The discriminant answers the question before you’ve finished the formula: two real roots, one repeated root, or two complex ones.
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Quadratic Equation Solver
Solve ax² + bx + c = 0 with full step-by-step working, real or complex roots, factored form, and the parabola's vertex.
a = 0 isn't a quadratic, it's linear (bx + c = 0), so it's solved that way below.
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What the discriminant actually tells you
b² minus 4ac decides everything about the roots before you even finish the quadratic formula. A positive discriminant means two distinct real roots, the parabola crosses the x-axis twice. Zero means exactly one repeated real root, the parabola's vertex sits precisely on the x-axis, touching but not crossing it. A negative discriminant means no real roots at all, the parabola never touches the x-axis, and the two roots are complex conjugates, still a mathematically valid answer, just not a point you can plot on a standard x-y graph.
Quadratic equation solver FAQ
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a, for any equation in the form ax² + bx + c = 0 where a is not zero. It solves every quadratic, whether the roots turn out to be real or complex.
What does a negative discriminant mean?
b² - 4ac being negative means the equation has no real roots, the parabola never crosses the x-axis. The two solutions are complex conjugates instead, a mathematically valid answer, just not a point you can plot on a standard graph.
How do I know if a quadratic factors nicely?
If both roots come out as whole numbers, the equation factors as a(x - root1)(x - root2), which this calculator shows automatically when it applies. Many quadratics, including anything with irrational or complex roots, simply don't factor into clean integer terms, and that's a normal, correct outcome, not a sign something went wrong.
What happens if a = 0?
The equation isn't quadratic anymore, it's linear (bx + c = 0), since the x² term disappears entirely. This calculator detects that case and solves the linear equation instead of dividing by zero.
What is the vertex of a parabola?
The turning point of the curve, where it changes from decreasing to increasing (if it opens upward) or increasing to decreasing (if it opens downward). Its x-coordinate is always -b/2a, which is also the parabola's axis of symmetry.
Is my data private?
Yes. Every calculation runs in your browser. Nothing you enter is sent to a server, logged, or stored.
Free quadratic equation solver by ANUPRESS
One formula, three different outcomes
The quadratic formula never fails to produce an answer, what changes is what kind of answer it gives. b² minus 4ac positive means the parabola crosses the x-axis at two distinct points. Exactly zero means it touches the axis at exactly one point, its vertex, and bounces back without crossing. Negative means it never reaches the axis at all, and the “roots” become a pair of complex conjugate numbers, a real, valid answer, just not one you can mark on an ordinary graph.
Why complex roots aren’t a mistake
A negative discriminant genuinely means the equation has no real solution, and reporting that honestly, with the actual complex values, matters more than most calculators bother with. x² + 2x + 5 = 0 has no real x that satisfies it, but it does have two valid complex solutions, -1 + 2i and -1 – 2i, and a tool that just says “no solution” is throwing away half the answer.
The vertex tells you more than either root alone
Every quadratic’s graph is a parabola, and its vertex, at x = -b/2a, is the single most useful point on the curve: the minimum if the parabola opens upward, the maximum if it opens downward. That’s the actual answer to a huge share of real “quadratic word problems”, the maximum height of a thrown object, the minimum cost at some production level, not the roots at all.