 Completing the square by area

Start with a square of side length $x>0$ and add two rectangles, each with side lengths $x$ and $3$. The total area is $x^2+6x$. Place the rectangles along two adjacent sides of the square: the missing corner is a $3$ by $3$ square, of area $9$. Filling it produces a square of side length $x+3$.

The area picture explains why the half-coefficient is squared. It also separates two operations that are easy to confuse: adding $9$ changes $x^2+6x$, whereas adding and subtracting $9$ preserves the expression. When solving an equation instead, adding $9$ to one side requires adding $9$ to the other side too.