Sign-Change Methods (WJEC Eduqas GCSE Maths): Revision Note

Sign-Change Methods

What is iteration?

• Iteration means finding an approximation to the solution using a method that gets closer and closer (converges) to the real solution

• It is often used for equations which don't have nice solutions

  • solutions themselves might be irrational numbers

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

• To find x₀ (the initial / starting value), you are often asked to show that there is a solution between two values

• For example, show that there is a solution to x³ + x = 7 between 1 and 2

• Method 1:

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

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

    • 1³ + 1 = 2 and 2³ + 2 = 10 which are below and above 7

    • A solution therefore lies between 1 and 2

• Method 2:

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

  • then substitute in x = 1 and x = 2, showing this gives values below and above 0, i.e. negative and positive

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

    • x³ + x - 7 = 0

    • Substitute x = 1 into the left-hand side: 1³ + 1 - 7 = -5 (negative)

    • Substitute x = 2 into the left-hand side:  2³ + 2 - 7 = 3 (positive)

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

How do I find an approximate solution to an equation?

Trial and improvement

• Write the equation in the form

• Substitute values into the equation until you get two values and where:

  • 

  • 

• Try values between and until you get two consecutive numbers at the correct degree of accuracy

  • e.g. if you need the answer to 1 decimal place then your two values should be 0.1 apart

• Test the midpoint of these values to decide which one is closer to the solution

  • e.g. if and then test

    • If then the solution is bigger than 3.25 so choose

    • If then the solution is smaller than 3.25 so choose

Decimal search

• Let’s say you know that a solution to an equation lies between 1 and 2 and you want to find it to 1 decimal place

  • The equation must be rearranged to have “= 0” on one side

• Search every decimal, by substituting 1.0, 1.1, 1.2, 1.3, 1.4, … into the equation until you see a change of sign

  • Let’s say there was a change of sign between 1.6 and 1.7

  • This is not yet enough information to be able to round the solution to 1 decimal place

• Substitute in 1.65, the value exactly in the middle of 1.6 and 1.7

  • If the sign change is between 1.6 and 1.65, the solution rounds down to 1.6 to 1 decimal place

  • If the sign change is between 1.65 and 1.7, the solution rounds up to 1.7 to 1 decimal place