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Quadratic equation solver

Enter a, b, and c to solve ax² + bx + c = 0 and get the discriminant, real or complex roots, the vertex, and the axis of symmetry.

Enter a, b, and c to solve the equation.

The discriminant b² − 4ac picks the branch: positive gives two real roots, zero gives one repeated root, and negative gives a complex conjugate pair. The vertex sits at x = −b/2a and the axis of symmetry is the vertical line through it. A zero discriminant is decided exactly; when it instead lands within twelve significant digits of zero, both roots are still reported with a warning that they nearly coincide, because floating-point arithmetic can blur that line. This tool refuses a = 0 rather than quietly solving a straight line, and it does not draw the parabola or factor higher-degree polynomials.
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