Quadratic Formula Calculator
Solve equations of the form ax² + bx + c = 0 using the quadratic
formula. The calculator shows both roots (real or complex), the discriminant,
the vertex of the parabola and a step-by-step derivation.
Enter coefficients
Enter coefficients a, b and c, then press Solve.
The quadratic formula
x = [ −b ± √(b² − 4ac) ] / (2a)
The expression under the square root is called the discriminant (Δ = b² − 4ac). Its sign tells you how many real roots exist.
| Discriminant | Roots |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One real root (repeated / double root) |
| Δ < 0 | Two complex conjugate roots (no real roots) |
Vertex of the parabola
x_vertex = −b / (2a)
y_vertex = c − b² / (4a)
The vertex is the minimum point of the parabola (if a > 0) or
the maximum point (if a < 0).
Worked examples
Example 1 — Two distinct real roots
Solve x² − 5x + 6 = 0.
a = 1, b = −5, c = 6. Δ = 25 − 24 = 1. x = (5 ± 1) / 2 → x = 3 or x = 2.
x = 2, 3
Example 2 — Double root
Solve x² − 6x + 9 = 0.
a = 1, b = −6, c = 9. Δ = 36 − 36 = 0. x = 6 / 2 = 3 (repeated root).
x = 3 (double root)
Example 3 — Complex roots
Solve x² + 4x + 13 = 0.
Δ = 16 − 52 = −36. √(−36) = 6i. x = (−4 ± 6i) / 2 → x = −2 ± 3i.
x = −2 ± 3i
Example 4 — Non-unit leading coefficient
Solve 2x² − 4x − 6 = 0.
a = 2, b = −4, c = −6. Δ = 16 + 48 = 64. x = (4 ± 8) / 4 → x = 3 or x = −1.
x = −1, 3
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Frequently asked questions
If a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0) with solution x = −c / b. This calculator requires a ≠ 0.
Complex roots appear as a pair of conjugates in the form p ± qi. They indicate the parabola does not intersect the x-axis, but the quadratic still has two roots in the complex number system.
The vertex is the highest or lowest point of the parabola. It occurs at x = −b / (2a). The sign of a determines whether the parabola opens up (a > 0, vertex is minimum) or down (a < 0, vertex is maximum).