Quadratic Equation Solver

Enter the coefficients a, b, and c — you get not just the answer but the full notebook-style working: the discriminant with numbers substituted, the root formula, and a Vieta check. Incomplete equations are solved the short way, with an explanation of why it works.

Equation:

How it works. A quadratic equation ax² + bx + c = 0 is solved via the discriminant D = b² − 4ac: when D is positive there are two roots, zero — one root, negative — no real roots. The calculator shows every substitution and intermediate computation, so the output can serve as a model write-up: copy the reasoning, not just the answer.

FAQ

What is the discriminant for?

The discriminant D = b² − 4ac tells you how many roots a quadratic equation has before you compute them: D > 0 — two roots, D = 0 — one, D < 0 — no real roots. The word literally means “distinguisher” — it distinguishes these three cases.

What if the discriminant is negative?

In a school course the answer is “no real roots”, and that is a complete answer. In higher mathematics such an equation has two complex roots of the form x = −b/2a ± i·√|D|/2a — the calculator shows them too.

How do you solve incomplete quadratic equations?

Faster than with the discriminant. If c = 0, factor out x: x(ax + b) = 0, giving roots 0 and −b/a. If b = 0, express x² = −c/a and take the square root. The calculator detects incomplete equations and shows the short method.

What are Vieta's formulas and why the check?

For ax² + bx + c = 0 the sum of the roots equals −b/a and the product equals c/a. It is a quick way to verify the answer: if the sum and the product match, the roots are correct. The calculator performs this check automatically.