Quadratic Formula Calculator

About the Quadratic Formula Calculator

A quadratic equation is any second-degree polynomial equation written in the standard form ax² + bx + c = 0, where a ≠ 0. Its graph forms a symmetric curve called a parabola.

Our Quadratic Formula Calculator computes the discriminant (Δ = b² − 4ac) and solves for both roots (x₁ and x₂), whether they are distinct real numbers, a repeated real root, or complex conjugates.

The Quadratic Formula & Discriminant

For any quadratic equation ax² + bx + c = 0:

How to Solve a Quadratic Equation Using the Quadratic Formula

1. Write the Equation in Standard Form: Move all terms to one side so it matches ax² + bx + c = 0 and identify coefficients a, b, and c.
2. Calculate the Discriminant: Evaluate Δ = b² − 4ac. If Δ > 0, there are 2 real roots; if Δ = 0, there is 1 repeated root; if Δ < 0, the roots are complex.
3. Substitute into the Formula: Compute (−b + √Δ) / (2a) and (−b − √Δ) / (2a).

Worked Example: For x² − 5x + 6 = 0 (a=1, b=−5, c=6), the discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1. Then x = (5 ± 1)/2, giving x₁ = 3 and x₂ = 2.

Frequently Asked Questions (FAQ)

What does the discriminant tell you about the graph of a quadratic?

A positive discriminant means the parabola crosses the x-axis at two points; a zero discriminant means its vertex touches the x-axis at exactly one point; a negative discriminant means the parabola never crosses the real x-axis.