Quadratic Formula Calculator

Mastering Quadratic Equations: Complete Guide

A quadratic equation is a second‑degree polynomial of the form ax² + bx + c = 0 (a ≠ 0). It appears throughout physics, engineering, finance, and optimization problems. This Professional Quadratic Formula Calculator solves any quadratic equation, providing the roots, discriminant, vertex, axis of symmetry, and an interactive graph—all with detailed step‑by‑step explanations.

The Quadratic Formula

The universal solution to any quadratic equation is given by the quadratic formula: x = [−b ± √(b² − 4ac)] / (2a). The expression under the square root, Δ = b² − 4ac, is called the discriminant and determines the nature of the roots.

The Discriminant and Nature of Roots

• Δ > 0: Two distinct real roots. The parabola crosses the x‑axis at two points.
• Δ = 0: One repeated real root (a double root). The parabola touches the x‑axis at the vertex.
• Δ < 0: Two complex conjugate roots. The parabola does not intersect the x‑axis.

Vertex and Axis of Symmetry

The vertex is the turning point of the parabola. Its x‑coordinate is x = −b/(2a). Substituting back gives the y‑coordinate. The axis of symmetry is the vertical line through the vertex. If a > 0, the parabola opens upward (vertex is a minimum); if a < 0, it opens downward (vertex is a maximum).

Vieta's Formulas

For roots r₁ and r₂, the sum is r₁ + r₂ = −b/a and the product is r₁·r₂ = c/a. These relationships provide a quick check of your solutions and are useful in symmetric root problems.