x² Quadratic Formula Calculator
Solve ax² + bx + c = 0 for real or complex roots. Also shows the discriminant and the vertex of the parabola.
Coefficients
Result
The Quadratic Formula
Discriminant: Δ = b² − 4ac
Vertex: x = −b / 2a, y = f(−b / 2a)
The discriminant determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one repeated real root; if it's negative, the roots are a complex conjugate pair.
Worked Example
The calculator loads with a = 1, b = −5, c = 6, for the equation x² − 5x + 6 = 0. The discriminant is Δ = (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, so there are two distinct real roots: x = (5 ± √1) / 2, giving x = 3 and x = 2. The vertex of the parabola is at x = −b/2a = 2.5, y = f(2.5) = (2.5, −0.25).
Frequently Asked Questions
What happens when the discriminant is negative?
When b² − 4ac is negative, the equation has no real roots — instead it has two complex conjugate roots of the form p ± qi.
What if a = 0?
If a = 0, the equation is not quadratic (it becomes linear: bx + c = 0). The calculator requires a non-zero value for a.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola y = ax² + bx + c, located at x = −b/2a. It represents the equation's axis of symmetry.