Quadratic Equation Calculator
Solve a quadratic (second-degree) equation ax² + bx + c = 0: enter a, b and c and get its solutions (two real, one repeated real, or complex), with the discriminant explaining which of the three cases applies.
How is it calculated?
General formula: x = (-b ± √(b² - 4ac)) ÷ 2a. The discriminant (Δ = b² - 4ac) determines the type of solution: if Δ > 0, there are two distinct real solutions; if Δ = 0, there's one repeated real solution; if Δ < 0, both solutions are complex (non-real) numbers.
Examples
x² − 3x + 2 = 0 (two real solutions)
x₁ = 2, x₂ = 1 (discriminant = 1).
x² − 4x + 4 = 0 (one repeated real solution)
x = 2 (discriminant = 0).
x² + 2x + 5 = 0 (complex solutions)
x = −1 ± 2i (discriminant = −16).
The three discriminant cases
| Discriminant (Δ) | Solution type | Graphical meaning |
|---|---|---|
| Δ > 0 | Two distinct real solutions | The parabola crosses the X axis at two points |
| Δ = 0 | One real (repeated) solution | The parabola touches the X axis at a single point (the vertex) |
| Δ < 0 | Two complex conjugate solutions | The parabola doesn't cross the X axis |