Discriminant Calculator

Reviewed by CalcMulti Editorial Team·Last updated: March 2026·← Algebra Hub

The discriminant D = b² − 4ac is the part under the square root in the quadratic formula. It instantly tells you how many real solutions ax² + bx + c = 0 has — without fully solving the equation.

Enter your a, b, c values below to compute the discriminant and get a clear explanation of the root type.

The three cases of the discriminant: When D > 0, the equation has two distinct real roots — the parabola crosses the x-axis at two different points. When D = 0, the equation has exactly one real root (a repeated root) — the parabola is tangent to the x-axis at its vertex. When D < 0, the equation has no real roots — the parabola does not cross the x-axis, and the roots are complex conjugate pairs of the form h ± ki.

The discriminant also determines whether a trinomial factors over the integers: if D is a perfect square (0, 1, 4, 9, 16, 25…), the trinomial factors with rational coefficients. If D is positive but not a perfect square, the roots are irrational (involving √D). If D < 0, the polynomial is irreducible over the reals.

Real-world examples: (1) Projectile motion — h(t) = −16t² + 48t + 5; discriminant = 48² − 4(−16)(5) = 2304 + 320 = 2624 > 0, so the projectile hits the ground at two times. (2) Break-even — profit = −x² + 10x − 25; discriminant = 100 − 100 = 0, meaning the business exactly breaks even at one production level. (3) Circuit analysis — when designing filters, discriminant < 0 indicates no resonant frequency in that range.