Math
Quadratic equation solver
Enter the three coefficients a, b, c — the calculator computes the discriminant, classifies the solution (two real roots, one repeated root, or complex conjugates), and returns the exact root values.
Formula
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any equation of the form ax² + bx + c = 0. The discriminant b² − 4ac tells you what kind of solution you'll get:
• > 0: two distinct real roots (parabola crosses x-axis twice) • = 0: one repeated real root (parabola touches x-axis at the vertex) • < 0: two complex conjugate roots (parabola doesn't cross x-axis)
The calculator also handles the degenerate case a = 0, which reduces to a linear equation bx + c = 0. If both a and b are zero, the equation has either no solution (c ≠ 0) or infinitely many (c = 0).
Examples
- 01x² − 5x + 6 = 0 (a=1, b=−5, c=6)→ D = 25 − 24 = 1 > 0 · x₁ = 3, x₂ = 2
- 02x² − 4x + 4 = 0 (perfect square)→ D = 16 − 16 = 0 · x = 2 (double root)
- 03x² + x + 1 = 0→ D = 1 − 4 = −3 < 0 · x = −0.5 ± 0.8660 i (complex)
- 04Linear case: 0·x² + 3x − 6 = 0→ Reduces to 3x = 6 → x = 2
FAQ
- The graph of y = ax² + bx + c is a parabola. The discriminant tells you how it intersects the x-axis: positive D = crosses twice (two roots), zero D = touches at the vertex (one root), negative D = doesn't touch (no real roots, only complex). The sign of a tells you which way it opens.
References
Note: Results are accurate to floating-point precision (~15 significant digits) for the standard formula shown above. See our terms of use for general usage information.