Quadratic Formula Calculator

How the Quadratic Formula Works

A quadratic equation is any polynomial of the form ax² + bx + c = 0, where a ≠ 0. The quadratic formula provides the exact solution for any such equation by expressing the roots (x-intercepts) in terms of the coefficients a, b, and c.

Enter the three coefficients — a (coefficient of x²), b (coefficient of x), and c (the constant) — and the calculator shows both roots, the discriminant, vertex coordinates, and step-by-step working.

The Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a Discriminant Δ = b² − 4ac Vertex: x = −b/2a, y = c − b²/4a

Interpreting the Discriminant

Discriminant	Nature of Roots	Graph Behavior
Δ > 0	Two distinct real roots	Parabola crosses x-axis at two points
Δ = 0	One repeated real root	Parabola touches x-axis at one point (vertex)
Δ < 0	Two complex (imaginary) roots	Parabola doesn't cross x-axis

Frequently Asked Questions

The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, solves any equation of the form ax² + bx + c = 0. It was known to ancient Babylonian mathematicians and provides exact solutions including complex roots when the discriminant is negative.

Factoring is faster when roots are integers or simple fractions (like x² − 5x + 6 = (x−2)(x−3)). The quadratic formula always works for any quadratic, regardless of root type, and is especially necessary when roots are irrational or complex. The formula is particularly useful in physics, engineering, and financial modeling.

When the discriminant (b² − 4ac) is negative, the square root of a negative number appears in the formula, producing complex (imaginary) roots. These are written as a ± bi where i = √(−1). Complex roots always come in conjugate pairs. They indicate the parabola doesn't intersect the x-axis.

The vertex is the highest or lowest point of the parabola. For ax² + bx + c, the vertex x-coordinate is −b/(2a) and the y-coordinate is c − b²/(4a). If a > 0, the vertex is the minimum point; if a < 0, it's the maximum. The vertex is also the axis of symmetry of the parabola.