Solve any quadratic equation ax² + bx + c = 0 — find real and complex roots with step-by-step working.
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How This Works
Quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a. Discriminant (b²−4ac): if positive, two distinct real roots; if zero, one repeated root; if negative, two complex conjugate roots. The vertex of the parabola is at x = −b/2a, y = c − b²/4a.
Complex roots appear in physics and engineering whenever systems exhibit oscillatory behaviour — electrical circuits (AC), spring-mass-damper systems, quantum mechanics, signal processing. In electrical engineering, complex numbers (as impedance) allow circuit analysis. The imaginary unit i represents a 90° phase shift in AC circuits. Despite the name "imaginary," complex numbers describe real physical phenomena.