GCSE Maths · Higher · Chapter 8: Algebraic manipulation

Simplifying, expanding, factorising and rearranging

Grades 4–9About 80 minutesNo calculator92 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Work out the value of $3 a - 2 b$ when $a = 4$ and $b = - 5$ .
[2 marks]
Answer

A2
Simplify $5 p + 3 q - 2 p + 4 q$ .
[1 mark]
Answer
A3
Expand $4 (2 x - 3)$ .
[1 mark]
Answer
A4
Factorise $6 y + 15$ .
[1 mark]
Answer
A5
Expand and simplify $(x + 8) (x + 7)$ .
[2 marks]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Factorise fully $12 x^{2} y - 18 x y^{2}$ .

Hint Take out the highest common factor of the numbers, then the lowest power of each letter that is in every term.

B1
Factorise fully $15 a^{3} b + 20 a^{2} b^{2}$ .
[2 marks]
Answer

Example 2

Expand and simplify $(2 x + 5) (x - 3)$ .

Hint Multiply each term in the first bracket by each term in the second. There are four products, then collect the two

x

terms.

B2
Expand and simplify $(3 x - 2) (2 x + 5)$ .
[2 marks]
Answer

Example 3

Factorise $2 x^{2} + 7 x + 3$ .

Hint Look for two numbers that multiply to

2 \times 3 = 6

and add to 7. Split the middle term with them.

B3
Factorise $3 x^{2} + 11 x - 4$ .
[2 marks]
Answer

Example 4

Make $w$ the subject of $P = 2 (l + w)$ .

Hint Undo the operations in the reverse order: divide by 2 first, then subtract

l

B4
Make $x$ the subject of $y = \frac{3 x - 5}{2}$ .
[2 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Work out the value of $\frac{a ^{2} - 3 b}{2 c}$ when $a = - 4$ , $b = - 2$ and $c = 5$ .
[2 marks]
Answer

C2
Simplify each expression.
(a)
$3 x^{2} y \times 4 x y^{3}$
[1 mark]
Answer
(b)
$\frac{20 a ^{5} b ^{2}}{4 a ^{2} b}$
[2 marks]
Answer
Total for C2: 3 marks
C3
Expand and simplify $3 (2 x - 5) - 2 (x - 4)$ .
[2 marks]
Answer
C4
Expand and simplify $(x - 6)^{2}$ .
[2 marks]
Answer
C5
Expand and simplify $(x + 2) (x - 3) (x + 4)$ .
[3 marks]
Answer
C6
Factorise each expression.
(a)
$x^{2} + x - 42$
[2 marks]
Answer
(b)
$x^{2} - 169$
[1 mark]
Answer
(c)
$4 x^{2} - 25 y^{2}$
[2 marks]
Answer
Total for C6: 5 marks
C7
Factorise $6 x^{2} - 7 x - 5$ .
[3 marks]
Answer
C8
Make $r$ the subject of $A = π r^{2} + 5$ .
[3 marks]
Answer
C9
A square of side $x$ cm is cut from the corner of a rectangle that measures $(x + 6)$ cm by $(x + 2)$ cm. The shaded part is left.
Write an expression for the shaded area. Simplify your answer.
[3 marks]
Not drawn accurately
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Show that $(n + 3)^{2} - (n - 3)^{2}$ is a multiple of 12 for every whole number $n$ .
[3 marks]

D2
Ella factorises $3 x^{2} - 27$ . She writes $3 (x^{2} - 9)$ and says, "That is factorised fully."
Explain why Ella is wrong. Then factorise $3 x^{2} - 27$ fully.
[3 marks]
Answer
D3
A cuboid has edges $(x + 6)$ cm, $(x + 2)$ cm and $(x - 3)$ cm.
Not drawn accurately
(a)
Show that the volume of the cuboid is $x^{3} + 5 x^{2} - 12 x - 36$ cm³.
[3 marks]
(b)
Work out the total surface area of the cuboid. Give your answer in the form $a x^{2} + b x + c$ .
[3 marks]
Answer
Total for D3: 6 marks
D4
Make $a$ the subject of $\frac{a + 2}{a - 3} = b$ .
[4 marks]
Answer
D5
Factorise fully $3 x^{2} - 12 x - 63$ .
[3 marks]
Answer
D6
Expand and simplify $(2 x - 3)^{2} - (x + 1) (x - 5)$ .
[3 marks]
Answer
D7
$a$ and $b$ are integers. $(2 x + a) (x + b) = 2 x^{2} + 3 x - 20$ .
Find the values of $a$ and $b$ .
[3 marks]
Answer $a = \dots\dots$ , $b = \dots\dots$
D8
The energy, $E$ joules, of a ball is given by $E = \frac{1}{2} m v^{2} + m g h$ .
(a)
Work out the value of $E$ when $m = 4$ , $v = 3$ , $g = 10$ and $h = 2$ .
[2 marks]
Answer
(b)
Make $h$ the subject of the formula.
[3 marks]
Answer
Total for D8: 5 marks
D9
The area of the rectangle is $x^{2} + 7 x - 30$ cm². One side is $(x - 3)$ cm.
Work out an expression for the perimeter of the rectangle. Simplify your answer.
[4 marks]
Not drawn accurately
Answer
D10
Make $x$ the subject of $y = \frac{x + 4}{3}$ .
[3 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Prove that the sum of the squares of three consecutive whole numbers is never a multiple of 3.
[4 marks]
Hint 1 · What to try
Call the numbers $n - 1$ , $n$ and $n + 1$ .
Hint 2 · The first line
$(n - 1)^{2} + n^{2} + (n + 1)^{2} = 3 n^{2} + 2$ .
Hint 3 · The full method
$3 n^{2}$ is a multiple of 3, so $3 n^{2} + 2$ is 2 more than a multiple of 3.

E2
Factorise fully $x^{4} - 81$ .
[3 marks]
Hint 1 · What to try
Write $x^{4}$ as $(x^{2})^{2}$ and 81 as $9^{2}$ .
Hint 2 · The first line
$x^{4} - 81 = (x^{2} + 9) (x^{2} - 9)$ .
Hint 3 · The full method
One of those two brackets is itself a difference of two squares.
Answer
E3
Show that $(x + 1) (x + 2) (x + 3) (x + 4) + 1$ is a perfect square. Write it in the form $(x^{2} + b x + c)^{2}$ .
[4 marks]
Hint 1 · What to try
Pair the outer brackets and the inner brackets.
Hint 2 · The first line
$(x + 1) (x + 4) = x^{2} + 5 x + 4$ and $(x + 2) (x + 3) = x^{2} + 5 x + 6$ .
Hint 3 · The full method
Write $y = x^{2} + 5 x$ . Then the expression is $(y + 4) (y + 6) + 1 = y^{2} + 10 y + 25$ .
Answer
E4
$a x^{2} + 17 x + 12 = (2 x + 3) (c x + 4)$ for every value of $x$ .
Find the value of $a$ .
[3 marks]
Hint 1 · What to try
Expand $(2 x + 3) (c x + 4)$ .
Hint 2 · The first line
$(2 x + 3) (c x + 4) = 2 c x^{2} + (8 + 3 c) x + 12$ .
Hint 3 · The full method
Compare the $x$ terms: $8 + 3 c = 17$ .
Answer $a = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can substitute numbers into an expression, collect like terms and take out a common factor.
I can expand two brackets, a squared bracket and three brackets.
I can factorise a quadratic, including one where the $x^{2}$ term is not 1 and a difference of two squares.
I can change the subject of a formula, including when the subject appears twice.

Answers and mark scheme

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