# Quadratic Equation Solver

URL: https://speedycalculator.com/calculators/math/quadratic-equation/
Last updated: 2026-10-25

> The quadratic equation solver finds the roots of ax² + bx + c = 0 with the quadratic formula. It shows the discriminant, which tells you whether there are two real roots, one repeated root or two complex roots, plus both roots and the vertex of the parabola.

## Inputs

- a (x² coefficient)
- b (x coefficient)
- c (constant)

## Results

- Discriminant (b² − 4ac)
- Root x₁ (real part)
- Root x₂ (real part)
- Imaginary part (±)
- Vertex x

## Formula

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

## Explanation

Any equation that can be rearranged to ax² + bx + c = 0 with a ≠ 0 is quadratic. The discriminant decides the type of solution before you compute it.

### Reading the discriminant

- **D > 0**: two different real roots; the parabola crosses the x-axis twice.
- **D = 0**: one repeated root; the vertex touches the axis.
- **D < 0**: two complex conjugate roots, shown as real part ± imaginary part.

### Check your answer

The roots always satisfy x₁ + x₂ = −b/a and x₁ × x₂ = c/a.

### Inputs explained

FieldWhat to entera (x² coefficient)A numberb (x coefficient)A numberc (constant)A number

### What each result means

- **Discriminant (b² − 4ac)** = [b (x coefficient)]^2 - 4×[a (x² coefficient)]×[c (constant)]
- **Vertex x** = -[b (x coefficient)] ÷ (2×[a (x² coefficient)])

### Example values

How the main result changes when **a** changes and everything else stays at the example values above:

a (x² coefficient)Discriminant (b² − 4ac)0.5290.75391491.25591.569289

## Worked example

Solve x² − 3x − 10 = 0 (a = 1, b = −3, c = −10).

D = 9 + 40 = 49, so √D = 7.

x = (3 ± 7) ÷ 2, giving **x₁ = 5** and **x₂ = −2**. Vertex at x = 1.5.

## Key terms

- **Discriminant**: b² − 4ac, which determines the number and type of roots.
- **Root**: A value of x that makes the equation equal zero.
- **Vertex**: The turning point of the parabola, at x = −b/2a.

## FAQ

**What if a = 0?**
The equation is linear, not quadratic; solve bx + c = 0 instead.

**How are complex roots shown?**
As a real part plus or minus an imaginary part.

**Can I factor instead?**
If the discriminant is a perfect square, the equation factors neatly over whole numbers.

**What formula does the Quadratic Equation Solver use?**
x = (−b ± √(b² − 4ac)) ÷ 2a.


## Sources

- [Wolfram MathWorld – Quadratic Equation](https://mathworld.wolfram.com/QuadraticEquation.html)