Solve any quadratic equation ax² + bx + c = 0. Enter decimals or fractions for a, b, and c; the calculator returns the real root, the pair of real roots, or the complex conjugate pair, plus the discriminant and optional worked steps.
Formula
A quadratic equation is a second-degree polynomial
ax² + bx + c = 0
with a ≠ 0. Completing the square yields the quadratic formula:
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant Δ = b² − 4ac classifies the solutions:
| Discriminant | Roots |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One real root (repeated), x = −b / (2a) |
| Δ < 0 | Two complex roots, α ± βi |
When a = 0 the equation is linear. If b ≠ 0 then x = −c / b.
Fractional values such as 1/4 and 3/4 can be typed directly. The default example a = 1, b = 1, c = 1/4 has Δ = 0, so there is a single real root x = −0.5.
Examples
Repeated root (default)
a = 1, b = 1, c = 1/4. Δ = 1 − 4 × 1 × 1/4 = 0, so x = −1 / 2 = −0.5.
Two real roots
a = 1, b = −3, c = 2. Δ = 9 − 8 = 1, so x = (3 ± 1) / 2, which is 2 or 1. The same pair appears for a = 1/2, b = −3/2, c = 1.
Complex roots
a = 1, b = 2, c = 5. Δ = 4 − 20 = −16, so x = −1 ± 2i. For a = 1, b = 1, c = 1 the pair is −0.5 ± 0.86602540378444i.
Linear fallback
a = 0, b = 1, c = 1 is the line x + 1 = 0, so x = −1.