Solves ax² + bx + c = 0 and shows the discriminant.
The equation ax² + bx + c = 0 is solved by x = (−b ± √(b² − 4ac)) / 2a. The expression under the root is the discriminant. The plus and minus signs give the two roots, and the coefficient a must be non-zero — otherwise the equation is not quadratic at all.
The discriminant D = b² − 4ac tells you how many roots there are before you compute them. Positive D means two distinct real roots; zero means one repeated root; negative means no real roots at all — the parabola never crosses the x-axis. One operation reveals the shape of the answer.
There is no solution among the real numbers, and that is a complete answer rather than an error. Geometrically the parabola lies entirely above or below the x-axis. Over the complex numbers the roots exist as a conjugate pair, but in a practical problem a negative discriminant usually means the state you were solving for cannot occur.
They link the roots to the coefficients without solving: the roots sum to −b/a and multiply to c/a. That gives you a fast check — if the sum and product do not match the coefficients, something is wrong. For simple equations the roots can often be guessed straight from those two relations.