Algebra
Grade 6-7Difference of Two Squares
The difference of two squares is a special factorising pattern that applies whenever one perfect square is subtracted from another, and recognising it turns what looks like a hard factorising problem into a very quick one. This lesson covers factorising a simple difference of two squares, factorising when both terms have coefficients, factorising with two variables, and using the technique to calculate expressions like 51² − 49² mentally.
What you need to know
- a squared minus b squared always factorises to (a + b)(a - b).
- It only works for a subtraction of two square terms with no middle term.
- x squared minus 9 is (x + 3)(x - 3); 4x squared minus 25 is (2x + 5)(2x - 5).
- Spotting this is often much faster than the standard method.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Factorising a simple difference of two squares
Any expression of the form x² − a², where a perfect square is subtracted from another, factorises as (x + a)(x − a).
x² − 16 is a difference of two squares, since 16 = 4², so it factorises as (x + 4)(x − 4).
Factorising when both terms have coefficients
Check whether each term is itself a perfect square, including any coefficient, by finding its square root, then apply the same pattern using these square roots.
4x² − 9 is a difference of two squares, since 4x² = (2x)² and 9 = 3², so it factorises as (2x + 3)(2x − 3).
Factorising with two variables
The same pattern applies when both terms involve different letters, as long as each term is a perfect square.
x² − y² factorises as (x + y)(x − y), since both x² and y² are perfect squares.
Using the difference of two squares to calculate mentally
A calculation like 51² − 49² can be treated as a difference of two squares and factorised into (51 + 49)(51 − 49), which is far quicker to work out than squaring each number separately.
51² − 49² = (51 + 49)(51 − 49) = 100 × 2 = 200.
Worked examples
Three exam-style questions, fully solved
Factorise x² − 16.
Easy- 1.Recognise this as a difference of two squares, since 16 = 4²
- 2.Write it as (x + 4)(x − 4)
Answer: (x + 4)(x − 4)
Factorise 4x² − 9.
Medium- 1.Find the square root of each term: 4x² = (2x)² and 9 = 3²
- 2.Write it as (2x + 3)(2x − 3)
Answer: (2x + 3)(2x − 3)
Use the difference of two squares to work out 51² − 49².
Hard- 1.Write it as a difference of two squares: (51 + 49)(51 − 49)
- 2.Work out each bracket: 51 + 49 = 100, and 51 − 49 = 2
- 3.Multiply the results: 100 × 2
Answer: 200
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Factorise x² − 16.
Factorise x² − 49.
Factorise 4x² − 25.
Factorise 9x² − 1.
Use the difference of two squares to work out 51² − 49² without a calculator.
Factorise x⁴ − 16 as a product of two brackets.
Avoid these
The mistakes students make most often
Trying to apply the difference of two squares to an addition, such as x² + 16, when the pattern only applies to a subtraction of two squares.
Not recognising that a term like 4x² or 25 is itself a perfect square, and instead treating the expression as if it could not be factorised this way.
Writing only one bracket, such as (x + 4), instead of both factors, (x + 4)(x − 4).
Forgetting to check for a common factor before applying the difference of two squares, leaving the expression only partly factorised.
When using the technique to calculate mentally, such as 51² − 49², making an arithmetic slip when adding or subtracting the two original numbers.
FAQ
Questions students and parents ask
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