Iteration repeatedly uses a formula to improve or generate values. xₙ₊₁ is the next value calculated from the current value xₙ; x₀ is the starting value.
To solve x2 = x + 2, one possible iteration is xₙ₊₁ = √(xₙ + 2). A rearrangement is chosen, then the result is fed back as the next input.
Worked example: With x₀ = 1, the successive values are x₁ ≈ 1.7321, x₂ ≈ 1.9319 and x₃ ≈ 1.9829; they approach the positive root 2. Keep full calculator precision between iterations.
Convergence means approaching a single fixed value; divergence means not approaching one. Values can also oscillate, moving back and forth, either towards a limit or without settling. A different rearrangement or starting value may behave differently, so do not assume repeated substitution always solves the equation.
The square-root iteration above cannot find the negative root −1. Always consider the domain and whether other roots may exist.
If asked for a value to a given number of decimal places, check that successive calculated values stabilise at that precision; this is numerical evidence, not a guarantee for every iteration scheme.
To locate a root by a sign change, use a function whose graph has no break or jump between the trial x-values. Find function values on opposite sides of zero. For f(x) = x3 − 2, f(1) = −1 and f(2) = 6, so at least one root lies between 1 and 2.
Refine the interval (or bracket) between the trial inputs using closer trial values. f(1.25) = −0.046875 and f(1.26) = 0.000376, so the positive root lies between 1.25 and 1.26. State your interval and show the signs.
Higher — quadratic inequalities
Find the roots of the associated quadratic equation, then inspect where the graph is above or below the x-axis. The roots divide the number line into intervals. Negative part of x squared minus five x plus six
Worked example: x2 − 5x + 6 < 0 has roots 2 and 3. The upward-opening parabola is negative between the roots, so 2 < x < 3.
For x2 − 5x + 6 ≥ 0, the solution is x ≤ 2 or x ≥ 3. There are two separate allowed intervals; joining them with “and” would be impossible.
With ≤ or ≥ include the roots; with < or > exclude them. Use filled/open endpoints accordingly.
If the leading coefficient is negative, the sign pattern reverses. For 6 − x − x2 > 0, roots are −3 and 2, and the allowed interval is −3 < x < 2.
Check one test value in each interval if unsure. Solving only the boundary equation does not answer an inequality question.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
Mind map
Use the branches to recall the ideas and explain their connections. Check the revision notes for the full detail.
View M20 mind mapOpen the full-size map to zoom. Use the printable view for A4 landscape or enlarge to A3.
Mr Kidman's Classroom
