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
Roots
x = 2, x = 3
Two real roots (discriminant > 0)
Discriminant (b²−4ac)
–
Vertex
–
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The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Discriminant: Δ = b² − 4ac
Vertex: x = −b / 2a, y = f(−b / 2a)
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.
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.