Quadratic Formula Calculator
Solve quadratic equations instantly. Calculates real and complex roots, the discriminant, and vertex. Ideal for students verifying algebra homework step-by-step.
About
A quadratic equation is a second-degree polynomial of the form ax2 + bx + c = 0. Solving these by factoring can be difficult or impossible when the roots are not integers. The Quadratic Formula provides a universal solution for any coefficients, determining where the parabola crosses the x-axis.
The nature of the solutions depends entirely on the Discriminant (Δ). If positive, there are two distinct real roots. If zero, there is one repeated real root. If negative, the roots are complex (imaginary numbers), meaning the parabola never touches the x-axis.
Formulas
The standard Quadratic Formula to find x is:
The term under the square root is the Discriminant Δ:
Reference Data
| Discriminant (Δ) | Value Type | Nature of Roots | Graph Behavior |
|---|---|---|---|
| Δ > 0 | Perfect Square | 2 Rational Roots | Intersects x-axis twice (Cleanly) |
| Δ > 0 | Non-Square | 2 Irrational Roots | Intersects x-axis twice (Decimals) |
| Δ = 0 | Zero | 1 Real Root (Repeated) | Touches x-axis once (Vertex) |
| Δ < 0 | Negative | 2 Complex Conjugates | Does not touch x-axis |
| a > 0 | Positive Coeff | Parabola opens UP | Has a Minimum |
| a < 0 | Negative Coeff | Parabola opens DOWN | Has a Maximum |
| c = 0 | Zero Constant | One root is 0 | Passes through Origin |
| b = 0 | Zero Linear | Roots are opposite (±) | Symmetrical on Y-axis |