Quadratic Formula Calculator

A quadratic equation is a second-order polynomial equation of the form ax² + bx + c = 0, where a ≠

• The discriminant Δ = b² - 4ac determines the nature of the roots:

• If Δ > 0, the equation has two distinct real roots.

• If Δ = 0, the equation has one repeated real root.

• If Δ < 0, the equation has two complex conjugate roots.

To use the Quadratic Formula Calculator, enter your Coefficient a, Coefficient b (x), Constant c. The calculator instantly computes Discriminant, Root Type and more. Results update in real time as you change any input — no submit button needed.

How to Use This Calculator

Enter your Coefficient a (x²) (e.g. 1), Coefficient b (x) (e.g. -5), and Constant c (e.g. 6). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a). Discriminant Δ = b² - 4ac. Vertex: h = -b/(2a), k = c - b²/(4a).

Worked Example

Solve x² - 5x + 6 =

• Discriminant: (-5)² - 4(1)(6) = 25 - 24 =

• Roots: x₁ =/2 = 3, x₂ = (5 - 1)/2 =

• Vertex: (2.5, -0.25).

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.