---
title: "Quadratic equation solver — free, runs in your browser"
description: "Solve ax² + bx + c = 0, including complex roots, with the working shown. Free, no sign-up, and nothing you enter leaves your browser."
url: https://atoneplace.net/tools/quadratic-equation-solver/
site: "At One Place"
---

# Quadratic equation solver

Solve ax² + bx + c = 0, including complex roots, with the working shown.

## How it works

Solves any quadratic, showing the discriminant, both roots, the vertex and the axis of symmetry.

The discriminant b² − 4ac decides everything before you compute a root. Positive gives two distinct real roots; zero gives one repeated root, where the parabola touches the axis rather than crossing it; negative gives a conjugate pair of complex roots, meaning the curve never meets the x-axis at all.

The tool reports complex roots rather than declaring no solution, because "no real roots" and "no roots" are different statements and the complex pair is often what is actually wanted.

A note on precision: when b² is much larger than 4ac, the standard formula loses accuracy for one of the roots through catastrophic cancellation — subtracting two nearly equal numbers. This solver uses the numerically stable form for that case, which is why its answers stay correct where a naive implementation drifts.

> This tool runs entirely in your browser. Nothing you enter is sent to a server, logged or stored, and the page keeps working with the network disconnected.

## Common questions

### What does the discriminant tell you?

b² − 4ac. Positive means two real roots, zero means one repeated root, negative means two complex roots and a curve that never crosses the x-axis.

### What if a is zero?

Then it is not a quadratic but a linear equation, and it has one root at −c/b. The tool handles that case rather than dividing by zero.

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## Sources

- Calculated on this page — At One Place (/about/)