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