Why do we add (b/2a)² when completing the square for a quadratic?

Completing the square works because we want to turn the expression into a perfect square like (x + k)². Recall that (x + k)² = x² + 2kx + k². So if we have x² + (b/a)x, we compare 2k with b/a, which gives k = b/(2a). To complete the square we need the k² term, which is (b/2a)². That is exactly why we add and subtract this value, it is the missing piece that makes a perfect square. For example, x² + 6x becomes x² + 6x + 9 − 9 = (x + 3)² − 9, where 9 = (6/2)². We subtract it back so the value of the expression stays unchanged. The logic is purely matching the perfect square identity.