Quadratic Equation Solver
Solve ax² + bx + c = 0 for real or complex roots. Shows the discriminant, the vertex, and the full quadratic-formula working.
How to use it
- Enter the coefficients a, b and c.
- The discriminant tells you what to expect: positive gives two real roots, zero gives one, negative gives a complex pair.
- The vertex and axis of symmetry are shown for sketching the parabola.
Examples
| a=1, b=-3, c=2 | x = 2 and x = 1 |
For any equation of the form ax² + bx + c = 0, this solves for x using the quadratic formula, and shows the discriminant, the vertex and the axis of symmetry alongside the roots — enough to sketch the parabola, not just read off two numbers.
The discriminant decides what kind of answer you get
b² − 4ac is calculated first because its sign tells you what to expect before the roots are even worked out. A positive discriminant means two distinct real roots — the parabola crosses the x-axis twice. Zero means exactly one real root (a repeated root), where the parabola just touches the x-axis at its vertex. A negative discriminant means no real roots — the parabola never crosses the x-axis at all, and the two solutions are a complex conjugate pair, shown here rather than hidden as an error.
Vertex and axis of symmetry
The axis of symmetry sits at x = −b ÷ 2a, and plugging that back in gives the vertex — the parabola's minimum (if a is positive) or maximum (if a is negative). This is the point graphing calculators and textbooks usually ask for separately, and it's included here automatically since it comes from the same coefficients as the roots.
Last updated 17 August 2026