Quadratic formula: x = [−b ± √(b² − 4ac)] ÷ 2a.
| Step | Working |
|---|---|
| 1 | Identify a = 1, b = -2, c = -1 |
| 2 | Δ = b² − 4ac = 4 − -4 = 8 |
| 3 | x = (−b ± √Δ) ÷ 2a = (2 ± √8) ÷ 2 |
| 4 | x = 2.414213562 and x = -0.4142135624 |
![Quadratic Roots formula: D = b² − 4ac; x = [ −b ± √D ] ÷ 2a](https://powercalculators.com/assets/img/calculators/quadratic-roots-1200x630.jpg?v=82fe2afedf "Quadratic Roots Calculator — formula explained")
The roots of a quadratic equation are the values of x that make ax² + bx + c equal zero — where the parabola crosses the x-axis.
Enter the coefficients to find both roots, whether they are real, repeated or complex, with the working from the quadratic formula.
| a (coefficient of x²) | 1 |
| b (coefficient of x) | -5 |
| c (constant term) | 6 |
| Roots | x = 3 and x = 2 |
| Equation | x² − 5x + 6 = 0 |
| Discriminant Δ = b² − 4ac | 1 |
| Nature of roots | Two distinct real roots (Δ > 0) |
| Factored form | (x − 3)(x − 2) |
| Vertex (turning point) | (2.5, -0.25) — minimum |
| Sum and product of roots | sum = −b/a = 5, product = c/a = 6 |
x = 3 and x = 2.
Try this example →| a (coefficient of x²) | 1 |
| b (coefficient of x) | -2 |
| c (constant term) | -1 |
| Roots | x = 2.414213562 and x = -0.4142135624 |
| Equation | x² − 2x − 1 = 0 |
| Discriminant Δ = b² − 4ac | 8 |
| Nature of roots | Two distinct real roots (Δ > 0) |
| Exact form | x = (2 ± √8) ÷ 2 |
| Factored form | (x − 2.41421)(x + 0.414214) (roots are irrational, shown to 6 figures) |
| Vertex (turning point) | (1, -2) — minimum |
| Sum and product of roots | sum = −b/a = 2, product = c/a = -1 |
x = 2.414213562 and x = -0.4142135624; x = (2 ± √8) ÷ 2.
Try this example →| a (coefficient of x²) | 1 |
| b (coefficient of x) | 0 |
| c (constant term) | 4 |
| Roots | x = 0 + 2i and x = 0 − 2i |
| Equation | x² + 4 = 0 |
| Discriminant Δ = b² − 4ac | -16 |
| Nature of roots | Two complex (imaginary) roots (Δ < 0) |
| Exact form | x = (0 ± i√16) ÷ 2 |
| Factored form | Does not factor over real numbers |
| Vertex (turning point) | (0, 4) — minimum |
| Sum and product of roots | sum = −b/a = 0, product = c/a = 4 |
x = 0 + 2i and x = 0 − 2i.
Try this example →x = [−b ± √(b² − 4ac)] ÷ 2a, for ax² + bx + c = 0 with a ≠ 0.
b² − 4ac > 0: two real roots; = 0: one repeated root; < 0: two complex roots.
Yes, when the discriminant is negative: x = p ± qi. The calculator shows them.
If the roots are r₁ and r₂, ax² + bx + c = a(x − r₁)(x − r₂).
Yes. The roots of the equation are the zeros of the function y = ax² + bx + c.