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This pdf covers the basics of expansions leading to the difference of two squares step by step with examples for better understanding.

1.
Factorising the difference of two squares

The technique of factorising a quadratic expression has been explained on the leaflet Factorising

quadratic expressions. There is a special case of quadratic expression known as the difference of

two squares. This leaflet explains what this means and how such expressions are factorised.

What is meant by the difference of two squares ?

A typical example of a quadratic expression which is the difference of two squares is x2 − 9. Note

that there is no x term and that the number 9 is itself a square number. A square number is one

which has resulted from squaring another number. In this case 9 is the result of squaring 3, (32 = 9),

and so 9 is a square number.

Hence x2 − 9 is the difference of two squares, x2 − 32 .

When we try to factorise x2 − 9 we are looking for two numbers which add to zero (because there

is no term in x), and which multiply to give −9. Two such numbers are −3 and 3 because

−3 + 3 = 0, and − 3 × 3 = −9

x2 − 9 = (x − 3)(x + 3)

It is always the case that x2 − a2 factorises to (x − a)(x + a).

The difference of two squares, x2 − a2 , always factorises to

x2 − a2 = (x − a)(x + a)

Factorise x2 − 25.

Note that x2 − 25 is the difference of two squares because 25 is a square number (25 = 52 ). So we

need to factorise x2 − 52 .

x2 − 52 = (x − 5)(x + 5)

Factorise y 2 − 81.

Note that y 2 − 81 is the difference of two squares because 81 is a square number (81 = 92 ). So we

need to factorise y 2 − 92 .

y 2 − 92 = (y − 9)(y + 9)

www.mathcentre.ac.uk 1 c mathcentre 2009

The technique of factorising a quadratic expression has been explained on the leaflet Factorising

quadratic expressions. There is a special case of quadratic expression known as the difference of

two squares. This leaflet explains what this means and how such expressions are factorised.

What is meant by the difference of two squares ?

A typical example of a quadratic expression which is the difference of two squares is x2 − 9. Note

that there is no x term and that the number 9 is itself a square number. A square number is one

which has resulted from squaring another number. In this case 9 is the result of squaring 3, (32 = 9),

and so 9 is a square number.

Hence x2 − 9 is the difference of two squares, x2 − 32 .

When we try to factorise x2 − 9 we are looking for two numbers which add to zero (because there

is no term in x), and which multiply to give −9. Two such numbers are −3 and 3 because

−3 + 3 = 0, and − 3 × 3 = −9

x2 − 9 = (x − 3)(x + 3)

It is always the case that x2 − a2 factorises to (x − a)(x + a).

The difference of two squares, x2 − a2 , always factorises to

x2 − a2 = (x − a)(x + a)

Factorise x2 − 25.

Note that x2 − 25 is the difference of two squares because 25 is a square number (25 = 52 ). So we

need to factorise x2 − 52 .

x2 − 52 = (x − 5)(x + 5)

Factorise y 2 − 81.

Note that y 2 − 81 is the difference of two squares because 81 is a square number (81 = 92 ). So we

need to factorise y 2 − 92 .

y 2 − 92 = (y − 9)(y + 9)

www.mathcentre.ac.uk 1 c mathcentre 2009

2.
1. Factorise the following.

a) x2 − 16 b) x2 − 36 c) x2 − 1 d) x2 − 121 e) x2 − 49

A different form

A slightly different form occurs if we now include a square number in front of the x2 term:

Suppose we wish to factorise 9x2 − 16.

Note that 9 is a square number, and so the term 9x2 can be written (3x)2 . So we still have a

difference of two squares

(3x)2 − 42

To factorise this we write

9x2 − 16 = (3x − 4)(3x + 4)

Note that when multiplying-out the brackets the x terms cancel out.

2. Factorise the following.

a) 9x2 − 1 b) 16x2 − 9 c) 49x2 − 1 d) 25x2 − 16

a) (x − 4)(x + 4) b) (x − 6)(x + 6) c) (x − 1)(x + 1) d) (x − 11)(x + 11)

1.

e) (x − 7)(x + 7)

2. a) (3x + 1)(3x − 1) bj) (4x + 3)(4x − 3) c) (7x + 1)(7x − 1) d) (5x + 4)(5x − 4)

www.mathcentre.ac.uk 2 c mathcentre 2009

a) x2 − 16 b) x2 − 36 c) x2 − 1 d) x2 − 121 e) x2 − 49

A different form

A slightly different form occurs if we now include a square number in front of the x2 term:

Suppose we wish to factorise 9x2 − 16.

Note that 9 is a square number, and so the term 9x2 can be written (3x)2 . So we still have a

difference of two squares

(3x)2 − 42

To factorise this we write

9x2 − 16 = (3x − 4)(3x + 4)

Note that when multiplying-out the brackets the x terms cancel out.

2. Factorise the following.

a) 9x2 − 1 b) 16x2 − 9 c) 49x2 − 1 d) 25x2 − 16

a) (x − 4)(x + 4) b) (x − 6)(x + 6) c) (x − 1)(x + 1) d) (x − 11)(x + 11)

1.

e) (x − 7)(x + 7)

2. a) (3x + 1)(3x − 1) bj) (4x + 3)(4x − 3) c) (7x + 1)(7x − 1) d) (5x + 4)(5x − 4)

www.mathcentre.ac.uk 2 c mathcentre 2009