Quadratic Equation Solver

x = (−b ± √(b² − 4ac)) / 2a. It gives both roots of ax² + bx + c = 0; the ± produces the two solutions.

The discriminant is b² − 4ac. If it is positive there are two real roots, if zero there is one repeated real root, and if negative there are two complex-conjugate roots.

Then it is not quadratic but linear: bx + c = 0 with the single solution x = −c/b. The tool detects this and solves the linear case instead.

The parabola y = ax² + bx + c has its turning point (vertex) at x = −b/2a, and that vertical line is the axis of symmetry. The vertex y-value is the minimum or maximum of the parabola.

Not quite, and deliberately. Computing both roots as (−b ± √D) / 2a is unstable: when b² dominates 4ac, one of the two roots subtracts nearly equal quantities and loses most of its significant digits. For x² + 10,000,000x + 12,345 = 0 the textbook form gives a small root of −0.001234 where the true value is −0.001235. This solver takes the root whose terms have the same sign and derives the other from the product of roots, which keeps both accurate; on ordinary coefficients the two approaches agree exactly.

They are Vieta's formulas — the roots always sum to −b/a and multiply to c/a — and they are shown because they are a free check that costs no extra arithmetic: add and multiply the two roots yourself, and if either disagrees with the value on the card, the coefficients are badly conditioned and the roots are not trustworthy to the digits displayed. They also run backwards: to build a quadratic with roots you have chosen, write x² − (their sum)x + (their product) = 0.