Quadratic Formula Calculator (ax² + bx + c = 0)

Quadratic Equations Guide

The Quadratic Formula

x = [−b ± √(b² − 4ac)] / (2a). For the equation ax² + bx + c = 0. The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0: two distinct real roots. Δ = 0: one repeated real root (touches the x-axis). Δ < 0: no real roots (complex conjugate pair). Example: 2x² − 3x − 2 = 0. a=2, b=−3, c=−2. Δ = 9 − 4×2×(−2) = 9 + 16 = 25. x = (3 ± 5)/4. x₁ = 8/4 = 2. x₂ = −2/4 = −0.5. Check: 2(4) − 3(2) − 2 = 8 − 6 − 2 = 0 ✓.

Methods for Solving Quadratics

Factorisation: fastest if integer factors exist. Find two numbers that multiply to ac and sum to b. 2x² − 3x − 2: ac = −4, numbers: −4 and +1... actually +4 and −1. Hmm. 2x² + x − 4x − 2 = x(2x+1) − 2(2x+1) = (x−2)(2x+1) = 0. x = 2 or x = −½. Completing the square: always works. Add and subtract (b/2a)². Useful for finding the vertex. Quadratic formula: works for all quadratics including those with non-integer roots. Best for examination use when other methods are unclear. Graph: parabola — y =

Vertex and Parabola Shape

Vertex (turning point): x-coordinate = −b/(2a). y-coordinate: substitute x back into equation. y = c − b²/(4a). For 2x² − 3x − 2: x = 3/4 = 0.75. y = 2(0.75)² − 3(0.75) − 2 = 1.125 − 2.25 − 2 = −3.125. Vertex = (0.75, −3.125). The parabola opens upward (a > 0 — minimum). Opens downward if a < 0 (maximum). Axis of symmetry: x = −b/(2a). Line of symmetry of the parabola.

Applications of Quadratics

Projectile motion: h = ut + ½at². Quadratic in t — solve for time of flight or time to reach height. Revenue maximisation: R = p × q(p) where q depends on price p — often quadratic. Area problems: field with fixed perimeter — area is a quadratic function of one dimension. Optimise by finding vertex. Pythagorean triples: finding integer solutions to a² + b² = c² involves quadratic Diophantine equations. Physics: oscillations, wave equations, electrical circuits with capacitors and inductors all p