Working with functions: algebra, linear, quadratic and reciprocal functions

Almost every later topic in Mathematics Advanced depends on fluent algebra, so this section gathers the core techniques: index laws, surds and the three methods for solving quadratic equations. Treat these as tools you should be able to use quickly and accurately without a calculator.

Index laws. For a base a > 0 and any real indices m and n: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ and a⁰ = 1. A negative index means a reciprocal, a⁻ⁿ = 1/aⁿ, and a fractional index means a root, a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ.

When evaluating a fractional power by hand, take the root first so the numbers stay small: 27^(2/3) = (∛27)² = 3² = 9. When simplifying expressions with different bases, rewrite each base as a power of a common prime where possible.

Expanding and factorising. Expanding uses the distributive law; useful patterns are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and (a + b)(a − b) = a² − b². Factorising reverses this. Always look for a common factor first, then a difference of two squares, then a quadratic trinomial, and for four terms try grouping in pairs.

Sums and differences of cubes are also useful: a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Surds. A surd is an irrational root such as √2 or ∛5. Simplify by extracting the largest square factor: √72 = √36 × √2 = 6√2. Like surds can be added (3√2 + 5√2 = 8√2), and products follow √a × √b = √(ab). Expand surd expressions exactly as you would algebraic brackets. To rationalise a denominator of the form √a, multiply top and bottom by √a.

For a binomial denominator such as a + √b, multiply by its conjugate a − √b, because (a + √b)(a − √b) = a² − b is rational.

Solving quadratics. Write the equation as ax² + bx + c = 0 first. Then choose a method: factorise if the roots look rational; complete the square by halving the coefficient of x and adding its square; or use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, which always works.

The discriminant Δ = b² − 4ac tells you the nature of the roots without solving: Δ > 0 gives two distinct real roots (rational if Δ is a perfect square), Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.