Discriminant of a Quadratic

Also known as b² − 4ac

\Delta = b^2 - 4ac

Worked example: 3x² − 4x − 2 = 0 → Δ = 40. Press Try an example to run it live, then adjust anything.

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Where Discriminant of a Quadratic fits

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The quadratic family8 formulas

Quadratic Formula (Negative Root): Both use coefficient of x² and quadratic coefficient b.Quadratic Formula (Positive Root): Both use coefficient of x² and quadratic coefficient b.Vertex y-Coordinate of a Parabola: Both use coefficient of x² and quadratic coefficient b.Product of the Roots of a Quadratic: Both use coefficient of x² and constant term.Sum of the Roots of a Quadratic: Both use coefficient of x² and quadratic coefficient b.Vertex x-Coordinate of a Parabola: Both use coefficient of x² and quadratic coefficient b.

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Discriminant of a Quadratic explained

The discriminant is the part of the quadratic formula that lives under the square root, and its sign alone answers "how many real solutions?" without solving anything. Positive means two distinct real roots (the parabola cuts the x-axis twice), zero means one repeated root (the parabola just kisses the axis at its vertex), and negative means no real roots at all (the parabola floats entirely above or below). Worked example: 3x² − 4x − 2 = 0 gives Δ = (−4)² − 4(3)(−2) = 16 + 24 = 40, positive but not a perfect square, so two real irrational roots.

That last test is the practical one for students: Δ a perfect square (and a, b, c whole numbers) means the quadratic factors neatly over the integers, so you can skip the formula entirely. Engineers read the same quantity as a stability check: the discriminant of a system's characteristic equation decides whether a damped oscillator returns smoothly to rest or overshoots and rings. The common trap is sign bookkeeping: with c negative, −4ac becomes an addition, which is why Δ = 40 above rather than −8. Square b first, then subtract, and keep the minus sign attached to c.

Discriminant of a Quadratic formula

\Delta = b^2 - 4ac

Where

$\Delta$ = Discriminant
$a$ = Coefficient of x²
$b$ = Coefficient of x
$c$ = Constant term

In this toolkit

The quadratic family

8 formulasThe two roots, the discriminant that predicts them, the vertex of the parabola and the sum-and-product shortcuts — all from ax² + bx + c.

Revised 31 October 2026

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formula.expert. (2026, October 31). Discriminant of a Quadratic Formula & Calculator. https://formula.expert/solvers/quadratic-discriminant

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