x^2 − 100 = (x + □)(x − □) Fill in each blank

10, 10. 100 = 10^2, so x^2 − 100 = x^2 − 10^2 = (x + 10)(x − 10).

x^2 − 144 can be factorised as (x + 12)(x − 12). True or false?

True. (x + 12)(x − 12) = x^2 − 144. Only two squares with a minus between make a difference of squares; with a plus it can't be factorised this way.

Work it out quickly with the difference of two squares: 87^2 − 73^2 = □

2240. 87^2 − 73^2 = (87 + 73)(87 − 73) = 160 × 14 = 2240.

Factorisation · Ages 13–14

Factorising a difference of two squares

Two squares with a minus between; a common factor first.

How to do it

Method

Two squares with a minus between; a common factor first

9x² − 16 = (3x + 4)(3x − 4)

Another way

Expand to check

(3x + 4)(3x − 4) = 9x² − 16

In full

Read the difference of squares backwards: a² − b² ≡ (a + b)(a − b). Spot two squares with a minus between: 9x² − 16 = (3x)² − 4² = (3x + 4)(3x − 4). With a plus between, x² + 16 can't be factorised this way. If every term shares a factor, take it out first: 2x² − 18 = 2(x² − 9) = 2(x + 3)(x − 3), and keep going until nothing more factorises. Or expand the answer to check: (3x + 4)(3x − 4) = 9x² − 16, back where it started. A quick sum: 57² − 43² = (57 + 43)(57 − 43) = 100 × 14 = 1400.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
x² − 100 = (x + )(x − )
Fill in each blank
Show the answer and working
Answer: 10, 10
100 = 10², so x² − 100 = x² − 10² = (x + 10)(x − 10).
Example 2
x² − 144 can be factorised as (x + 12)(x − 12).
True or false?
Show the answer and working
Answer: True
(x + 12)(x − 12) = x² − 144. Only two squares with a minus between make a difference of squares; with a plus it can't be factorised this way.
Example 3
Work it out quickly with the difference of two squares: 87² − 73² =
Show the answer and working
Answer: 2240
87² − 73² = (87 + 73)(87 − 73) = 160 × 14 = 2240.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Factorise 4x² − 9. What do you get?
- A. (2x + 3)(2x − 3)
- B. (2x − 3)²
- C. (4x + 9)(4x − 9)
- D. (2x + 3)²
Choose the right answer
Answer:
2.x² + 81 can be factorised as (x + 9)(x − 9).
True or false?
Answer:
3.x² − 225 = (x + )(x − )
Fill in each blank
Answer:
4.Work it out quickly with the difference of two squares: 60² − 30² =
Answer:
5.Factorise 9x² − 4y². What do you get?
- A. (3x − 2y)²
- B. (3x + 2y)(3x − 2y)
- C. (3x + 2y)²
- D. (2x + 3y)(2x − 3y)
Choose the right answer
Answer:

All the answers

1. A. (2x + 3)(2x − 3)
2. False
3. 15, 15
4. 2700
5. B. (3x + 2y)(3x − 2y)

Free worksheet: Factorising a difference of two squaresWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake