# Quadratic Equation Solver — Roots, Discriminant & Graph

> Solve ax²+bx+c=0 with step-by-step working: real or complex roots, discriminant, vertex, axis of symmetry, and a plotted parabola. Free quadratic formula calculator.

- **Category:** Math
- **Interactive calculator:** https://youcalc.com/en/math/quadratic-equation/
- **Price:** Free, no sign-up required

## Overview

This calculator solves any quadratic equation of the form ax²+bx+c=0, finding real or complex roots via the quadratic formula. Enter the three coefficients and instantly see the roots, discriminant, vertex, axis of symmetry, and a plotted parabola.

## How to read your result

The result card shows the roots at the top — either two distinct real values, one repeated root, or a complex conjugate pair. Below the roots you will find the discriminant (which tells you the root type), the vertex coordinates, the axis of symmetry, and the y-intercept. A small parabola chart plots the curve, marking real x-intercepts as filled dots and the vertex as an open circle. The step-by-step breakdown underneath shows each stage of the quadratic formula.

## Method

The roots are found using the quadratic formula x = (−b ± √(b²−4ac)) / (2a), as documented by Wolfram MathWorld and Khan Academy. The discriminant D = b²−4ac is computed first; its sign determines whether the square root is real or imaginary. The vertex is (−b/2a, f(−b/2a)), and the axis of symmetry is x = −b/2a. The y-intercept is always c, and real x-intercepts are reported only when D ≥ 0.

## Example

- **Setup:** Enter a = 1, b = −5, c = 6 (solving x² − 5x + 6 = 0).
- **Result:** The discriminant is 1 (positive), so there are two distinct real roots: x₁ = 3 and x₂ = 2. The vertex sits at (2.5, −0.25) and the axis of symmetry is x = 2.5. The parabola crosses the x-axis at both roots.

## Frequently asked questions

### What does the discriminant tell me?

The discriminant D = b²−4ac determines how many real roots the equation has. When D is positive the parabola crosses the x-axis twice; when D equals zero it just touches it at one repeated root; when D is negative the roots are complex numbers and the parabola never crosses the x-axis.

### What are complex roots and when do they appear?

Complex roots appear when the discriminant is negative. They come in conjugate pairs of the form p ± qi, where i is the imaginary unit. Although they are not visible as x-intercepts on the real plane, they are still valid solutions to the equation.

### Can I use this for equations where a is not 1?

Yes. Enter any non-zero value for a. The calculator applies the full quadratic formula x = (−b ± √(b²−4ac)) / (2a), so coefficients like 2, −3, or 0.5 work just as well as 1.

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## Sources

- https://mathworld.wolfram.com/QuadraticFormula.html
- https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-formula-a1/a/quadratic-formula-explained-article

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Interactive version: https://youcalc.com/en/math/quadratic-equation/ · From YouCalc — https://youcalc.com