Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 11: Special Factorizations

Lesson 2: Difference of Squares

In this Grade 4 AMC math lesson from AoPS: Introduction to Algebra, students learn the difference of squares factorization — the identity a² − b² = (a − b)(a + b) — and practice applying it to expressions like 4t² − 121 and z⁴ − 1. The lesson extends this technique to solving Diophantine equations by factoring expressions such as (m − n)(m + n) = 105 to find all integer solution pairs. Part of Chapter 11: Special Factorizations, this lesson builds algebraic reasoning skills essential for AMC 8 and AMC 10 competition preparation.

Section 1

Difference of Squares Pattern

Property

Difference of Two Squares.

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

Section 2

Consecutive Squares and Difference Patterns

Property

The difference between consecutive squares follows the pattern:

(n+1)^2 - n^2 = 2n + 1

This means the difference between any two consecutive perfect squares is always an odd number.

Section 3

Difference of Squares Factorization

Property

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

Examples

Section 4

Solving Diophantine Equations Using Difference of Squares

Property

When a Diophantine equation can be written as $x^2 - y^2 = n$ , factor it as $(x-y)(x+y) = n$ and find integer solutions by examining all factor pairs of $n$ .

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Difference of Squares Pattern

Property

Difference of Two Squares.

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

Section 2

Consecutive Squares and Difference Patterns

Property

The difference between consecutive squares follows the pattern:

(n+1)^2 - n^2 = 2n + 1

This means the difference between any two consecutive perfect squares is always an odd number.

Section 3

Difference of Squares Factorization

Property

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

Examples

Section 4

Solving Diophantine Equations Using Difference of Squares

Property

When a Diophantine equation can be written as $x^2 - y^2 = n$ , factor it as $(x-y)(x+y) = n$ and find integer solutions by examining all factor pairs of $n$ .

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Difference of Squares Pattern

Property

Difference of Two Squares.

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

Section 2

Consecutive Squares and Difference Patterns

Property

The difference between consecutive squares follows the pattern:

(n+1)^2 - n^2 = 2n + 1

This means the difference between any two consecutive perfect squares is always an odd number.

Section 3

Difference of Squares Factorization

Property

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

Examples

Section 4

Solving Diophantine Equations Using Difference of Squares

Property

When a Diophantine equation can be written as $x^2 - y^2 = n$ , factor it as $(x-y)(x+y) = n$ and find integer solutions by examining all factor pairs of $n$ .