Quadratic Equation Solver — Quadratic Formula Calculatorax² + bx + c = 0 · x = (−b ± √(b²−4ac)) / 2a · Real & Complex Roots
Use this free Quadratic Equation Solver to instantly find all roots of any quadratic equation in the standard form ax² + bx + c = 0 using the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Enter your three quadratic coefficients — a (leading coefficient) · b (linear coefficient) · c (constant term) — to automatically compute: both roots (x₁ and x₂), discriminant value (Δ = b² − 4ac), vertex coordinates (h, k), axis of symmetry (x = −b/2a), and nature of roots — two distinct real roots (Δ > 0) · one repeated real root (Δ = 0) · two complex conjugate roots (Δ < 0) — with full step-by-step working shown.
This quadratic formula calculator is trusted across a wide range of academic and professional applications: GCSE, A-Level, SAT, ACT, GRE, JEE, and NEET algebra exam preparation, university calculus and polynomial root finding, physics — projectile motion and kinematic equations, engineering — stress analysis and structural load equations, economics — revenue maximization and cost function analysis, and computer graphics — parabolic curve and trajectory rendering. Beyond just solving ax² + bx + c = 0, this tool also identifies the parabola vertex, x-intercepts (zeros), and y-intercept — making it a complete quadratic function analyzer for both real and complex number solutions.
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Quadratic Solver — Real and Complex Roots With Full Discriminant Analysis
The quadratic equation ax² + bx + c = 0 has roots given by the quadratic formula x = (-b ± √(b²-4ac)) / 2a. The discriminant b²-4ac determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated real root (tangency), negative gives two complex conjugate roots. A ball thrown upward with initial velocity 20 m/s from height 1.5m reaches zero height when -4.9t² + 20t + 1.5 = 0, giving t = 4.15 seconds (landing) and t = -0.074 seconds (unphysical negative time). The quadratic solver computes both roots and identifies the physical solution.
Completing the square — the algebraic technique that the quadratic formula is derived from — reveals the vertex of the parabola y = ax² + bx + c. The vertex (h, k) where h = -b/2a and k = c - b²/4a is the minimum (if a > 0) or maximum (if a < 0) of the parabola. For profit maximization, cost minimization, or optimum angle problems, the vertex gives the optimal x value directly. The calculator reports the vertex coordinates alongside the roots so optimization problems are fully solved.
Complex roots occur when the discriminant is negative. For ax² + bx + c = 0 with negative discriminant, the roots are x = (-b ± i√(4ac-b²)) / 2a — complex conjugate pairs. These roots have no real-number meaning but are essential in electrical engineering (impedance), control systems (damping), and signal processing (filter poles and zeros). The calculator presents complex roots in standard a ± bi form and confirms they are conjugate pairs, which is a consistency check for the calculation.