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

Solving quadratics and completing the square

70 marks

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Date

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Quick questions to start.

A1
Needed today
Factorise $x^{2} - 3 x - 70$ .
[2 marks]
Answer
A2
Last lesson
Simplify $108 - 12$ .
[2 marks]
Answer
A3
GCSE Higher
Expand and simplify $(x - 6)^{2}$ .
[2 marks]
Answer
A4
From chapter 1
Factorise completely $2 x^{3} - 18 x$ .
[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

Solve (a) $10 x^{2} = 9 - 9 x$ (b) $4 x^{2} = 7 x$ .

Hint Make one side

0

first. (a)

a c = - 90

: use

15

and

- 6

. (b) Factorise; do not divide by

x

B1
Solve (a) $8 x^{2} = 2 x + 15$ (b) $3 x^{2} = 11 x$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Solve $2 x^{2} - 7 x - 3 = 0$ . Give your answers exactly, then correct to $3$ significant figures.

Hint

a = 2

b = - 7

c = - 3

, so

b^{2} - 4 a c = 49 + 24

B2
Solve $3 x^{2} + 4 x - 5 = 0$ . Give exact answers in the form $\frac{p \pm q}{3}$ , then answers to $3$ s.f.
[4 marks]
Answer $\dots\dots$ ; $\dots\dots$

Example 3

Complete the square for (a) $x^{2} - 9 x + 3$ (b) $3 x^{2} - 24 x + 41$ .

Hint Halve the coefficient of

x

. In (b) take out

3

from the first two terms.

B3
Complete the square for (a) $x^{2} + 5 x + 1$ (b) $2 x^{2} - 20 x + 9$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 4

Solve, giving exact answers, (a) $(3 x + 1)^{2} = 20$ (b) $x^{2} - 8 x + 4 = 0$ by completing the square.

Hint Square-root with

\pm

. In (b)

(x - 4)^{2} - 16 + 4 = 0

, so

(x - 4)^{2} = 12

B4
Solve, giving exact answers, (a) $(2 x - 5)^{2} = 18$ (b) $2 x^{2} - 10 x + 3 = 0$ by completing the square.
[5 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Practice

A-Level

C1
Solve $\frac{x + 2}{x - 1} = 2 x$ .
[4 marks]
Answer $x = \dots\dots$

C2
The diagram shows a right-angled triangle. Its sides are $x$ cm, $(x + 3)$ cm and $(x + 6)$ cm. Find $x$ .
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$
C3
The solutions of $x^{2} + b x + c = 0$ are $x = 5 \pm 2$ . Find $b$ and $c$ .
[3 marks]
Answer $b = \dots$ $c = \dots$

Exam-style questions

Edexcel exam style

D1
Mia is asked to write $2 x^{2} + 12 x + 7$ in the form $a (x + b)^{2} + c$ . She writes: $2 x^{2} + 12 x + 7 = 2 (x + 3)^{2} - 9 + 7 = 2 (x + 3)^{2} - 2$ .
Mia has made a mistake.
(a)
Explain the mistake Mia has made.
[1 mark]
(b)
Write $2 x^{2} + 12 x + 7$ correctly in the form $a (x + b)^{2} + c$ .
[2 marks]
Answer
(c)
Hence solve $2 x^{2} + 12 x + 7 = 0$ , giving your answers in the form $p \pm \frac{q}{2}$ .
[2 marks]
Answer
Total for D1: 5 marks

D2
$f (x) = 18 + 12 x - 2 x^{2}$ .
(a)
Write $f (x)$ in the form $p - q (x - r)^{2}$ , where $p$ , $q$ and $r$ are integers.
[3 marks]
Answer
(b)
Hence solve $f (x) = 0$ , giving your answers in the form $a \pm b 2$ .
[2 marks]
Answer
(c)
Hence write down the maximum value of $f (x)$ , and the value of $x$ at which it occurs.
[2 marks]
Answermaximum $\dots\dots$ at $x = \dots\dots$
(d)
Hence solve $f (x) = 4$ .
[2 marks]
Answer $x = \dots\dots$
Total for D2: 9 marks
D3
A rectangular lawn measures $(2 x + 3)$ m by $(x + 1)$ m. Its area is $50$ m².
Not drawn accurately
(a)
Show that $2 x^{2} + 5 x - 47 = 0$ .
[2 marks]
(b)
Hence find the value of $x$ , correct to $3$ significant figures.
[3 marks]
Answer $x = \dots\dots$
(c)
Explain why the equation gives only one value of $x$ for this lawn.
[1 mark]
(d)
Find the perimeter of the lawn, correct to $3$ significant figures.
[2 marks]
Answer $\dots\dots$ m
Total for D3: 8 marks
D4
Solve the equation $x - 4 = x + 8$ .
(a)
Show that squaring both sides gives $x^{2} - 9 x + 8 = 0$ .
[2 marks]
(b)
Hence solve $x^{2} - 9 x + 8 = 0$ .
[2 marks]
Answer $x = \dots\dots$
(c)
Check each value in the original equation, and so solve $x - 4 = x + 8$ .
[2 marks]
Answer $x = \dots\dots$
Total for D4: 6 marks

Extension

Stretch

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

E1
A rectangle has a perimeter of $30$ cm and a diagonal of $12$ cm. Find the exact lengths of its sides.
[6 marks]
Hint 1 · What to try
Call the sides $x$ and $y$ . Write one equation from the perimeter and one from the diagonal.
Hint 2 · The first line
$y = 15 - x$ , so $x^{2} + (15 - x)^{2} = 144$ .
Hint 3 · The full method
$2 x^{2} - 30 x + 81 = 0$ . Use the formula and simplify $252$ .
Answer $\dots\dots$ cm and $\dots\dots$ cm

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve a quadratic by factorising or by square-rooting.
I can use the quadratic formula and give exact or rounded answers.
I can complete the square and use it to solve an equation.
I can form a quadratic from a problem and reject roots that do not fit.

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