Completing the square feels like a magic trick the first time: x² + 6x + 5 becomes (x + 3)² − 4, and suddenly the vertex (−3, −4) is just sitting there in the open. That transformation — from standard form to vertex form y = a(x − h)² + k — is the whole point of this corner of algebra. The squared term is never negative, so the smallest the expression can be is k, achieved exactly at x = h. For a downward-opening parabola the logic mirrors: the vertex is the maximum, the best you can do.

The shortcut x = −b/(2a) is completing the square with the algebra pre-done, and it's what most students actually use under test pressure. But knowing the square-completion behind it pays off: it explains why the axis of symmetry exists, it derives the quadratic formula, and it converts circle equations into center-radius form too. This calculator evaluates the vertex computation: enter your two known values and read the result. Whether you're finding the cheapest production level on a cost curve or the peak of a water fountain's arc, the workflow is identical — locate (h, k), check the sign of a to name it max or min, and the problem's headline answer is written.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Quadratic Vertex Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.