What does the discriminant tell you?

For D = b² − 4ac, a positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives a complex-conjugate pair.

Why must coefficient a be non-zero?

If a is zero, the x² term disappears and the equation is linear rather than quadratic.

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Enter coefficients from ax² + bx + c = 0. Results update immediately.

abc

Graph formy = x² - 3x + 2

Discriminant: 1
Real-root case: Two distinct roots
Roots: 1, 2
Vertex: (1.5, -0.25)
Axis of symmetry: x = 1.5
Parabola: Opens upward

Calculated numerically from the displayed coefficients.

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Reviewed examples

See how the discriminant changes the answer

D > 0

x² − 3x + 2 = 0

Two distinct real roots: 1 and 2. The parabola crosses the x-axis twice.

D = 0

x² + 2x + 1 = 0

One repeated root: −1. The vertex touches the x-axis at exactly one point.

D < 0

x² + 1 = 0

No real roots. The real graph does not meet the x-axis; its roots are ±i.

Interpretation

Reading the result

The vertex is the parabola’s turning point. Its x-coordinate defines the vertical axis of symmetry.

A positive a opens upward and has a minimum; a negative a opens downward and has a maximum.

Common errors

Checks before you solve

Keep coefficient signs: x² − 3x + 2 means b = −3.
Use a non-zero value for a.
Do not assume a negative discriminant means an error—it means the roots are complex.