# Quadratic Formula Calculator

> Solve any quadratic equation with the quadratic formula: real or complex roots, the discriminant and the vertex of the parabola.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/quadratic-formula-calculator

Enter a, b and c from ax² + bx + c = 0. The discriminant decides whether there are two real roots, one repeated root or two complex roots.

Also called: Quadratic equation solver, Quadratic solver.

## Inputs
- a (coefficient of x²) (default 1)
- b (coefficient of x) (default -5)
- c (constant) (default 6)

## Results
- Solutions
- Type of roots
- Discriminant (b² − 4ac)
- x₁
- x₂
- Vertex of y = ax² + bx + c

## Formula

```
x = (−b ± √(b² − 4ac)) ÷ 2a;  Discriminant D = b² − 4ac;  Vertex = (−b ÷ 2a, c − b² ÷ 4a)
```

Where:
- a, b, c: the coefficients, with a ≠ 0
- D: discriminant: > 0 two real roots, = 0 one repeated root, < 0 two complex roots

## Worked example
1. D = (−5)² − 4 × 1 × 6 = 25 − 24 = 1.
2. x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, so x = 3 or x = 2.
3. Check: (x − 3)(x − 2) = x² − 5x + 6.

**Result:** x² − 5x + 6 = 0 has two real solutions, x = 3 and x = 2.

## What the result means

The real roots are where the parabola y = ax² + bx + c crosses the x-axis. Complex roots mean it never crosses; the vertex is then the closest it gets.

## Assumptions and limits
- Coefficients are real numbers and a is not 0.
- Roots are shown to 10 significant digits. The calculator uses the numerically stable form of the formula, so a tiny root next to a huge one is still accurate.

## Questions

### What if a is 0?

Then there is no x² term and the equation is linear: bx + c = 0 gives x = −c ÷ b.

### What does i mean in the answer?

i is the imaginary unit, √−1. A negative discriminant gives two complex roots p + qi and p − qi, which are conjugates of each other.

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Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/quadratic-formula-calculator