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.
Verify the roots by substitution
For x² − 5x + 6 = 0, the discriminant is 25 − 24 = 1. The formula gives (5 ± 1) ÷ 2, so the roots are 3 and 2. Substitute them into the original equation: 3² − 5·3 + 6 = 0 and 2² − 5·2 + 6 = 0. This check detects a copied sign or coefficient error that a plausible-looking decimal answer may hide.
Read roots, intercepts, and vertices separately
The roots are the x-values where the quadratic equals zero; they are not the vertex unless a repeated root makes those points coincide. For the same example, the axis is x = 2.5 and the vertex is (2.5, −0.25). If the equation originally came from a model, a mathematically valid root may still be outside its allowed domain, such as negative time or length. Retain the original equation and units, then reject or interpret solutions using the model’s constraints.
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.