Difference of Two Squares

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Difference Of Two Squares

Difference of two squares

a² - b² factorises to (a + b)(a - b)
- This can be shown by expanding the brackets
  - $(a + b) (a - b) = a^{2} - a b + b a - b^{2} = a^{2} - b^{2}$
- The brackets can swap order
  - a² - b² = (a + b)(a - b) = (a - b)(a + b)
  - (but terms inside a bracket cannot swap order)
For example, $x^{2} - 9 = (x + 3) (x - 3)$
- This is the same as $(x - 3) (x + 3)$
- But not the same as $(3 + x) (3 - x)$
  - which expands to $9 - x^{2}$

You may find it used with numbers
- 7² - 3² = (7+3) (7-3) = (10) (4) = 40
You can use a combination of square numbers and squared variables
- 4m² - 9n² = (2m)² - (3n)² = (2m + 3n)(2m - 3n)
You can use other powers which can be written as a difference of two squares
- r⁸ - t⁶ = (r⁴)² - (t³)² = (r⁴+ t³) (r⁴- t³)
You can use the difference of two squares more than once to fully factorise expressions
- a⁴ - b⁴ = (a²)² - (b²)² = (a²+ b²) (a²- b²) = (a²+ b²)(a+ b)(a- b)
You may also need to take out a common factor first
- $2 x^{2} - 18 = 2 (x^{2} - 9)$ giving $2 (x + 3) (x - 3)$
  - The 2 comes out in front

Using the difference of two squares to expand is quicker than expanding double brackets and collecting like terms
Brackets of the form (a + b)(a - b) expand to a² - b²
- For example $(2 x + 3) (2 x - 3)$ expands to $4 x^{2} - 9$

Worked example

(a) Factorise $9 x^{2} - 16$ .

Answer:

Recognise that $9 x^{2}$ and $16$ are both squared terms

Therefore you can factorise using the difference of two squares

Rewrite as a difference of two squared terms

$9 x^{2} - 16 = {(3 x)}^{2} - {(4)}^{2}$

Use the rule $a^{2} - b^{2} = (a + b) (a - b)$

$(3 x + 4) (3 x - 4)$

(b) Factorise $4 r^{2} - t^{4}$ .

Answer:

Recognise that $4 r^{2}$ and $t^{4}$ are both squared terms

Therefore you can factorise using the difference of two squares

Rewrite as a difference of two squared terms

$4 r^{2} - t^{4} = {(2 r)}^{2} - {(t^{2})}^{2}$

Use the rule $a^{2} - b^{2} = (a + b) (a - b)$

$(2 r + t^{2}) (2 r - t^{2})$

Answer:

This does not appear to be in the form $a^{2} - b^{2}$

There is a common factor of 2, so take this factor out

$2 (y^{2} - 25)$

You can now see $y^{2} - 25$ which has the form $y^{2} - 5^{2}$

Use the rule $a^{2} - b^{2} = (a + b) (a - b)$

$y^{2} - 25 = (y + 5) (y - 5)$

Now multiply this answer by 2 (leaving the 2 on the outside)

$2 (y + 5) (y - 5)$