Questions students ask, answered step by step

Composite functions: domain, range and when fg exists

How to find fg(x) in the right order, when a composite function exists, and its domain and range, with worked examples from Cambridge exams.

How to complete the square

Complete the square step by step, including when the x² coefficient is not 1 or is negative, with worked examples and how to read the turning point.

How to factorise a cubic without a calculator

Factorise any cubic by hand: find a root with the factor theorem, divide it out, then factorise the quadratic that is left. Worked example step by step.

How to find stationary points and their nature

Find stationary points by setting dy/dx = 0, then decide whether each is a maximum or minimum with the second derivative. Worked example on a cubic.

How to find the discriminant of a quadratic

The discriminant b² − 4ac tells you how many real roots a quadratic has before you solve it. How to find it, what each sign means, and how to use it for k.

How to find the equation of a tangent to a curve

Find the equation of a tangent (and normal) to a curve in three steps: the point, the gradient from dy/dx, then y − y₁ = m(x − x₁). Worked examples included.

How to find the inverse of a function

Find the inverse of any function in three steps, including functions with a fraction, with worked examples and a check that catches mistakes.

How to find the nth term of a quadratic sequence

Find the nth term of a quadratic sequence from its second difference, in four steps, with two worked examples and a lab that shows the differences.

How to find the range of a function

Find the range of a function without a graph: quadratics by completing the square, restricted domains, fractions and square roots, with worked examples.

How to solve logarithms with different bases

Use the change of base rule to solve log equations with different bases, and solve exponential equations like 2ˣ = 7. Worked examples with checks.

How to solve quadratic inequalities

Solve quadratic inequalities in three steps: find the roots, sketch the curve, and read off where it is above or below the axis. Worked examples included.

How to solve simultaneous equations

Solve simultaneous equations by elimination, by substitution, and when one equation is quadratic. Step-by-step examples, and what the answer means on a graph.