At the Oliver School (Lawrence) Focus on Mathematics study group we discussed the following geometric interpretation of "completing the square." The basic idea is to figure out what needs to be added to x² + b x to create a perfect square.

This figure assumes b > 0.

This process is used in the context of solving a quadratic equation such as

x² + 5 x = -3

Since b = 5, b/2 = 2.5 and so you add 2.5² = 6.25 to both sides of the equation. Then you can take the square root of both sides since the left side is the square of x - 2.5

Question: How would you visualize this process if b < 0?