Quadratic Formula Calculator

Solve ax² + bx + c = 0 for x. Returns real or complex roots plus the discriminant and parabola vertex.

Solution

Roots

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Discriminant (b² − 4ac)

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Nature

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Vertex

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The formula

x = (−b ± √(b² − 4ac)) / (2a)

What the discriminant tells you

b² − 4ac > 0 — two distinct real roots; the parabola crosses the x-axis twice.
b² − 4ac = 0 — one repeated real root; the parabola touches the x-axis at exactly one point.
b² − 4ac < 0 — two complex conjugate roots; the parabola never touches the x-axis.

If a = 0

The equation becomes linear (bx + c = 0) — one root: x = −c/b. The calculator handles this case automatically.