Math calculator

Quadratic Formula Calculator

Enter coefficients a, b and c for ax² + bx + c = 0 to see real or complex roots and the turning point.

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No account. Formula and assumptions are shown below.

Calculator inputs

Quadratic equation details

Formula & disclaimer

x = (−b ± √(b² − 4ac)) ÷ 2a; Vertex = (−b ÷ 2a, f(−b ÷ 2a))

View assumptions

Formula reviewCalcs at Hand Editorial TeamVerified 31 August 2026 · automated checks included

Three useful checks, kept deliberately short.

What this result includes

The roots are x = 2 and x = 3, and the graph crosses the x-axis at both points.

A practical example

x² − 5x + 6 = 0: Calculate the discriminant · Apply the quadratic formula · Find the vertex.

Common input mistake

The graph displays the real-valued parabola even when the roots are complex.

x = (−b ± √(b² − 4ac)) ÷ 2a; Vertex = (−b ÷ 2a, f(−b ÷ 2a))

A positive discriminant gives two real roots, zero gives one repeated real root and a negative value gives two complex roots.

Keep in mind: Results are rounded for display; retain additional precision when using the roots in later calculations.

Produced byCalcs at Hand Editorial Team

Last formula check31 August 2026

Review methodKnown examples + boundary cases