AQA Level 2 Further Maths · Level 2 Certificate · Chapter 4: Algebra IV · Lesson 1

Quadratic equations

About 85 minutesCalculator allowed65 marks

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Quick questions to start.

A1
Needed today
Factorise $6 x^{2} - 7 x - 10$ .
[2 marks]
Answer
A2
GCSE Higher
Write $252$ in the form $a b$ , where $a$ and $b$ are integers and $b$ is as small as possible.
[2 marks]
Answer
A3
Last lesson
$f (x) = {x + 2, 7 - x^{2}, x < 1 x \geq 1$
Work out $f (- 3)$ and $f (2)$ .
[2 marks]
Answer
A4
From chapter 2
Write $x^{2} - 4 x + 9$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer

Example, then your turn

Level 2

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

Example 1

Solve $6 x^{2} = 7 x + 20$ .

Hint First move every term to one side, so the other side is

0

. Then factorise.

B1
Solve $10 x^{2} + 9 = 21 x$ .
[3 marks]
Answer $x = \dots\dots\dots$ or $x = \dots\dots\dots$

Example 2

Solve $x^{2} - 6 x - 11 = 0$ by completing the square. Give your answers in exact form.

Hint Halve the coefficient of

x

and square it. Write the left side as a square, then square root both sides, remembering

\pm

B2
Solve $x^{2} + 8 x - 2 = 0$ by completing the square. Give your answers in exact form.
[3 marks]
Answer $x = \dots\dots\dots$ or $x = \dots\dots\dots$

Example 3

Solve $3 x^{2} + 5 x - 4 = 0$ . Give your answers (a) in exact form, (b) correct to $2$ decimal places.

Hint It does not factorise. Write down

a

b

and

c

, then use

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

B3
Solve $5 x^{2} - 2 x - 4 = 0$ .
(a)
Give your answers in the form $\frac{a \pm b}{c}$ , where $a$ , $b$ and $c$ are integers and $b$ is as small as possible.
[2 marks]
Answer $x = \dots\dots\dots$
(b)
Give your answers correct to $2$ decimal places.
[2 marks]
Answer $x = \dots\dots\dots$ or $x = \dots\dots\dots$
Total for B3: 4 marks

Example 4

Solve $\frac{4}{x - 1} - \frac{3}{x} = 1$ .

Hint Multiply every term by

x (x - 1)

to clear the fractions. Then collect all terms on one side.

B4
Solve $\frac{5}{x - 3} + \frac{6}{x + 1} = 2$ .
[4 marks]
Answer $x = \dots\dots\dots$ or $x = \dots\dots\dots$

Example 5

A picture $20$ cm wide and $14$ cm high is in a frame of the same width $x$ cm all the way round. The total area of the picture and frame is $432$ cm². Work out $x$ .

Not drawn accurately

Hint Write the outside width and height in terms of

x

, then multiply them. A width cannot be negative.

B5
A right-angled triangle has shorter sides of $x$ cm and $(x + 4)$ cm. Its area is $25$ cm².
Not drawn accurately
(a)
Show that $x^{2} + 4 x - 50 = 0$ .
[2 marks]
(b)
Work out the value of $x$ . Give your answer correct to $2$ decimal places.
[2 marks]
Answer $x = \dots\dots\dots$
Total for B5: 4 marks

Practice

Level 2

C1
Solve $9 x^{2} + 30 x + 25 = 0$ . How many different solutions are there?
[2 marks]
Answer
C2
Work out the value of $b^{2} - 4 a c$ for the equation $3 x^{2} + 4 x + 2 = 0$ . Hence write down the number of real roots of the equation.
[2 marks]
Answer

C3
Two positive numbers differ by $3$ . The sum of their squares is $117$ . Work out the two numbers.
[3 marks]
Answer $\dots\dots$ and $\dots\dots$
C4
A stone is thrown up from the top of a wall. After $t$ seconds its height is $h$ metres, where $h = 2 + 19 t - 5 t^{2}$ .
(a)
Work out the two times when the stone is $14$ m high.
[3 marks]
Answer $t = \dots\dots$ and $t = \dots\dots$
(b)
Work out the time when the stone reaches the ground. Give your answer correct to $2$ decimal places.
[2 marks]
Answer $t = \dots\dots$
Total for C4: 5 marks
C5
The formula $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ is used for a lens. For one lens $f = 12$ , and $v$ is $10$ more than $u$ . Work out $u$ , which is positive.
[3 marks]
Answer $u = \dots\dots$
C6
Sam used the quadratic formula and correctly found $x = \frac{- 3 \pm 29}{2}$ . Write down a quadratic equation $x^{2} + b x + c = 0$ that Sam might have solved.
[2 marks]
Answer

Exam-style questions

AQA exam style

D1
The equation $x^{2} + 6 x + c = 0$ has one repeated root.
Circle the value of $c$ .
$36936$
[1 mark]

D2
Leo is solving $(x - 2) (x + 5) = 30$ . He writes: " $x - 2 = 30$ or $x + 5 = 30$ , so $x = 32$ or $x = 25$ ."
Leo is wrong.
(a)
Show that $x = 32$ is not a solution.
[1 mark]
(b)
Solve $(x - 2) (x + 5) = 30$ correctly.
[3 marks]
Answer $x = \dots\dots\dots$ or $x = \dots\dots\dots$
Total for D2: 4 marks
D3
The diagram shows a trapezium. Its area is $24$ cm².
Not drawn accurately
(a)
Show that $3 x^{2} - x - 24 = 0$ .
[2 marks]
(b)
Hence work out the length of the longer parallel side.
[3 marks]
Answer $\dots\dots$ cm
Total for D3: 5 marks
D4
Solve $\frac{2}{x - 1} - \frac{1}{x + 2} = \frac{1}{3}$ .
Give your answers to $3$ significant figures. You must show your working.
[5 marks]
Answer $x = \dots\dots\dots$ or $x = \dots\dots\dots$
D5
The solutions of the equation $x^{2} + b x + c = 0$ are $x = 2 + 7$ and $x = 2 - 7$ .
Work out the values of $b$ and $c$ .
[3 marks]
Answer $b = \dots\dots$ , $c = \dots\dots$

Extension

Stretch

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

E1
Solve $(x^{2} - 3 x)^{2} - 14 (x^{2} - 3 x) + 40 = 0$ .
[4 marks]
Hint 1 · What to try
Do not expand. The same expression appears twice: give it a single letter.
Hint 2 · The first line
Let $u = x^{2} - 3 x$ . Then $u^{2} - 14 u + 40 = 0$ , so $(u - 4) (u - 10) = 0$ .
Hint 3 · The full method
$x^{2} - 3 x = 4$ gives $(x - 4) (x + 1) = 0$ ; $x^{2} - 3 x = 10$ gives $(x - 5) (x + 2) = 0$ . So $x = - 2$ , $- 1$ , $4$ or $5$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve a quadratic equation by factorising, by completing the square or with the formula.
I can give the answers in exact form or to a given accuracy.
I can turn an equation with fractions, or a problem in context, into a quadratic and solve it.
I can tell from $b^{2} - 4 a c$ how many real roots an equation has.

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