# Discriminant Calculator

URL: https://speedycalculator.com/calculators/math/discriminant/
Last updated: 2026-10-30

> The discriminant calculator computes the discriminant of a quadratic, cubic or quartic polynomial and explains what it says about the roots.

## Inputs

- Polynomial degree
- a₄ (x⁴)
- a₃ (x³)
- a₂ (x²)
- a₁ (x)
- a₀

## Results

- Discriminant
- What it means
- Number of distinct real roots

## Formula

```
Quadratic: Δ = b² − 4ac
Cubic: Δ = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd
Quartic: the degree-4 formula in a, b, c, d, e
```

## Explanation

The discriminant is a single number built from the coefficients that tells you, without solving, whether a polynomial has repeated roots and how many of its roots are real. For quadratics it is the expression under the square root in the quadratic formula.

### How to use the Discriminant Calculator

- Choose the **degree** of each polynomial – the matching coefficient fields appear.
- Enter the **coefficients** from the highest power down; leave a field at 0 for a missing power.
- Read the discriminant and its interpretation.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The calculator evaluates the standard discriminant formula for the chosen degree and also counts the real roots numerically, so you can see the classification in action.

## Worked example

Inputs — Polynomial degree: 2; a₂ (x²): 1; a₁ (x): -3; a₀: 2.

Results — **Discriminant = 1**; What it means = Positive: two distinct real roots; Number of distinct real roots = 2.

## Key terms

- **Discriminant**: A function of the coefficients that is zero exactly when the polynomial has a repeated root.

## FAQ

**What does a negative discriminant mean for a quadratic?**
The two roots are complex conjugates; the parabola does not cross the x-axis.

**What does a zero discriminant mean?**
The polynomial has a repeated root.

**Is there a discriminant for higher degrees?**
Yes, every polynomial has one, but the formulas grow quickly – the quartic one has 16 terms.

**Why does the cubic rule differ from the quadratic?**
For a cubic, Δ > 0 means three real roots and Δ < 0 means one real root.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [Wolfram MathWorld – Polynomial discriminant](https://mathworld.wolfram.com/PolynomialDiscriminant.html)