Quadratic Equation Solver
Discriminant and roots of ax² + bx + c = 0.
How it works
Δ = b² − 4ac; x = (−b ± √Δ) ÷ 2a
The discriminant sets the number of roots: Δ > 0 two distinct real roots; Δ = 0 a double root; Δ < 0 no real root (shown as "—"). a must not be 0.
FAQ
Why no real roots when Δ < 0?
You would take the square root of a negative number, which has no real solution — the roots are complex conjugates.
Worked example
Solve x² − 5x + 6 = 0: Δ = 25 − 24 = 1, x = (5 ± 1) ÷ 2, so x = 3 or 2.
If Δ < 0 (e.g. x² + 1 = 0) there is no real solution.
Notes and limitations
The discriminant decides the nature of the roots: Δ > 0 two distinct reals, Δ = 0 one repeated root, Δ < 0 no real roots (complex pair). If a = 0 it is linear, and the quadratic formula does not apply.