Difference of Two Squares

Statement

The difference of two squares is a special factorisation rule. It states that the difference between the squares of two numbers can always be expressed as a product:

\[ a^2 - b^2 = (a - b)(a + b) \]

This identity is extremely useful for simplifying algebraic expressions and solving equations.

Why it’s true

• Start with the expansion of \((a - b)(a + b)\).
• By distributive law: \(a \cdot a + a \cdot b - b \cdot a - b \cdot b\).
• This simplifies to \(a^2 - b^2\), since the middle terms cancel out.

Recipe (how to use it)

1. Look for an expression in the form \(x^2 - y^2\).
2. Factorise it as \((x - y)(x + y)\).
3. Check by expanding back to ensure accuracy.

Spotting it

You can use this rule when an expression has exactly two terms, both are perfect squares, and they are subtracted.

Common pairings

• Quadratic factorisation problems.
• Simplifying algebraic fractions.
• Equation solving, especially when rearranging quadratics.

Mini examples

1. Given: \(x^2 - 9\). Factorise: \((x - 3)(x + 3)\).
2. Given: \(16y^2 - 25\). Factorise: \((4y - 5)(4y + 5)\).

Pitfalls

• Forgetting that it only works for subtraction, not addition. \(a^2 + b^2\) cannot be factorised this way.
• Missing square recognition: numbers like 49 or 121 are perfect squares (7² and 11²).
• Incorrectly handling coefficients inside the square term (e.g., \((3x)^2 = 9x^2\), not \(3x^2\)).

Exam strategy

• First, check if both terms are squares and connected by subtraction.
• Factorise using the rule quickly to save time.
• Always expand your answer to double-check correctness.

Summary

The difference of two squares is a quick and powerful factorisation method: \(a^2 - b^2 = (a - b)(a + b)\). Recognising this pattern helps simplify problems, especially in quadratic factorisation and algebraic manipulation.