Try a short problem first. Then use the topic map to plan what to learn next.
Jump to the full topic map · Continue the four-subject practice plan
Worked example: an inequality from a square
For positive real numbers a and b, prove a/b + b/a ≥ 2. Before manipulating an inequality, check the domain: ab is positive, so multiplication by ab preserves the direction.
The desired inequality is equivalent to a² + b² ≥ 2ab, or (a − b)² ≥ 0. Every real square is nonnegative, so the proof is complete. Equality holds precisely when a = b.
The positivity assumption matters. For a = 1 and b = −1, the original expression is −2, so the claim would fail if arbitrary nonzero real numbers were allowed.
Two steps further
- If x and y are real and x + y = 10, prove xy ≤ 25. When does equality hold?
- For positive a,b,c, prove (a + b + c)(1/a + 1/b + 1/c) ≥ 9.
Solutions
1. Since (x − y)² ≥ 0, we have (x + y)² ≥ 4xy. Substitute x + y = 10 to obtain 100 ≥ 4xy. Thus xy ≤ 25, with equality exactly at x = y = 5.
2. Expand the product into 3 + (a/b + b/a) + (a/c + c/a) + (b/c + c/b). Each bracket is at least 2 by the worked example. The total is at least 9. Equality in all three brackets requires a = b = c.
These are teaching examples written for this guide, not past-paper questions or an official marking scheme.
Topic map and learning goals
| Unit | Core content | Advanced or extension content |
|---|---|---|
| Expressions and identities | Factorisation, completing the square, symmetric expressions, substitutions | Homogenisation, cyclic expressions, sum-of-squares decompositions |
| Equations and systems | Linear and quadratic equations, discriminants, parameter conditions, elimination | Root transformations, integer-root constraints, cleverly chosen substitutions |
| Polynomials | Division algorithm, remainder and factor theorems, roots and coefficients, Vieta’s formulas | Polynomial identities, integer-coefficient polynomials, interpolation and finite differences |
| Inequalities | Non-negativity of squares, AM–GM, Cauchy–Schwarz, equality conditions | Rearrangement, Chebyshev, Schur, Hölder, convexity and Jensen, smoothing |
| Sequences and finite sums | Arithmetic and geometric progressions, telescoping, induction, basic recurrences | Bounding recurrences, monotonicity, periodicity, recursively defined sequences |
| Functional equations | Substitution, finding special values, testing candidates, domain awareness | Injectivity/surjectivity arguments, composition, integer versus real domains, additional regularity assumptions |
| Complex numbers | Arithmetic, conjugates, modulus, polar form | Roots of unity and geometric applications |
| Binomial methods | Binomial theorem, coefficient identities, combinatorial interpretation | Coefficient comparison and more elaborate finite identities |
Learning outcomes: students should know why a transformation is valid, find all solutions rather than a few examples, and verify equality or exceptional cases. In functional equations, a real-valued additive function cannot simply be assumed linear without an appropriate hypothesis.
Suggested order: expressions → equations → polynomials → basic inequalities → sequences → functional equations. Introduce complex numbers after algebraic fluency. Do not make every advanced inequality a beginner prerequisite.