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Discriminant Calculator

Calculate Δ = b² - 4ac and classify quadratic roots

a =

b =

c =

Discriminant Formula

Δ = b² - 4ac

The discriminant is the expression under the square root in the quadratic formula. Its value tells you what kind of solutions the equation has before you calculate the roots. A positive discriminant means the square root is real and nonzero, zero means the square root disappears, and a negative discriminant means real-number solutions are not available. This makes Δ a fast diagnostic for quadratic equations.

⚠Note: a must not be 0 for a quadratic equation.

What the Discriminant Tells You

The discriminant is the part under the square root in the quadratic formula. Its sign tells you the number and type of solutions before solving the equation.

Positive

Two distinct real roots.

Zero

One repeated real root at the vertex of the parabola.

Negative

No real roots; the roots are complex conjugates.

Graph Meaning

The sign tells whether the parabola crosses, touches, or misses the x-axis.

💡 Example: For x²-5x+6=0, Δ=(-5)²-4×1×6=1, so there are two real roots.

Applications

QuadraticsRoot TypeGraphing

Frequently Asked Questions

What is a discriminant calculator?▼

A discriminant calculator computes Δ=b²-4ac for ax²+bx+c=0 and identifies the type of roots.

What does Δ greater than 0 mean?▼

If Δ>0, the quadratic equation has two distinct real roots.

What does Δ equal to 0 mean?▼

If Δ=0, the quadratic equation has one repeated real root.

What does Δ less than 0 mean?▼

If Δ<0, the quadratic equation has no real roots and has two complex conjugate roots.

Where is the discriminant used?▼

The discriminant is used in quadratic equations, graphing parabolas, factoring, and checking root behavior before solving.

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