Quadratic Formula Calculator
- Equation: x² − 5x + 6 = 0
- Discriminant Δ = b² − 4ac = -5² − 4(1)(6) = 1
- Two real roots: x = (−b ± √Δ) / (2a)
- x₁ = 3, x₂ = 2
| Property | Value |
|---|---|
| Equation | x² − 5x + 6 = 0 |
| Discriminant Δ | 1 |
| Root x1 | 3 |
| Root x2 | 2 |
| Vertex | (2.5, -0.25) |
| Axis of symmetry | x = 2.5 |
| Y-intercept | (0, 6) |
ax² + bx + c = 0
The Quadratic Formula
A quadratic equation ax² + bx + c = 0 (with a ≠ 0) has at most two real roots. The quadratic formula x = (−b ± √Δ) / (2a) finds them, where the discriminant Δ = b² − 4ac determines the nature of the roots.
If Δ > 0, two distinct real roots exist — for example x² − 5x + 6 = 0 gives Δ = 1, x = 2 or x = 3. If Δ = 0, one repeated real root: x² − 4x + 4 = 0 gives x = 2 twice. If Δ < 0, roots are complex conjugates: x² + 1 = 0 gives x = ±i.
The vertex of the parabola y = ax² + bx + c occurs at x = −b/(2a). Substituting back gives the y-coordinate. The axis of symmetry is the vertical line x = −b/(2a). The y-intercept is simply (0, c).
Step-by-step working shows the discriminant calculation and root formula substitution. The parabola diagram plots y = ax² + bx + c with real roots, vertex, and axis marked when applicable.
Enter coefficients a, b, and c to solve any quadratic instantly — ideal for algebra classes, physics projectile equations, and optimization problems.
Examples
| Example | Result |
|---|---|
| x² − 5x + 6 = 0 | x = 2, 3 |
| x² + 1 = 0 | x = ±i |
| x² − 4x + 4 = 0 | x = 2 (double) |
| 2x² + 3x − 2 = 0 | x = 0.5, −2 |
Frequently asked questions
The equation is not quadratic. a must be non-zero for this formula to apply.
As a ± bi, for example 0 + 1i and 0 − 1i for x² + 1 = 0.
Δ > 0: two real roots. Δ = 0: one repeated root. Δ < 0: two complex roots.