GCSE Maths · Higher · Chapter 17: Quadratic equations

Quadratic graphs, factorising, the formula and simultaneous equations

Grades 4–9About 90 minutesCalculator allowed98 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Factorise $x^{2} + 9 x + 14$ .
[1 mark]
Answer

A2
Expand and simplify $(x - 4) (x + 9)$ .
[1 mark]
Answer
A3
Work out the value of $2 x^{2} - 3 x$ when $x = - 4$ .
[2 marks]
Answer
A4
Solve $3 x - 7 > 11$ .
[1 mark]
Answer
A5
Write down the coordinates of the point where the line $y = 2 x - 6$ crosses the $x$ -axis.
[1 mark]
Answer $(\dots\dots, \dots\dots)$

Examples, then your turn

Grades 5–6

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

Example 1

Solve $x^{2} - 3 x - 40 = 0$ .

Hint Find two numbers that multiply to

- 40

and add to

- 3

. Then set each bracket equal to 0.

B1
Solve $x^{2} + 3 x - 28 = 0$ .
[2 marks]
Answer

Example 2

Solve $3 x^{2} + 10 x - 8 = 0$ by factorising.

Hint The brackets start

(3 x) (x)

. Try pairs of numbers that multiply to

- 8

until the middle term is

10 x

B2
Solve $2 x^{2} - 7 x - 15 = 0$ by factorising.
[3 marks]
Answer

Example 3

Solve $2 x^{2} + 5 x - 4 = 0$ . Give your answers correct to 2 decimal places.

Hint Use the quadratic formula with

a = 2

b = 5

and

c = - 4

. Work out

b^{2} - 4 a c

first.

B3
Solve $3 x^{2} - 4 x - 2 = 0$ . Give your answers correct to 2 decimal places.
[3 marks]
Answer

Example 4

Solve the simultaneous equations
$y = x^{2} - 7 y = 6 x$

Hint Both equations say what

y

is, so set

x^{2} - 7 = 6 x

. Solve, then find each

y

B4
Solve the simultaneous equations
$y = x^{2} + 4 y = 5 x + 18$
[4 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
The graph of $y = x^{2} - 2 x - 2$ is drawn on the grid.
(a)
Write down the coordinates of the turning point of the curve.
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(b)
Use the graph to find estimates of the solutions of $x^{2} - 2 x - 2 = 0$ .
[2 marks]
Answer
(c)
Use the graph to solve $x^{2} - 2 x - 2 = 1$ .
[2 marks]
Answer
Total for C1: 5 marks

C2
Solve $5 x^{2} = 20 x$ .
[2 marks]
Answer
C3
Work out the value of the discriminant, $b^{2} - 4 a c$ , of $2 x^{2} - 3 x + 5 = 0$ .
Hence state how many real solutions the equation has.
[2 marks]
Answer $\dots\dots$ number of solutions $\dots\dots$
C4
Write $x^{2} + 6 x - 11$ in the form $(x + p)^{2} + q$ .
Hence write down the coordinates of the turning point of $y = x^{2} + 6 x - 11$ .
[3 marks]
Answer $\dots\dots\dots\dots$ $(\dots\dots, \dots\dots)$
C5
Solve $x^{2} - x - 20 \leq 0$ .
[3 marks]
Answer
C6
Use the quadratic formula to solve $x^{2} + 6 x + 4 = 0$ . Give your answers in the form $a \pm b$ .
[2 marks]
Answer
C7
Solve the simultaneous equations
$x^{2} + y^{2} = 40 y = x + 4$
[5 marks]
Answer
C8
The curve $y = x^{2} + b x + c$ crosses the $x$ -axis at $(1, 0)$ and $(7, 0)$ .
Find the value of $b$ , the value of $c$ , and the coordinates of the turning point.
[3 marks]
Answer $b = \dots\dots$ $c = \dots\dots$ $(\dots\dots, \dots\dots)$
C9
$n$ is an integer and $n^{2} < 2 n + 24$ .
Write down all the possible values of $n$ .
[3 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
A rectangle is $(2 x + 3)$ cm long and $(x - 1)$ cm wide. Its area is 33 cm².
(a)
Show that $2 x^{2} + x - 36 = 0$ .
[2 marks]
(b)
Work out the perimeter of the rectangle.
[3 marks]
Answer
Total for D1: 5 marks

D2
Lena says, "The solution of $x^{2} > 16$ is $x > 4$ ."
Lena is not right. Explain why, and write down the full solution.
[2 marks]
Answer
D3
A ball is thrown from the top of a wall. Its height, $h$ metres, above the ground $t$ seconds later is
$h = 2 + 14 t - 5 t^{2}$
Work out the time it takes for the ball to reach the ground. Give your answer correct to 3 significant figures.
[3 marks]
Answer
D4
Show that the line $y = 2 x - 1$ touches the curve $y = x^{2}$ at exactly one point, and find the coordinates of that point.
[3 marks]
Answer $(\dots\dots, \dots\dots)$
D5
Solve $\frac{8}{x + 1} - \frac{3}{x} = 1$ .
You must show your working.
[4 marks]
Answer
D6
A rectangular lawn is 15 m long and 10 m wide. A path of the same width, $x$ metres, goes all the way round the lawn. The area of the path is 84 m².
Not drawn accurately
(a)
Show that $2 x^{2} + 25 x - 42 = 0$ .
[3 marks]
(b)
Work out the width of the path. You must show your working.
[3 marks]
Answer
Total for D6: 6 marks
D7
Solve $2 x^{2} - x - 15 > 0$ .
[3 marks]
Answer
D8
The diagram shows a sketch of the curve $y = x^{2} + b x + c$ .
The curve meets the $y$ -axis at $(0, - 18)$ and the $x$ -axis at $(- 3, 0)$ and $(k, 0)$ .
Find the value of $b$ and the value of $k$ .
[3 marks]
Not drawn accurately
Answer $b = \dots\dots$ $k = \dots\dots$
D9
The sides of a right-angled triangle are $x$ cm, $(x + 7)$ cm and $(2 x - 3)$ cm. The longest side is $(2 x - 3)$ cm.
Work out the value of $x$ . Give your answer correct to 2 decimal places.
[4 marks]
Not drawn accurately
Answer $x = \dots\dots$
D10
Solve the simultaneous equations
$y = x^{2} - 4 x + y = 3$
Give your values of $x$ and $y$ correct to 2 decimal places.
[5 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Find the values of $k$ for which the equation $x^{2} + (k + 1) x + 4 = 0$ has no real solutions.
[4 marks]
Hint 1 · What to try
An equation has no real solutions when its discriminant, $b^{2} - 4 a c$ , is negative.
Hint 2 · The first line
$(k + 1)^{2} - 16 < 0$
Hint 3 · The full method
$(k + 1)^{2} < 16$ , so $- 4 < k + 1 < 4$ , which gives $- 5 < k < 3$ .
Answer

E2
The sum of the whole numbers from 1 to $n$ is $\frac{1}{2} n (n + 1)$ .
Find the smallest value of $n$ for which this sum is more than 500.
[3 marks]
Hint 1 · What to try
Write an inequality and turn it into a quadratic.
Hint 2 · The first line
$n^{2} + n - 1000 > 0$
Hint 3 · The full method
The positive root of $n^{2} + n - 1000 = 0$ is about 31.1, so $n = 32$ . Check: the sum to 31 is 496 and the sum to 32 is 528.
Answer
E3
Solve $x - 7 x + 12 = 0$ .
[3 marks]
Hint 1 · What to try
$x$ is $(x)^{2}$ . Give $x$ a letter of its own.
Hint 2 · The first line
With $y = x$ : $y^{2} - 7 y + 12 = 0$ , so $(y - 3) (y - 4) = 0$ .
Hint 3 · The full method
$x = 3$ gives $x = 9$ , and $x = 4$ gives $x = 16$ . Both work in the original equation.
Answer
E4
The line $y = x + c$ is a tangent to the circle $x^{2} + y^{2} = 8$ .
Find the possible values of $c$ .
[4 marks]
Hint 1 · What to try
Substitute the line into the circle. A tangent meets the circle at exactly one point.
Hint 2 · The first line
$2 x^{2} + 2 c x + c^{2} - 8 = 0$
Hint 3 · The full method
One point means the discriminant is 0: $4 c^{2} - 8 (c^{2} - 8) = 0$ , so $64 - 4 c^{2} = 0$ and $c = 4$ or $c = - 4$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can read the roots, the intercept and the turning point off a quadratic graph.
I can solve a quadratic by factorising, including when the coefficient of $x^{2}$ is not 1.
I can use the quadratic formula and the discriminant.
I can complete the square to find a turning point.
I can solve a linear and a quadratic equation together.
I can solve a quadratic inequality.

Answers and mark scheme

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