Quadratic Equation Solver

Free online quadratic equation solver. Find real and complex roots of ax² + bx + c = 0 using the quadratic formula. Shows discriminant and step-by-step results.

A quadratic equation in the form ax² + bx + c = 0 can have two, one, or no real solutions depending on the discriminant (Δ = b² − 4ac). This solver applies the quadratic formula to find all roots instantly, including complex (imaginary) results when the discriminant is negative.

The Quadratic Formula

The roots are given by x = (−b ± √(b² − 4ac)) / (2a). When Δ > 0, there are two distinct real roots. When Δ = 0, there is exactly one real root (a repeated root). When Δ < 0, both roots are complex numbers of the form p ± qi where q = √(−Δ) / (2a).

Real-world Applications

  • Projectile motion — Find when an object hits the ground (h = 0)
  • Profit maximization — Determine the optimal production quantity
  • Area problems — Calculate dimensions when the area is known
  • Geometry — Solve for unknown sides in right triangles with Pythagorean theorem extensions

Solving Quadratic Equations: Three Methods Explained

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