$x^2 + bx$ is an unfinished square. Add the right constant and it snaps shut into $\left(x + \tfrac{b}{2}\right)^2$ — and the right constant is always $k = \left(\tfrac{b}{2}\right)^2$. Read aloud: k equals b over two, squared. Half FIRST, square second — squaring first is the classic slip, and it misses by a factor of four.

Why bother? Because the finished square hands you the vertex by eye: $(x - 3)^2 - 4$ turns at x = 3, height −4, no formula consulted. Completing the square IS last lesson's $-\tfrac{b}{2a}$ in slow motion — do it twice and you will believe the vertex formula forever. One warning: $(x + \tfrac{b}{2})^2$ turns where the bracket DIES, at $x = -\tfrac{b}{2}$. The square is honest; read it carefully.