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Unit F3 Section 2
Expanding Brackets

Equations or formulae may contain brackets, for example

P = 8(x − 7)   or   y = (2x − 3)(x − 8)

Removing brackets from these types of expressions is known as expanding.

When evaluating or manipulating expressions it is sometimes important to be able to use the original equation, whereas at other times the expanded version might be helpful.

The reverse of expanding, the process of factorisation, is even more important and becomes easier once you have gained confidence in expanding brackets.

Worked Examples

1

Expand 2x(5x − 8).

2x(5x − 8) = 2x × 5x − 2x × 8
= 10x2 − 16x
2

Expand −4(x − 6).

−4(x − 6) = −4 × x + (−4) × (−6)
= −4x + 24

Sometimes situations will arise where a bracket has to be multiplied by another bracket, as in the next example.
3

Expand (x + 2)(x + 5).

The first bracket (x + 2) is split so that each if its terms, (x) and (+2), can multiply the other bracket.

(x + 2)(x + 5) = x(x + 5) + 2(x + 5)
= x2 + 5x + 2x + 10
= x2 + 7x + 10   (collecting like terms)
4

Expand (4x − 3)(2x − 7).

Splitting the first bracket and multiplying the other bracket by each of its terms gives,

(4x − 3)(2x − 7) = 4x(2x − 7) − 3(2x − 7)
= 8x2 − 28x − 6x + 21
= 8x2 − 34x + 21
5

Expand (x + 6)2.

First note that

(x + 6)2 = (x + 6)(x + 6)

Then the brackets can be expanded.

(x + 6)(x + 6) = x(x + 6) + 6(x + 6)
= x2 + 6x + 6x + 36
= x2 + 12x + 36
6

Expand (3x − 2n)(4x + 5n).

(3x − 2n)(4x + 5n) = 3x(4x + 5n) − 2n(4x + 5n)
= 12x2 + 15xn − 8nx − 10n2
= 12x2 + 7xn − 10n2

Note that xn is the same as nx.

Exercises

Simplify each expression below, expanding the brackets as necessary.

(a)
x(x + 1)
(b)
−x(3x − 2)
(c)
3a(2a − 5)
(d)
x2(1 − x)
(e)
4x(x − 7)
(f)
5 + 3(x + 2)
(g)
4x + x(x − 6)
(h)
x(1 + x2)
(i)
5 − 4(x2 + 2)
(j)
x2 − x(x + 5)
(k)
x2 − 4x(5 − x)
(l)
x2 + 4(x2 − 2)
(m)
x3 + x2(5 − 2x)
(n)
4(x + 4) − x
(o)
5x − x(3x − 2)

Expand and simplify each of the following.

(a)
(x + 1)(x + 6)
(b)
(x + 4)(x + 7)
(c)
(x − 2)(x + 8)
(d)
(x − 1)(x + 4)
(e)
(x − 7)(x − 1)
(f)
(2x + 1)(3x − 1)
(g)
(4x + 3)(2x + 7)
(h)
(5x − 2)(2x + 3)
(i)
(6x − 1)(4x + 3)
(j)
(3x − 1)(4x − 3)
(k)
(5x − 6)(3x − 8)
(l)
(3a − 7)(5a + 2)
(m)
(4a − 5)(3a − 5)
(n)
(6n + 1)(4n − 2)
(o)
(4x + 3)(5x − 8)

Expand and simplify each of the following.

(a)
(x + 1)2
(b)
(x − 1)2
(c)
(x + 8)2
(d)
(2x + 1)2
(e)
(3x + 4)2
(f)
(6x − 1)2
(g)
(2x − 5)2
(h)
(6x − 7)2
(i)
(5x + 9)2
(a)

Expand and simplify

(i)
(x + 1)(x − 1)
(ii)
(x + 4)(x − 4)
(iii)
(x + 6)(x − 6)
(b)

Without expanding the brackets, write down what you would expect to get if you expanded

(x + 5)(x − 5)

(c)

Expand and simplify

(a + b)(a − b)

(d)

What would you expect to obtain if you expanded

(5x + 2)(5x − 2)?

Find the area of each rectangle.

(a)
(b)
(c)