Quadratic Equation Calculator

About Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The solutions (roots) represent the x-values where the parabola y = ax² + bx + c crosses the x-axis. The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) solves any quadratic equation, regardless of whether the roots are rational, irrational, or complex.

The discriminant Δ = b² − 4ac tells you the nature of the roots without fully solving: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one repeated root; if Δ < 0, the roots are complex conjugates of the form p ± qi. Complex roots occur when the parabola does not intersect the x-axis at all.

The vertex of the parabola occurs at x = −b/(2a), y = −Δ/(4a). If a > 0 the parabola opens upward and the vertex is the minimum; if a < 0 it opens downward and the vertex is the maximum. Quadratic equations model projectile motion, profit maximization, signal processing, and many other real-world phenomena.