Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown

Quadratic functions are a cornerstone of A-Level Mathematics, appearing in everything from projectile motion to optimisation problems. This topic covers the full toolkit for analysing and solving quadratics: sketching graphs, using the discriminant to determine the nature of roots, completing the square to find turning points, and solving equations—including those where the unknown appears inside a function like sin²x or e²ˣ. Mastering these techniques is essential for calculus, sequences, and even complex numbers later in the course.

The discriminant (Δ = b² – 4ac) is a powerful shortcut: it tells you whether a quadratic has two distinct real roots, one repeated root, or no real roots. Completing the square not only reveals the vertex of the parabola but also helps solve equations and derive the quadratic formula. You'll also learn to solve disguised quadratics—for example, equations like x⁴ – 5x² + 4 = 0 or 2²ˣ – 3·2ˣ + 2 = 0—by substituting a function of the unknown (e.g., u = x² or u = 2ˣ). These skills are frequently tested in both pure and applied contexts.

In the Edexcel A-Level, this topic is assessed across all papers, often as part of larger problems involving graphs, inequalities, or modelling. A solid grasp of quadratics will make topics like differentiation, integration, and trigonometric equations much more manageable. The key is to practise fluency in all methods—factorising, completing the square, and using the formula—so you can choose the most efficient approach in an exam.