Enter the coefficients a, b, and c of any quadratic ax²+bx+c=0. This tool computes the discriminant Δ = b²−4ac and tells you exactly how many solutions the equation has — with step-by-step work and a graph of the parabola.
Δ = b² − 4ac
For any quadratic equation ax²+bx+c=0, the discriminant tells you how many real solutions exist — before you do any solving.
Δ > 0 — The parabola crosses the x-axis at two distinct points. There are two distinct real solutions.
Δ = 0 — The parabola just touches the x-axis at its vertex. There is exactly one real solution (a repeated root).
Δ < 0 — The parabola never reaches the x-axis. There are no real solutions (two complex/imaginary solutions).
x = (−b ± √b²−4ac) / 2a
The discriminant IS the expression under the radical sign (√) in the quadratic formula. That's why it controls the solution type:
If b²−4ac is positive, √Δ is a real number and ± gives two distinct values.
If b²−4ac is zero, √0 = 0 and ± makes no difference — both give the same root: x = −b/2a.
If b²−4ac is negative, you'd need √(negative) — which is imaginary. No real solutions exist.
One-on-one Algebra 1 tutoring builds rock-solid intuition for the discriminant, the quadratic formula, and everything in between — so tests feel easy instead of stressful.