Solve ax² + bx + c = 0 for x, including complex roots.
The sign of b² − 4ac decides the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex (imaginary) roots — that's why the calculator shows a different result layout for each case.
If a = 0 there's no x² term left, so the equation becomes linear (bx + c = 0), not quadratic — the quadratic formula divides by 2a, which would be division by zero.
It's the turning point of the parabola. h = −b/2a is the x-coordinate (also the axis of symmetry), and k is the equation evaluated at that x — together they show the parabola's minimum or maximum point.