GCSEMathsWJECMaths & Numeracy (Double Award)FoundationRevision NotesAlgebraTrial & ImprovementTrial & Improvement

Trial & Improvement (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Jamie Wood

Written by: Jamie Wood

Reviewed by: Mark Curtis

Updated on

WJEC Maths & Numeracy (Double Award): FoundationWJEC Maths & Numeracy (Double Award): Higher

Trial & Improvement

What is trial and improvement?

  • Trial and improvement means taking an educated guess at the answer, and then refining your guess to make it more accurate

  • It is useful for equations where you do not know an algebraic method for solving it

    • E.g. When solving a cubic equation of the form a x cubed plus b x squared plus c x plus d equals 0

  • It is commonly used for equations which do not have "nice" answers

    • E.g. the answer is a long decimal or irrational number

  • You might be asked to do one of the following:

    • Show that there is a solution between two values

    • Find an approximate solution to a given level of accuracy

    • Show that a given solution is correct to a given level of accuracy

How do I show that there is a solution between two values?

  • This is best shown through an example

  • Show that there is a solution to x cubed plus x equals 7 between 1 and 2

  • Method 1:

    • Leave a constant term (e.g. the 7) on the right

    • Substitute x equals 1 and x equals 2 into the left and show that this gives values below and above 7

      • 1 cubed plus 1 equals 2 which is below 7

      • 2 cubed plus 2 equals 10 which is above 7

      • Therefore a solution lies between 1 and 2

  • Method 2:

    • Use "0" as your constant term on the right (by rearranging the equation into "... = 0")

    • Substitute in x equals 1 and x equals 2 and show that this gives values below and above 0,

      • i.e. negative and positive

    • This is called a change of sign between 1 and 2

      • x cubed plus x minus 7 equals 0

      • Substitute x equals 1 into the left-hand side: 1 cubed plus 1 minus 7 equals negative 5 (negative)

      • Substitute x = 2 into the left-hand side:  2 cubed plus 2 minus 7 equals 3 (positive)

      • A solution lies between 1 and 2 as there is a change of sign

How do I find a solution using trial and improvement?

  • Write the equation in the form "something = 0"

    • i.e. Rearrange the equation so that all the terms are on one side, and the other side is zero

    • E.g. x cubed plus 2 x minus 7 equals 0

  • Substitute values into the equation

    • Try different values of x until you find two that give answers with opposite signs

      • i.e. One positive and one negative

    • If x equals a gives a negative result when substituted in,

    • and x equals b gives a positive result when substituted in,

    • then the solution lies between a and b

  • Narrow down the interval

    • Keep testing values between a and b until you find two numbers that give results with opposite signs to the level of accuracy required

    • E.g. If you need the answer to 1 decimal place, then your two numbers should be 0.1 apart

    • If you need the answer to 2 decimal places, then your numbers should be 0.01 apart

  • Test the midpoint

    • To decide which number of the two is closest to the solution, test the value halfway between them

    • E.g. If x equals 3.2 gives a negative result and x equals 3.3 gives a positive result, then test the midpoint, x equals 3.25

    • If substituting in x equals 3.25 produces a negative then the solution must be between x equals 3.25 and x equals 3.3

      • So the answer would be x equals 3.3 to 1 decimal place

    • If substituting in x equals 3.25 produces a positive then the solution must be between x equals 3.2 and x equals 3.25

      • So the answer would be x equals 3.2 to 1 decimal place

    • The key thing to remember is that the answer lies between the two values of x which produce a sign change

Examiner Tips and Tricks

In an exam you must show how you confirm which of your values is the correct solution, using the above process. An answer without supporting working will not receive full marks.

Worked Example

Find the solution to x cubed plus 2 x equals 50 to 1 decimal place using trial and improvement.

Show clearly how you reach your answer.

Answer:

Set the equation equal to zero by subtracting 50 from both sides

x cubed plus 2 x minus 50 equals 0

Start "guessing" some values which could be close to a solution

You are looking for values of x which when substituted in result in answers with opposite signs

Try x equals 2

2 cubed plus 2 open parentheses 2 close parentheses minus 50 equals negative 38

Try x equals 3

3 cubed plus 2 open parentheses 3 close parentheses minus 50 equals negative 17

These are both negative, so the solution is not between them

Try x equals 4

4 cubed plus 2 open parentheses 4 close parentheses minus 50 equals 22

This is positive, so you now know the solution is between x equals 3 and x equals 4

Must be between x equals 3 and x equals 4

You are looking for the solution to 1 decimal place, so your two values of x need to be separated by just 0.1

Try x equals 3.5

3.5 cubed plus 2 open parentheses 3.5 close parentheses minus 50 equals negative 0.125

So the answer is between x equals 3.5 (negative answer) and x equals 4 (positive answer)

Try x equals 3.6

3.6 cubed plus 2 open parentheses 3.6 close parentheses minus 50 equals 3.856

So the answer is between x equals 3.5 (negative answer) and x equals 3.6 (positive answer)

Now that you have consecutive values of x separated by 0.1 which produce a sign change, you just need to check which one is the closest answer

Test the midpoint of x equals 3.5 and x equals 3.6, which is x equals 3.55

3.55 cubed plus 2 open parentheses 3.55 close parentheses minus 50 equals 1.838875

Summarise what you have found to make the decision

x equals 3.5 produces a negative value

x equals 3.55 produces a positive value

x equals 3.6 produces a positive value

The answer must be where the sign change is

The solution lies between x equals 3.5 and 3.55

Write a final conclusion to the given level of accuracy

Therefore the solution is x equals 3.5 to 1 decimal place

Examiner Tips and Tricks

You could present your working to the above worked example in a table. It would look as below.

x

x cubed plus 2 x minus 50

2

-38

3

-17

4

22

3.5

-0.125

3.6

3.856

3.55

1.838875

Therefore the solution is x equals 3.5 to 1 decimal place.

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.