Discriminant Calculator
b² − 4ac tells you how many roots a quadratic has.
Enter your values
ⓘ Live results
Results
- Discriminant Δ
- 1
- Nature of roots
- Two distinct real roots
- Roots
- 2 and 3
- Perfect square?
- Yes — the quadratic factors over the rationals
- Vertex x
- 2.5
Quick answer
The discriminant b² − 4ac decides the roots: positive means two real roots, zero means one repeated root, negative means two complex roots.
Δ = b² − 4ac
How to use the Discriminant Calculator
- 1Enter the coefficients a, b and c.
- 2Read the discriminant's sign to see the root count.
- 3Check whether it is a perfect square to know if the quadratic factors neatly.
Worked example
Using the values the calculator loads with:
Inputs
- a1
- b-5
- c6
Results
- Discriminant Δ1
- Nature of rootsTwo distinct real roots
- Roots2 and 3
- Perfect square?Yes — the quadratic factors over the rationals
- Vertex x2.5
Frequently asked questions
What does a discriminant of zero mean graphically?
The parabola just touches the x-axis at its vertex — a double root.
Does the discriminant work for cubics?
There is a cubic discriminant, but the b² − 4ac form applies only to quadratics.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.