Difference of Two Squares

The difference of two squares is one of the most elegant patterns in algebra. It lets you factorise expressions instantly and even perform tricky mental arithmetic. This identity appears frequently on both Foundation and Higher papers across AQA, Edexcel, and OCR, so recognising it quickly is a real time-saver. On this page you will learn what the pattern is, how to apply it, and how to use it for clever numerical calculations.

What Is the Difference of Two Squares?

The difference of two squares (often abbreviated DOTS) is a factorising rule that applies whenever you subtract one perfect square from another. The word "difference" means subtraction, and "two squares" means both terms are perfect squares.

The identity states:

This works because when you expand (a + b)(a − b), the middle terms cancel: a² − ab + ab − b² = a² − b².

Key Formulas

To use this pattern, both terms must be perfect squares and they must be separated by a minus sign. If there is a plus sign, this identity does not apply.

Step-by-Step Method

1. Check the expression has exactly two terms separated by a minus sign.
2. Confirm each term is a perfect square. For numbers, check they are 1, 4, 9, 16, 25, 36, … For variables, check the power is even (x², x⁴, y⁶, etc.).
3. Take the square root of each term.
4. Write the answer as (√first + √second)(√first − √second).
5. Expand to verify your factorisation is correct.

Worked Example 1 — Foundation Level

Question: Factorise x² − 49.

Working:

• Both terms are perfect squares: x² = (x)² and 49 = (7)².
• The terms are separated by a minus sign.
• Square roots: x and 7.
• Write as two brackets: (x + 7)(x − 7).
• Check: x² − 7x + 7x − 49 = x² − 49 ✓

Answer: (x + 7)(x − 7)

Worked Example 2 — Higher Level

Question: Factorise 4x² − 25y².

Working:

• 4x² = (2x)² and 25y² = (5y)². Both are perfect squares.
• Separated by a minus sign — DOTS applies.
• Square roots: 2x and 5y.
• Write as (2x + 5y)(2x − 5y).
• Check: 4x² − 10xy + 10xy − 25y² = 4x² − 25y² ✓

Answer: (2x + 5y)(2x − 5y)

Worked Example 3 — Exam Style

Question: Without a calculator, work out 99² − 1².

Working:

Using DOTS: a² − b² = (a + b)(a − b).

99² − 1² = (99 + 1)(99 − 1) = 100 × 98 = 9800.

You can verify: 99² = 9801 and 1² = 1, so 9801 − 1 = 9800 ✓

Answer: 9800

Common Mistakes

• Using DOTS when there is a plus sign. The expression a² + b² does not factorise using this method. DOTS only works with subtraction.
• Forgetting to take out a common factor first. For example, 2x² − 18 should first become 2(x² − 9), then 2(x + 3)(x − 3). Missing the 2 means you have not factorised fully.
• Not recognising disguised squares. Expressions like 16x⁴ − 1 use DOTS because 16x⁴ = (4x²)² and 1 = (1)², giving (4x² + 1)(4x² − 1).

Exam Tips

• When a question says "factorise fully," always check for a common factor before applying DOTS.
• DOTS can appear inside larger problems — for instance, simplifying algebraic fractions or solving equations.
• For numerical questions like 47² − 3², use DOTS to avoid long multiplication: (47 + 3)(47 − 3) = 50 × 44 = 2200.
• Look out for expressions such as (x + 1)² − (x − 1)² — treat each bracket as your a and b.

Practice Questions

Q1 (Foundation): Factorise x² − 64.

Q2 (Foundation): Work out 50² − 48² without a calculator.

Q3 (Higher): Factorise fully 3x² − 75.

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Related Topics

• Factorising Expressions
• Factorising Quadratics
• Expanding Double Brackets

Summary

• The difference of two squares states a² − b² = (a + b)(a − b).
• Both terms must be perfect squares separated by a minus sign.
• Always check for a common factor before applying DOTS.
• The pattern can be used for quick mental arithmetic with large numbers.
• DOTS is tested on both Foundation and Higher papers and often appears within multi-step questions.