Quadratic equation solver

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.