Showing a solution is between two values
- If the value of an expression changes sign between x = a and x = b, the equation 'expression = 0' has a solution between a and b.
- For x³ + x − 4 = 0: when x = 1 the value is −2; when x = 2 it is 6. The sign changes, so there is a solution between 1 and 2.
Iterative formulae
- Rearrange the equation into the form x = …, for example x³ − 2x − 5 = 0 becomes x³ = 2x + 5, so x = ∛(2x + 5).
- This gives the iterative formula xₙ₊₁ = ∛(2xₙ + 5). Put in the starting value x₀ to get x₁, put x₁ in to get x₂, and so on.
- With x₀ = 2: x₁ = 2.0801, x₂ = 2.0924, x₃ = 2.0942 (4 d.p.). The values settle towards the solution.
Calculator method
- Type the starting value and press =. Then type the formula using the ANS key in place of xₙ. Each press of = gives the next value.
- Keep the full calculator values and only round your final answer.
- If the values settle to a fixed number, the iteration converges. If they spread out, it does not.
Key terms
- Iteration
- Repeating a process, using each output as the next input.
- Iterative formula
- A formula such as xₙ₊₁ = ∛(2xₙ + 5) that gives the next value from the previous one.
- Converge
- Settle towards a fixed value.
- Change of sign
- When a value goes from negative to positive, or positive to negative, showing a solution in between.
- Starting value
- The first value x₀ used in an iteration.