Robinson Tuition
Grade 5 · Foundation and Higher · free GCSE maths worksheet

Trial and improvement

Solving equations by successive approximation.

9 original questions, 39 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

f(x) = x2 − 6x + 2

  1. Show that the equation f(x) = 0 has a root between x = 0 and x = 1

  2. Show that the equation x2 − 6x + 2 = 0 can be rearranged to give

    x = x2 + 26

4 marks
Show answer
a) f(0) = 2 > 0 and f(1) = -3 < 0, sign change so a root lies between 0 and 1 b) 6x = x2 + 2 so x = (x2 + 2)/6
2

A square has an area of 70 cm2.

Use trial and improvement to find the length of a side, correct to 1 decimal place. You must show all your trials.

3 marks
Show answer
8.4 cm
3

f(x) = x3 + 4x − 9

  1. Show that the equation f(x) = 0 has a root between x = 1 and x = 2

  2. Show that the equation x3 + 4x − 9 = 0 can be rearranged to give

    x = 3√(9 − 4x)

  3. Starting with x0 = 1.5, use the iteration formula xn+1 = 3√(9 − 4xn) to work out the values of x1, x2 and x3. Give each value correct to 4 decimal places.

6 marks
Show answer
a) f(1) = -4 and f(2) = 7; the sign change means a root lies between 1 and 2 b) x3 = 9 - 4x, so x = cube root of (9 - 4x) c) x1 = 1.4423, x2 = 1.4784, x3 = 1.4560
4

Use trial and improvement to find the cube root of 300, correct to 1 decimal place. Do not use the cube root key on your calculator. You must show all your trials.

3 marks
Show answer
6.7
5

The equation x3 + 2x = 50 has a solution between 3 and 4.

Use trial and improvement to find this solution correct to 1 decimal place. You must show all your trials.

4 marks
Show answer
3.5
6

The equation x3 − 3x = 10 has a solution between 2 and 3.

Use trial and improvement to find this solution correct to 1 decimal place. You must show all your trials.

4 marks
Show answer
2.6
7

A cuboid has a width of x cm, a length of (x + 2) cm and a height of (x + 5) cm. The volume of the cuboid is 200 cm3.

Use trial and improvement to find the value of x, correct to 1 decimal place. You must show all your trials.

4 marks
Show answer
3.9
8

f(x) = x2 + 2x − 5

  1. Show that the equation f(x) = 0 has a root between x = 1 and x = 2

  2. Show that the equation x2 + 2x − 5 = 0 can be rearranged to give

    x = 5x + 2

  3. Starting with x0 = 1.5, use the iteration formula xn+1 = 5xn + 2 to work out the values of x1, x2 and x3. Give each value correct to 4 decimal places.

  4. Write down an estimate of this root of f(x) = 0 correct to 2 decimal places.

6 marks
Show answer
a) f(1) = -2 and f(2) = 3; the sign change means a root lies between 1 and 2 b) x2 + 2x = 5, so x(x + 2) = 5 and x = 5/(x + 2) c) x1 = 1.4286, x2 = 1.4583, x3 = 1.4458 d) 1.45
9

The equation x3 − x = 20 has a solution between 2 and 3.

Use trial and improvement to find this solution correct to 2 decimal places. You must show all your trials.

5 marks
Show answer
2.84

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

All Grade 5 topics