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9.2 Expanding brackets

9.2 Expanding brackets

To expand brackets, multiply each term inside the brackets Expanding brackets is sometimes
by the term outside the brackets. called multiplying out brackets.

Worked example 9.2
a Expand these expressions. i 3(b + 5) ii a(a − 3)
2
b Expand and simplify this expression. 4(2x + 3x ) − x(6 + x)
a i 3(b + 5) = 3 × b + 3 × 5 Multiply 3 by b then 3 by 5.
= 3b + 15 Simplify 3 × b to 3b and 3 × 5 to 15.
ii a(a − 3) = a × a − a × 3 Multiply a by a then a by 3.
2
2
= a − 3a Simplify a × a to a and a × 3 to 3a.
b 4(2x + 3x ) − x(6 + x) Start by multiplying out both brackets and simplify each term.
2
2
2
= 8x + 12x − 6x − x So, 4 × 2x = 8x, 4 × 3x = 12x , −x × 6 = −6x, −x × x = −x .
2
2
2
2
2
2
= 2x + 11x Collect like terms, so 8x − 6x = 2x and 12x − x = 11x .
2
✦ Exercise 9.2
1 Expand each expression.
a 4(x + 6) b 3(y + 7) c 7(z − 2) d 2(w − 4) e 2(5 + a) f 8(9 + g)
g 5(8 − b) h 6(6 − d) i 3(2a + 8) j 12(3 + 4b) k 5(2c − 1) l 6(3 − 4e)
m 2(2p + 3q) n 4(5c + 4d) o 9(6t − 2s) p 3(2ab + 3c) q 7(6xy − 2z) r 5(2x + y + 4)
2 Expand and simplify each expression.
a 2(x + 3) + 3(x + 4) b 4(y + 5) + 2(2y + 2) c 8(z + 3) + 5(4 + 3z)
d 5(2w + 3) − 6(w + 2) e 6(5 + 4v) − 4(3v + 7) f 3(5a + 3b) − 2(3a − 5b)
3 Expand each expression.
a x(3y + 2) b y(y + 8) c z(2w − 1) d m(m − 4) e n(2n + 5) f n(9 − 8n)
g a(1 − 3b) h c(5 − d) i e(2e + 7f) j g(3h + 7g) k h(2h − 5k) l d(3c − 5e)
m 2x(x + 3y) n 3y(5y + 6) o 4b(6b − 2a) p 6h(1 + 3h) q 5k(6m − 8k) r 2f(2f + g − 3)
4 Expand and simplify each expression.
a x(x + 2) + x(x + 5) b z(2z + 1) + z(4z + 5)
c u(2u + 5) − u(u + 3) d w(6w + 2x) − 2w(2w − 9x) Remember that −2 × −9 = +18
i
5 This is part of Shen’s homework. He has made a mistake in every question.

Question Expand and simplify: 1. 8(x + 5) – 3(2x + 7)
2. a(2b + c) + b(3c – 2a)
3. 2y(y + 5x) + x(3x + 4y)
A
A
A
A 1. 8(x + 5) – 3(2x + 7) = 8x + 40 – 6x + 21 = 2x + 61
A
Answer
A
A
2. a(2b + c) + b(3c – 2a) = 2ab + ac + 3bc – 2ab = ac + 3bc = 3abc 2
2
2
2
2
3. 2y(y + 5x) + x(3x + 4y) = 2y + 10xy + 9x + 4xy = 9x + 2y + 14xy
a Explain what he has done wrong.
b Work out the correct answers.
98 9 Simplifying expressions and solving equations