GCSE Maths · Higher · Chapter 15: Equations and inequalities · Lesson 4

Trial and improvement

Grades 5–8About 50 minutesCalculator allowed42 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
Work out $3 x^{2} + 5$ when $x = - 2$ .
[1 mark]
Answer
A2
Last lesson
Solve $3 x + 2 < 14$ .
[1 mark]
Answer
A3
From chapter 11
A triangle has sides 20, 21 and 30. Is it right-angled?
[1 mark]
Answer
A4
From chapter 13
Two fair dice are rolled. Find P(total is 9).
[1 mark]
Answer

Grades 5–6

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

Example 1

Show that $x^{3} + 4 x = 60$ has a solution between $3$ and $4$ .

Hint Work out the left-hand side at

3

and at

4

B1
Between which two consecutive whole numbers is the solution of $x^{3} - x = 100$ ?
[2 marks]
Answer

Example 2

Use trial and improvement to solve $x^{3} + 4 x = 60$ . Give your answer to 1 decimal place.

Hint After trapping it between two tenths, try the value halfway.

B2
Use trial and improvement to solve $x^{3} - x = 100$ . Give your answer to 1 decimal place. You must show your working.
[3 marks]
Answer

Example 3

Solve $x^{3} = 5 x + 30$ by trial and improvement, to 1 decimal place.

Hint Take

5 x

from both sides first:

x^{3} - 5 x = 30

B3
Solve $x^{3} = 8 x + 15$ by trial and improvement. Give your answer to 1 decimal place.
[3 marks]
Answer

Grades 6–7

C1
The equation $x^{3} + 9 x = 88$ has a solution between $3.7$ and $3.8$ . Find it correct to 2 decimal places.
[3 marks]
Answer
C2
Use trial and improvement to solve $3^{x} = 40$ . Give your answer to 1 decimal place.
[3 marks]
Answer

C3
The equation $x^{3} + 5 x = - 20$ has a solution between $- 3$ and $- 2$ . Find it to 1 decimal place.
[3 marks]
Answer
C4
A rectangle is $3$ cm longer than twice its width. Its area is $50$ cm². Use trial and improvement to find its width $w$ , to 1 decimal place.
[3 marks]
Not drawn accurately
Answer

Grades 6–8

D1
Jo solves $x^{3} - 7 x = 57$ . She finds that $4.4$ gives $54.384$ and $4.5$ gives $59.625$ . She says, "The answer to 1 decimal place is $4.4$ , because $54.384$ is closer to $57$ ."
Is Jo right? You must show your working.
[3 marks]

D2
A cylinder has radius $r$ cm, height $(r + 3)$ cm and volume $400$ cm³. Use trial and improvement to find $r$ to 1 decimal place. Show your working.
[4 marks]
Not drawn accurately
Answer
D3
A square sheet of card has side $20$ cm. A square of side $x$ cm is cut from each corner, and the sides are folded up to make an open box. The volume of the box is $400$ cm³.
Not drawn accurately
(a)
Show that $x^{3} - 20 x^{2} + 100 x = 100$ .
[3 marks]
(b)
The smaller value of $x$ that works lies between $1$ and $2$ . Use trial and improvement to find it. Give your answer to 1 decimal place. You must show your working.
[4 marks]
Answer
Total for D3: 7 marks

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 trap a root between two whole numbers.
I can close in on it one decimal place at a time.
I can test the halfway value to round correctly.

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