A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 1: Algebraic methods · Lesson 2

Algebraic fractions

112 marks

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Date

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Quick questions to start.

A1
Needed today
Factorise $2 x^{2} + 5 x - 3$ .
[2 marks]
Answer
A2
Needed today
Factorise fully $x^{3} - 9 x$ .
[2 marks]
Answer
A3
Common slip
Jay says $\frac{x + 6}{x + 2} = \frac{6}{2} = 3$ . Is he right? Give a reason.
[2 marks]
A4
Needed today
Write $\frac{2}{x} + \frac{3}{x + 1}$ as a single fraction.
[2 marks]
Answer
A5
Common slip
Simplify $3 (x + 1) - (x - 4)$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Simplify fully $\frac{x ^{2} - 9}{2 x ^{2} + 5 x - 3}$ .

Hint Factorise the top and the bottom first. Only a factor that multiplies the whole top and the whole bottom can be cancelled.

B1
Simplify fully $\frac{3 x ^{2} - 12}{x ^{3} + 2 x ^{2}}$ .
[3 marks]
Answer

Example 2

Simplify fully $\frac{x ^{2} + 3 x}{x ^{2} - 4} \div \frac{x ^{2} - 9}{x + 2}$ .

Hint Dividing by a fraction is the same as multiplying by it turned upside down. Then factorise everything and cancel.

B2
Simplify fully $\frac{2 x ^{2} - x - 1}{x ^{2} - 1} \times \frac{x ^{2} + 2 x + 1}{4 x ^{2} - 1}$ .
[4 marks]
Answer

Example 3

Write $\frac{3}{x + 1} - \frac{x - 5}{x ^{2} - 1}$ as a single fraction in its simplest form.

Hint

x^{2} - 1 = (x - 1) (x + 1)

, so that is the common denominator. Put the subtracted top in a bracket.

B3
Write $\frac{2}{x - 3} - \frac{x + 9}{x ^{2} - 9}$ as a single fraction in its simplest form.
[4 marks]
Answer

Example 4

Simplify $\frac{1 - \frac{4}{x ^{2}}}{1 + \frac{2}{x}}$ .

Hint Multiply the top and the bottom of the big fraction by

x^{2}

. This clears the small fractions.

B4
Simplify $\frac{\frac{1}{x} - \frac{1}{3}}{9 - x ^{2}}$ .
[4 marks]
Answer

Example 5

Solve $\frac{6}{x} - \frac{3}{x + 2} = 1$ .

Hint Multiply every term by

x (x + 2)

. Then check that no answer makes a denominator zero.

B5
Solve $\frac{4}{x - 1} + \frac{5}{x + 2} = 3$ .
[5 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Practice

A-Level

C1
Ravi simplifies $\frac{x ^{2} + 5 x}{x ^{2} + 3 x - 10}$ by cancelling the $x^{2}$ terms, and gets $\frac{5 x}{3 x - 10}$ . Show that Ravi is wrong, and find the correct answer.
[4 marks]
Answer

C2
Mia writes $\frac{2 x}{x + 3} - \frac{x - 1}{x + 3} = \frac{2 x - x - 1}{x + 3} = \frac{x - 1}{x + 3}$ . Explain her mistake. Then write $\frac{3 x + 1}{x ^{2} - 4} - \frac{x - 1}{x ^{2} - 2 x}$ as a single fraction in its simplest form.
[5 marks]
Answer
C3
Simplify fully $\frac{x ^{2} - 16}{x ^{2} + x - 12} \div \frac{x ^{2} - 4 x}{x ^{2} - 9}$ .
[4 marks]
Answer
C4
$f (x) = \frac{x ^{2} - 2 x - 8}{x ^{2} - 16}$ . Simplify $f (x)$ , and hence show that the equation $f (x) = \frac{3}{4}$ has no solution.
[5 marks]
Answer
C5
Write $\frac{2 x ^{2} + x - 6}{x ^{2} - 4} + \frac{x}{x + 2}$ as a single fraction in its simplest form.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Solve $\frac{3 x}{x + 2} - \frac{x + 14}{x ^{2} - x - 6} = 2$ , giving your answers in the form $a \pm b 2$ , where $a$ and $b$ are integers.
[7 marks]
Answer $x = \dots\dots$

D2
$f (x) = \frac{x + 1}{x - 2} - \frac{2}{x ^{2} - 4}$ , $x \neq = \pm 2$ .
(a)
Show that $f (x) = \frac{x ( x + 3 )}{x ^{2} - 4}$ .
[3 marks]
(b)
Solve $f (x) = 1$ .
[3 marks]
Answer
(c)
Solve $f (x) = 2$ , giving your answers in surd form.
[3 marks]
Answer
Total for D2: 9 marks
D3
Leo says: " $\frac{3}{x - 2} - \frac{x + 4}{x ^{2} - 4} = \frac{3 - x - 4}{x ^{2} - 4} = \frac{- x - 1}{x ^{2} - 4}$ ."
Leo writes $\frac{3}{x - 2} - \frac{x + 4}{x ^{2} - 4}$ over the denominator $x^{2} - 4$ .
(a)
By substituting $x = 0$ , show that Leo is wrong.
[2 marks]
(b)
Write $\frac{3}{x - 2} - \frac{x + 4}{x ^{2} - 4}$ correctly as a single fraction in its simplest form.
[3 marks]
Answer
(c)
Solve $\frac{3}{x - 2} - \frac{x + 4}{x ^{2} - 4} = \frac{1}{2}$ , giving your answers in the form $a \pm b 3$ .
[3 marks]
Answer
Total for D3: 8 marks
D4
The curve $C$ has equation $y = \frac{x ^{2} + x - 6}{x ^{2} - 3 x + 2}$ .
(a)
Show that $y = 1 + \frac{4}{x - 1}$ , and state the two values of $x$ for which $y$ is not defined.
[4 marks]
(b)
The graph of $y = 1 + \frac{4}{x - 1}$ passes through a point that is missing from $C$ . Find its coordinates.
[3 marks]
Answer
(c)
Find the set of values of $x$ for which $y > 3$ on $C$ .
[4 marks]
Answer
Total for D4: 11 marks
D5
$f (x) = \frac{1}{x} - \frac{2}{x + 1} + \frac{1}{x + 2}$ , $x \neq = 0, - 1, - 2$ .
(a)
Show that $f (x) = \frac{2}{x ( x + 1 ) ( x + 2 )}$ .
[4 marks]
(b)
Hence solve $f (x) = \frac{1}{3 x}$ .
[3 marks]
Answer
(c)
Hence show that $f (x) < \frac{2}{x ^{3}}$ for all $x > 0$ .
[2 marks]
Total for D5: 9 marks
D6
Given that $\frac{x ^{2} + a x + b}{x ^{2} - 5 x + 6}$ simplifies to $\frac{x + 4}{x - 3}$ , find the values of the constants $a$ and $b$ .
[6 marks]
Answer $a = \dots\dots$ , $b = \dots\dots$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Find all the pairs of positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{4}$ .
[9 marks]
Hint 1 · What to try
Multiply through by $4 x y$ to clear the fractions.
Hint 2 · The first line
You get $x y - 4 x - 4 y = 0$ . Add $16$ to both sides and factorise the left side.
Hint 3 · The full method
$(x - 4) (y - 4) = 16$ . Why must both brackets be positive? Then list the ways to write $16$ as a product.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify an algebraic fraction by cancelling common factors.
I can multiply, divide, add and subtract algebraic fractions.
I can solve equations with algebraic fractions and check for values that are not allowed.
I can use a single fraction to solve an equation or show an inequality.

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