Quadratic Equation Solver

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0, and every one of them is solved by the same recipe: compute the discriminant D = b² − 4ac, then apply the quadratic formula x = (−b ± √D)/(2a). This solver performs that recipe for your coefficients and writes out each step — the substituted discriminant, the formula with your numbers in it, the simplified radical and the final roots — so you can follow the method, not just copy an answer.

The arithmetic is exact. Coefficients are handled in arbitrary-precision integer math (decimals are scaled to whole numbers first), so the discriminant is never off by a rounding error. When D is a perfect square the roots come out as reduced fractions and the trinomial is factored — x² − 3x + 2 = 0 yields x = 1 and x = 2 with (x − 1)(x − 2). When D is positive but not a perfect square, the radical is simplified and common factors are cancelled, giving textbook forms such as x = (1 ± √5)/2 for x² − x − 1 = 0. When D is negative you get the complex conjugate pair a ± bi, like −1 ± 2i for x² + 2x + 5 = 0. Decimal approximations are printed alongside every exact answer.

Beyond the roots, the solver reports the parabola's vertex at (−b/2a, −D/4a) together with the vertex form y = a(x − h)² + k, and states how the sign of the discriminant classifies the equation: two distinct real roots, one double root, or no real roots. That makes it useful for graphing homework and analysis questions, not only for root finding.

Everything runs locally in your browser: nothing is uploaded, there is no registration, and you can solve as many equations as you like on desktop and mobile alike. Related tools on this site pick up neighbouring jobs — the ratio calculator for proportions and the fraction calculator for plain fraction arithmetic.