Quadratic formula calculator

Using the quadratic formula

For an equation ax² + bx + c = 0 with a not equal to zero, the solutions are x = (−b ± √(b² − 4ac))/(2a). The expression b² − 4ac is the discriminant. This calculator substitutes the entered coefficients, evaluates the discriminant, and reports both roots.

What the discriminant tells you

A positive discriminant produces two distinct real roots. A zero discriminant produces one repeated real root. A negative discriminant produces a conjugate pair of complex roots, shown with the imaginary unit i. Very small values near zero are normalized to reduce misleading floating-point noise.

Vertex and axis of symmetry

The parabola’s axis of symmetry is x = −b/(2a). Substituting that x-coordinate into the quadratic gives the vertex y-coordinate. When a is positive the vertex is a minimum; when a is negative it is a maximum. Decimal results use finite floating-point approximations rather than symbolic radicals.

The coefficient a must be nonzero; otherwise the equation is linear. Decimal rounding can hide tiny numerical differences, especially with extremely large, small, or nearly cancelling coefficients.