Sheet № 136 · Foundation + Higher · AQA · Edexcel · OCR
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
§Key definitions
Question:
Factorise x² − 49.
Answer:
(x + 7)(x − 7)
Q1 (Foundation):
Factorise x² − 64.
Q2 (Foundation):
Work out 50² − 48² without a calculator.
Q3 (Higher):
Factorise fully 3x² − 75.
§Formulas to memorise
a² − b² = (a + b)(a − b)
Take the square root of each term. — 4. Write the answer as (√first + √second)(√first − √second).
Expand to verify — your factorisation is correct.
Worked example
Factorise x² − 49.
Working:
⚠ 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.