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

Functions and their roots

71 marks

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Quick questions to start.

A1
Needed today
Solve $5 x^{2} + 9 x - 2 = 0$ .
[2 marks]
Answer $x = \dots\dots$
A2
Last lesson
Write $x^{2} - 16 x + 71$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer
A3
From chapter 1
Write $2 5^{x}$ as a power of $5$ .
[2 marks]
Answer
A4
GCSE Higher
Solve $(2 x - 1)^{2} = 49$ .
[2 marks]
Answer $x = \dots\dots$

Example, then your turn

A-Level

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

Example 1

$f (x) = x^{2} - 6 x$ , $g (x) = 2 x + 1$ . Find (a) $f (- 3) + g (4)$ (b) $f (3)$ (c) both values of $a$ with $f (a) = 16$ .

Hint Put the number in place of

x

everywhere. In (c) write

a^{2} - 6 a = 16

and make one side

0

B1
$f (x) = 2 x^{2} + 5 x$ , $g (x) = 5 x - 1$ . Find (a) $f (- 2) + g (2)$ (b) $f (2)$ (c) both values of $a$ with $f (a) = 3$ .
[4 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 2

(a) Find the roots of $h (x) = x^{3} - 2 x^{2} - 15 x$ . (b) Solve $f (x) = g (x)$ , where $f (x) = x^{2} + 4 x - 9$ and $g (x) = 7 x + 9$ .

Hint A root is a value of

x

that makes the function

0

. In (a) take out

x

first. In (b) put

f (x)

equal to

g (x)

B2
(a) Find the roots of $k (x) = 3 x^{3} - 5 x^{2} - 2 x$ . (b) Solve $p (x) = q (x)$ , where $p (x) = x^{2} - 3 x + 1$ and $q (x) = 2 x + 37$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $x = \dots\dots$

Example 3

$f (x) = x^{2} + 12 x + 33$ . (a) Write $f (x)$ as $(x + p)^{2} + q$ . (b) Find its roots in surd form. (c) State its minimum value and where it occurs.

Hint Halve the

12

. A square is never negative, so the smallest it can be is

0

B3
$g (x) = x^{2} + 18 x + 70$ . (a) Write $g (x)$ as $(x + p)^{2} + q$ . (b) Find its roots in surd form. (c) State its minimum value and where it occurs.
[5 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 4

Find all the real roots of (a) $f (x) = x^{6} - 26 x^{3} - 27$ (b) $g (x) = 3 x - 7 x - 6$ .

Hint (a) Let

u = x^{3}

, so

x^{6} = u^{2}

. (b) Let

u = x

, so

x = u^{2}

. A square root cannot be negative.

B4
Find all the real roots of (a) $j (x) = x^{6} + 19 x^{3} - 216$ (b) $k (x) = x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} - 4$ .
[5 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Practice

A-Level

C1
The functions $f$ and $g$ are given by $f (x) = 3 x^{3} + 5 x$ and $g (x) = 16 x^{2}$ , $x \in R$ . Find the three values of $x$ for which $f (x) = g (x)$ .
[3 marks]
Answer $x = \dots\dots$

C2
$f (x) = 3 x^{2} - 12 x + 17$ . Write $f (x)$ in the form $a (x + p)^{2} + q$ . Hence explain why $f (x) > 0$ for every value of $x$ .
[3 marks]
Answer $f (x) = \dots\dots$
C3
Solve $5^{2 x + 1} - 26 (5^{x}) + 5 = 0$ .
[4 marks]
Answer $x = \dots\dots$

Exam-style questions

Edexcel exam style

D1
Sam solves $x - 3 x - 28 = 0$ . He writes: let $u = x$ , so $u^{2} - 3 u - 28 = 0$ , $(u - 7) (u + 4) = 0$ , $u = 7$ or $u = - 4$ , so $x = 49$ or $x = 16$ .
Sam's answer is not correct.
(a)
Explain Sam's mistake, and give the correct solution.
[2 marks]
Answer $x = \dots\dots$
(b)
Solve $2 x + 3 x - 20 = 0$ .
[3 marks]
Answer $x = \dots\dots$
Total for D1: 5 marks

D2
The function $f$ is defined by $f (x) = x^{2} - 12 x + 1$ , $x \in R$ .
(a)
Write $f (x)$ in the form $(x + p)^{2} + q$ , where $p$ and $q$ are integers.
[2 marks]
Answer
(b)
Hence write down the minimum value of $f (x)$ , and the value of $x$ at which it occurs.
[2 marks]
Answerminimum $\dots\dots$ at $x = \dots\dots$
(c)
Hence find the roots of $f (x)$ , giving your answers in surd form.
[2 marks]
Answer $x = \dots\dots$
(d)
Hence find the four roots of $g (x) = x^{4} - 12 x^{2} + 1$ , giving exact answers.
[3 marks]
Answer $x = \dots\dots$
Total for D2: 9 marks
D3
The function $f$ is defined by $f (x) = 4^{x} - 12 (2^{x}) + 32$ , $x \in R$ .
(a)
Write $f (x)$ in the form $(2^{x} - a) (2^{x} - b)$ , where $a$ and $b$ are integers.
[2 marks]
Answer
(b)
Hence find the two roots of $f (x)$ .
[2 marks]
Answer $x = \dots\dots$
(c)
Show that $f (x) = (2^{x} - 6)^{2} - 4$ , and hence write down the least value of $f (x)$ .
[3 marks]
Answerleast value $\dots\dots$
(d)
Explain why the equation $f (x) = - 5$ has no solutions.
[1 mark]
Total for D3: 8 marks
D4
The function $f$ is defined by $f (x) = 3 x^{2} - 8 x$ , $x \in R$ .
(a)
Find $f (- 2)$ .
[1 mark]
Answer
(b)
Given that $f (p) = 3$ , find the two possible values of $p$ .
[3 marks]
Answer $p = \dots\dots$
(c)
Find $f (2)$ . Hence find the value of $q$ , other than $2$ , for which $f (q) = f (2)$ .
[3 marks]
Answer $q = \dots\dots$
Total for D4: 7 marks

Extension

Stretch

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

E1
Find all four roots of $f (x) = (x^{2} + 2 x)^{2} - 11 (x^{2} + 2 x) + 24$ .
[6 marks]
Hint 1 · What to try
The same bracket appears twice. Call it $u$ .
Hint 2 · The first line
With $u = x^{2} + 2 x$ , the equation is $u^{2} - 11 u + 24 = 0$ , so $u = 3$ or $u = 8$ .
Hint 3 · The full method
Now solve $x^{2} + 2 x = 3$ and $x^{2} + 2 x = 8$ .
Answer $x = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out values of a function and find inputs that give an output.
I can find the roots of a function and solve $f (x) = g (x)$ .
I can use a completed square to find roots and a least value.
I can solve an equation that is a quadratic in disguise.

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