AQA Level 2 Further Maths · Level 2 Certificate · Chapter 2: Algebra II

Factorising, rearranging formulae, algebraic fractions and completing the square

About 90 minutesNo calculator103 marks

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Date

Do Now

These bring back work from chapter 1 and from GCSE Higher that this chapter uses.

A1
Needed today
Factorise $x^{2} - 11 x y + 28 y^{2}$ .
[2 marks]
Answer

A2
From chapter 1
Expand and simplify $(x - 1) (x + 2) (x + 4)$ .
[2 marks]
Answer
A3
From chapter 1
Work out $(5 + 3) (5 - 3)$ .
[2 marks]
Answer
A4
GCSE Higher
Make $x$ the subject of $5 (x - y) = 2 y + 7$ .
[2 marks]
Answer
A5
GCSE Higher
Solve $x^{2} - 6 x = 16$ .
[2 marks]
Answer

Examples, then your turn

Level 2

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

Example 1

Make $x$ the subject of $y = \frac{3 x - 1}{x + 2}$ .

Hint Multiply both sides by

x + 2

. Then collect every

x

term on one side and take

x

out as a common factor.

B1
Make $a$ the subject of $k = \frac{5 a + 2}{4 - a}$ .
[3 marks]
Answer

Example 2

Solve $\frac{2}{x - 1} + \frac{3}{x + 5} = 1$ .

Hint Multiply every term by

(x - 1) (x + 5)

. You get a quadratic: put everything on one side.

B2
Solve $\frac{3}{x - 2} + \frac{6}{x + 4} = 1$ .
[4 marks]
Answer

Example 3

Write $5 + 12 x - 3 x^{2}$ in the form $a - b (x - c)^{2}$ .

Hence write down the greatest value of $5 + 12 x - 3 x^{2}$ .

Hint Take out

- 3

from the

x

terms:

5 - 3 (x^{2} - 4 x)

. Complete the square inside the bracket.

B3
Write $1 + 12 x - 2 x^{2}$ in the form $a - b (x - c)^{2}$ .
Hence write down the greatest value of $1 + 12 x - 2 x^{2}$ .
[4 marks]
Answer

Example 4

Factorise fully $x^{4} - 37 x^{2} + 36$ .

Hint Treat

x^{2}

as one letter:

x^{4} - 37 x^{2} + 36

is like

y^{2} - 37 y + 36

. Then look for the difference of two squares.

B4
Factorise fully $2 x^{4} - 10 x^{2} + 8$ .
[3 marks]
Answer

Practice

Level 2

Each question asks for something different. Show your working.

C1
Factorise $4 x y - 6 x + 10 y - 15$ .
[2 marks]
Answer

C2
Factorise $12 x^{2} - x - 20$ .
[2 marks]
Answer
C3
Factorise fully $3 x^{3} y - 27 x y^{3}$ .
[2 marks]
Answer
C4
Use factorising to work out $63. 5^{2} - 36. 5^{2}$ without a calculator.
[2 marks]
Answer
C5
Make $x$ the subject of $y = 4 + \frac{3}{x - 2}$ .
[3 marks]
Answer
C6
Simplify fully $\frac{5 x ^{2} + 13 x - 6}{25 x ^{2} - 4}$ .
[3 marks]
Answer
C7
Write $\frac{5}{x + 3} - \frac{2}{x - 1}$ as a single fraction in its simplest form.
[3 marks]
Answer
C8
Solve $\frac{x + 1}{x - 2} = \frac{x + 4}{x - 1}$ .
[3 marks]
Answer
C9
This question is about $x^{2} - 10 x + 22$ .
(a)
Write $x^{2} - 10 x + 22$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer
(b)
Hence solve $x^{2} - 10 x + 22 = 0$ . Give your answers in surd form.
[2 marks]
Answer
Total for C9: 4 marks

Exam-style questions

AQA exam style

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

D1
Sam simplifies $\frac{x ^{2} + 6 x}{x ^{2} - 36}$ . He writes " $\frac{x ^{2} + 6 x}{x ^{2} - 36} = \frac{6 x}{- 36} = - \frac{x}{6}$ ".
Sam is wrong. Explain why, and simplify $\frac{x ^{2} + 6 x}{x ^{2} - 36}$ fully.
[3 marks]
Answer

D2
Ali drives $120$ km at an average speed of $x$ km/h. He drives back the same $120$ km at an average speed $20$ km/h faster, and the journey back takes $1$ hour less.
(a)
Show that $x^{2} + 20 x - 2400 = 0$ .
[3 marks]
(b)
Solve $x^{2} + 20 x - 2400 = 0$ to work out Ali's average speed on the way out.
[2 marks]
Answer
(c)
Work out the time Ali's journey back takes.
[1 mark]
Answer
Total for D2: 6 marks
D3
$y = 2 x^{2} - 12 x + 23$
(a)
Write $2 x^{2} - 12 x + 23$ in the form $a (x + b)^{2} + c$ .
[3 marks]
Answer
(b)
Hence write down the coordinates of the turning point of the graph of $y = 2 x^{2} - 12 x + 23$ .
[1 mark]
Answer
(c)
Hence solve $2 x^{2} - 12 x + 23 = 13$ .
[2 marks]
Answer
Total for D3: 6 marks
D4
$a = \frac{b + 2 c}{b - c}$
(a)
Make $b$ the subject of the formula.
[3 marks]
Answer
(b)
Hence work out the value of $b$ when $a = 4$ and $c = 3$ .
[1 mark]
Answer
Total for D4: 4 marks
D5
This question is about $\frac{3}{x - 4} - \frac{24}{x ^{2} - 16}$ .
(a)
Show that $\frac{3}{x - 4} - \frac{24}{x ^{2} - 16} = \frac{3}{x + 4}$ .
[3 marks]
(b)
Hence solve $\frac{3}{x - 4} - \frac{24}{x ^{2} - 16} = \frac{1}{2}$ .
[1 mark]
Answer
Total for D5: 4 marks
D6
$n$ is an integer.
Show that $(2 n + 5)^{2} - (2 n + 1)^{2}$ is a multiple of $8$ .
[3 marks]
D7
This question is about $x^{3} + 2 x^{2} - 9 x - 18$ .
(a)
Factorise fully $x^{3} + 2 x^{2} - 9 x - 18$ .
[3 marks]
Answer
(b)
Hence solve $x^{3} + 2 x^{2} - 9 x - 18 = 0$ .
[1 mark]
Answer
Total for D7: 4 marks
D8
Solve $2 x^{2} + 6 x - 3 = 0$ by completing the square.
Give your answers in the form $\frac{a \pm b}{c}$ , where $a$ , $b$ and $c$ are integers.
[4 marks]
Answer
D9
$16 x^{2} - 40 x + k \equiv (p x + q)^{2} + 3$ , where $p > 0$ .
Work out the values of $p$ , $q$ and $k$ .
[3 marks]
Answer $p = \dots\dots q = \dots\dots k = \dots\dots$
D10
A rectangular lawn is $8$ m long and $5$ m wide. A path of width $x$ m goes all the way round the lawn. The area of the path is $30$ m².
Not drawn accurately
(a)
Show that $2 x^{2} + 13 x - 15 = 0$ .
[2 marks]
(b)
Work out the width of the path.
[2 marks]
Answer
Total for D10: 4 marks

Extension

Stretch

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
$x$ and $y$ are positive whole numbers and $x^{2} - y^{2} = 15$ .
Work out every pair of values of $x$ and $y$ .
[3 marks]
Hint 1 · What to try
Factorise the left-hand side.
Hint 2 · The first line
$(x - y) (x + y) = 15$ , and $x + y$ is the bigger factor. The factor pairs of $15$ are $1 \times 15$ and $3 \times 5$ .
Hint 3 · The full method
$x - y = 1$ , $x + y = 15$ gives $x = 8$ , $y = 7$ . $x - y = 3$ , $x + y = 5$ gives $x = 4$ , $y = 1$ .
Answer

E2
$k$ is a constant. The expression $x^{2} + 2 k x + 9$ is positive for every value of $x$ .
Work out the range of possible values of $k$ .
[3 marks]
Hint 1 · What to try
Complete the square, keeping $k$ as a letter.
Hint 2 · The first line
$x^{2} + 2 k x + 9 = (x + k)^{2} + 9 - k^{2}$ . The smallest value of $(x + k)^{2}$ is $0$ .
Hint 3 · The full method
The expression is always positive when $9 - k^{2} > 0$ , so $k^{2} < 9$ .
Answer
E3
Simplify fully $\frac{16 x ^{4} - 1}{8 x ^{3} + 4 x ^{2} + 2 x + 1}$ .
[4 marks]
Hint 1 · What to try
Factorise the top as a difference of two squares, and the bottom by grouping the terms in pairs.
Hint 2 · The first line
$16 x^{4} - 1 = (4 x^{2} - 1) (4 x^{2} + 1)$ and $8 x^{3} + 4 x^{2} + 2 x + 1 = 4 x^{2} (2 x + 1) + (2 x + 1)$ .
Hint 3 · The full method
$\frac{( 2 x - 1 ) ( 2 x + 1 ) ( 4 x ^{2} + 1 )}{( 2 x + 1 ) ( 4 x ^{2} + 1 )}$ . Cancel the common factors.
Answer
E4
$a > 1$ and $a + \frac{1}{a} = 3$ .
Work out the value of $a^{2} + \frac{1}{a ^{2}}$ . Hence work out the exact value of $a - \frac{1}{a}$ .
[4 marks]
Hint 1 · What to try
Square both sides of $a + \frac{1}{a} = 3$ .
Hint 2 · The first line
$(a + \frac{1}{a})^{2} = a^{2} + 2 + \frac{1}{a ^{2}}$ , so $a^{2} + \frac{1}{a ^{2}} = 7$ .
Hint 3 · The full method
$(a - \frac{1}{a})^{2} = a^{2} - 2 + \frac{1}{a ^{2}} = 5$ . When $a > 1$ , $a - \frac{1}{a}$ is positive.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can factorise quadratics, quartics in x², four terms by grouping and the difference of two squares.
I can change the subject of a formula, including when the subject appears twice.
I can simplify, add, subtract and solve equations with algebraic fractions.
I can complete the square and use it to solve equations and find greatest and least values.

Answers and mark scheme

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