# Quadratic Equation Calculator

- **URL:** https://codeasystem.com/calculators/math/quadratic-equation-calculator/
- **Category:** math
- **Description:** Solve ax² + bx + c = 0 for real or complex roots, showing the discriminant and vertex so the answer never arrives unexplained.
- **Primary output:** Two real roots: x = 3 and x = 2

## Inputs
- Coefficient a (x²) (name: `a`, type: number, example: 1)
- Coefficient b (x) (name: `b`, type: number, example: -5)
- Constant c (name: `c`, type: number, example: 6)

## Outputs
- Two real roots: x = 3 and x = 2
- Discriminant (b² − 4ac): 1
- Axis of symmetry (x = −b/2a): 2.5

## Formula / methodology
```
x = (−b ± √(b² − 4ac)) / 2a     Discriminant D = b² − 4ac
```

The discriminant b² − 4ac is computed first because it classifies everything: positive yields two distinct real roots via the quadratic formula, zero collapses to one repeated root at −b/2a, and negative produces a complex conjugate pair written a ± bi. The axis of symmetry row shows where the parabola pivots.

## Assumptions & limitations
- Coefficient a must be non-zero or the equation is not quadratic.
- Complex results are formatted as x = real ± imaginary·i.
- Roots display up to six decimal places.

## How to use
1. **Enter all three coefficients**; Include signs: x² − 5x + 6 means a=1, b=−5, c=6.
2. **Check the discriminant row**; It predicts two roots, one double root, or a complex pair before you read them.
3. **Read the roots**; Real solutions appear as plain numbers; complex ones as real ± imaginary·i.

## Example
Solving x² − 5x + 6 = 0 gives discriminant 1 and two real roots: x = 3 and x = 2; it factors as (x−3)(x−2).

Result for these inputs:

```
Two real roots: x = 3 and x = 2
```

## About this calculator
### The discriminant is the story

Before any root appears, b² − 4ac already tells you whether the parabola slices the x-axis twice, kisses it once, or floats entirely above or below. Engineers use it to predict feasibility; students use it to check their algebra before it goes wrong.

### What complex roots mean physically

A negative discriminant does not make the problem unsolvable; it means no real number satisfies it. In oscillating systems those complex parts encode damping and frequency, which is why electrical engineers treat complex roots as features rather than failures.

### Why a = 0 breaks everything

The quadratic formula divides by 2a, so zero there is undefined. Worse, without an x² term the graph is a line with exactly one root −c/b; a different tool for a different shape, and the calculator says so instead of erroring cryptically.

## FAQs
### How do I enter a missing term?

Enter 0 for that coefficient; e.g. x² − 9 = 0 uses a=1, b=0, c=−9.

### What does “double root” mean?

The parabola touches the x-axis at exactly one point, so both formula branches produce the same value.

### Can I get decimal roots?

Yes; irrational roots render to six decimals, enough for checking homework or engineering estimates.

### Why is a=0 rejected?

Without the squared term the equation is linear, and the quadratic formula would divide by zero.

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Last updated: 2026-08-23 · Version: 1.0.0 · [HTML version](https://codeasystem.com/calculators/math/quadratic-equation-calculator/)