Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Returns roots, discriminant, vertex and axis of symmetry.
Written by Rahman · Last reviewed
Quick answer
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac decides what you get: positive gives two distinct real roots, zero gives one repeated root, and negative gives a pair of complex conjugates with no real solution.
Solve a quadratic equation of the form ax² + bx + c = 0.
Results are estimates for general guidance only. Double-check any figure you rely on for an important decision.
About the Quadratic Equation Solver
A quadratic equation has the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The solutions (called roots) are the x-values that make the equation true. Depending on the discriminant (b² − 4ac), there may be two real roots, one repeated real root, or two complex roots.
This solver applies the quadratic formula and returns the roots, the discriminant, the vertex of the parabola y = ax² + bx + c, and the axis of symmetry. It also handles edge cases where a = 0 (degenerate linear or constant equations).
x = ( −b ± √(b² − 4ac) ) / (2a)
If b² − 4ac > 0 → two real roots. If = 0 → one (double) root. If < 0 → two complex roots.
Frequently Asked Questions
What is the discriminant?
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The expression b² − 4ac under the square root in the quadratic formula. Its sign tells you what kind of roots to expect without solving the equation.
What if a = 0?
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Then it is no longer a quadratic: it becomes a linear equation bx + c = 0 (if b ≠ 0) or a constant equation. The calculator handles both cases gracefully.
What is the vertex of a parabola?
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The lowest or highest point on the curve y = ax² + bx + c. If a > 0 the vertex is a minimum; if a < 0 it is a maximum. The x-coordinate is −b/2a.
How do complex roots work?
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When the discriminant is negative, the square root is imaginary. The roots come in conjugate pairs of the form p + qi and p − qi, where i = √(−1).