Proof
- Deduction, exhaustion, disproof by counter-example, and proof by contradiction (assume the negation, reach an absurdity - e.g. root 2 is irrational). State assumptions and finish with a clear conclusion.
Surds and indices
- Index laws (a^m a^n = a^(m+n), a^(-n) = 1/a^n, a^(m/n) = nth root of a^m); simplify surds and rationalise denominators using the conjugate.
Quadratics and equations
- Completing the square (vertex, min/max), the quadratic formula, and the discriminant b^2 - 4ac (number of real roots).
- Solve simultaneous equations (linear + quadratic by substitution) and inequalities (linear and quadratic - sketch to identify the region; represent with interval notation).
Algebraic manipulation
- Polynomial division, the factor theorem (f(a) = 0 means (x - a) is a factor) and remainder theorem; partial fractions; simplifying algebraic fractions.
A concluding statement is required to earn full proof marks, and the discriminant is your tool whenever real roots or a line-meets-curve condition is involved.
Diagram
Identify topic
Choose theorem/formula
Rearrange or sketch
Solve algebraically
Check restrictions
Given information
Working line
Final answer
Reasonableness check