GCSE Maths · Higher · Chapter 17: Quadratic equations · Lesson 5

One linear and one non-linear equation

Grades 6–8About 50 minutesCalculator allowed36 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Solve $x + y = 15$ and $x - y = 3$ .
[1 mark]
Answer
A2
Last lesson
Write $x^{2} + 4 x - 4$ in the form $(x + p)^{2} + q$ .
[1 mark]
Answer
A3
From chapter 9
A circle has diameter 6 cm. Find its area in terms of $π$ .
[1 mark]
Answer
A4
From chapter 16
Work out $3 \times 27$ .
[1 mark]
Answer

Grades 6–7

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

Example 1

Solve the simultaneous equations $y = x^{2} + 2 x$ and $y = 3 x + 6$ .

Hint Both equal

y

, so set

x^{2} + 2 x = 3 x + 6

. Find each

x

, then put it into the LINE to get

y

B1
Solve the simultaneous equations $y = x^{2} + 5 x$ and $y = x + 12$ .
[3 marks]
Answer $x = \dots, y = \dots$ and $x = \dots, y = \dots$

Example 2

Solve the simultaneous equations $x^{2} + y^{2} = 10$ and $y = x + 2$ .

Hint Replace

y

in the circle by

(x + 2)

, expand, and collect everything on one side.

B2
Solve the simultaneous equations $x^{2} + y^{2} = 58$ and $y = x + 4$ .
[3 marks]
Answer $x = \dots, y = \dots$ and $x = \dots, y = \dots$

Grades 6–7

C1
The graph of $y = x^{2} - 4 x$ is drawn. Find the equation of the straight line to draw on it to solve $x^{2} - x - 5 = 0$ .
[2 marks]
Answer $y = \dots\dots$
C2
Two numbers add up to $15$ . The sum of their squares is $113$ . Find the two numbers.
[3 marks]
Answer $\dots\dots$ and $\dots\dots$

C3
Show that the line $y = x + 6$ does not meet the circle $x^{2} + y^{2} = 16$ .
[2 marks]
C4
Solve the simultaneous equations $x^{2} + y^{2} = 28$ and $y = 2 x - 1$ . Give your answers correct to 2 decimal places.
[4 marks]
Answer $x = \dots, y = \dots$ and $x = \dots, y = \dots$

Grades 7–8

D1
Tom solves $y = x^{2} + 6$ and $y = 9 x - 8$ together. He writes " $x^{2} - 9 x + 14 = 0$ , so $x = 2$ or $x = 7$ ."
Explain why Tom's answer is not complete, and write down the full solution.
[3 marks]
Answer $x = \dots, y = \dots$ and $x = \dots, y = \dots$

Grade 8 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve a linear and a quadratic equation together.
I can find where a line meets a circle.
I can choose a line to solve a quadratic on a graph.

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