply to mental math problems
  • • Identify whether expressions can or cannot be factored

For Advanced Students:

  • • Factor expressions requiring multiple applications like x8−1x^8 - 1
  • • Explore the pattern (a2−b2)(a2+b2)=a4−b4(a^2 - b^2)(a^2 + b^2) = a^4 - b^4
  • • Connect to the complex number factorization of a2+b2a^2 + b^2
Standards Alignment
  • HSA.SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)

    Use the structure of an expression to identify ways to rewrite it

  • HSA.SSE.B.3 (CCSS.MATH.CONTENT.HSA.SSE.B.3)

    Choose and produce an equivalent form of an expression to reveal and explain properties

  • 8.EE.A.2 (CCSS.MATH.CONTENT.8.EE.A.2)

    Use square root and cube root symbols to represent solutions to equations

Lesson Resources
  • visualArea Model Visualization

    Students see how (a+b)(a−b)(a+b)(a-b) creates the difference of two square areas

  • activityMental Math Challenge

    Practice using difference of squares for quick multiplication

  • worksheetFactor or Not?

    Identify which expressions can be factored as difference of squares

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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.

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

Can I factor x2+9x^2 + 9?

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.

Why does the middle term disappear?

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.

How do I recognize a perfect square?

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

More in This Topic

Introduction to Factoring

Intermediate25 min

Factoring Out the GCF

Intermediate25 min

Factoring by Grouping

Intermediate25 min

Factoring Trinomials (a=1)

Intermediate25 min