Revision notes, worked examples and methods for iteration and quadratic inequalities.
Notes and quizzes ready · 50 questions · Video coming soon.
This topic is Higher-only.
Revise the key ideas
Higher — iteration
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.
An iteration can converge, diverge or oscillate. 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 iterates stabilise at that precision; this is numerical evidence, not a guarantee for every iteration scheme.
To locate a continuous function's root by a sign change, find 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 bracket using 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.
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