Quadratic Equation Solver

A quadratic equation solver is a mathematics tool that finds the roots (solutions) of any equation in the standard form ax² + bx + c = 0, where a, b, and c are real-number coefficients. Quadratic equations appear throughout algebra, physics, engineering, and finance — from calculating projectile trajectories and optimising profit functions to finding intersection points of parabolas.

This solver works through the equation using three methods and displays the full working for each, making it a useful learning aid as well as a quick computation tool. The quadratic formula method, x = (-b ± √(b² - 4ac)) / 2a, always works regardless of the discriminant value and handles all real and complex root scenarios. Factoring attempts to express the equation as (px + q)(rx + s) = 0 when integer or simple rational factors exist, which is the technique most commonly taught in secondary school algebra. Completing the square rewrites the equation in the form (x + h)² = k, providing geometric insight into the vertex of the parabola.

The discriminant (b² - 4ac) determines the nature of the roots: a positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex conjugate roots. The solver displays the discriminant value and its interpretation alongside all three solution methods, helping students understand why the method produces the result it does.