GCSE Maths · Higher · Chapter 17: Quadratic equations · Lesson 6

Quadratic inequalities

Grades 6–8About 55 minutesCalculator allowed48 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 $15 - 4 x \geq - 9$ .
[1 mark]
Answer
A2
From chapter 16
A mass is 170 g to the nearest 10 g. Write down its upper bound.
[1 mark]
Answer
A3
From chapter 17
How many real solutions has $2 x^{2} + 4 x + 6 = 0$ ? Use $b^{2} - 4 a c$ .
[1 mark]
Answer
A4
Last lesson
Solve $y = x^{2}$ and $y = 36$ together.
[1 mark]
Answer

Grades 6–7

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

Example 1

Solve $x^{2} - 6 x - 16 < 0$ .

Hint Solve

x^{2} - 6 x - 16 = 0

for the critical values. The curve is below the axis BETWEEN them.

B1
Solve $x^{2} - 7 x - 60 < 0$ .
[3 marks]
Answer $\dots\dots < x < \dots\dots$

Example 2

Solve $x^{2} - 8 x - 65 \geq 0$ .

Hint The critical values are

- 5

and

13

. The curve is above the axis OUTSIDE them, so there are two parts joined by "or".

B2
Solve $x^{2} + 5 x - 50 > 0$ .
[3 marks]
Answer $x < \dots\dots$ or $x > \dots\dots$

Grades 6–7

C3
Solve $x^{2} > 10 x + 39$ .
[3 marks]
Answer
C4
List the integers $n$ for which $n^{2} - 17 n + 70 < 0$ .
[3 marks]
Answer
C5
Solve $x^{2} - 8 x + 3 \leq 0$ . Give the critical values correct to 2 decimal places.
[3 marks]
Answer
C6
Solve $3 x^{2} - 11 x - 4 < 0$ . Give your answer using set notation.
[3 marks]
Answer ${x : \dots\dots}$
C7
The $n$ th term of a sequence is $n^{2} - 16 n + 28$ . How many terms of the sequence are negative?
[3 marks]
Answer
C8
The sketch shows the graph of $y = x^{2} - x - 90$ . It crosses the $x$ -axis at $- 9$ and $10$ . Use the graph to solve $x^{2} - x - 90 > 0$ .
[2 marks]
Not drawn accurately
Answer

Grades 7–8

D1
Mia solves $x^{2} < 7 x$ . She divides both sides by $x$ and writes " $x < 7$ ".
Explain why Mia's answer is wrong, and solve $x^{2} < 7 x$ correctly.
[3 marks]
Answer

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 find the critical values of a quadratic inequality.
I can decide whether the answer is between the critical values or outside them.
I can write the answer with the right signs, using "or" for two parts.

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