What does it mean to "complete the square"?

Completing the square is a useful method of rearranging a quadratic equation so the unknown ("x") only appears in it once. The theory of completing the square tells us that we can take a quadratic equation: x² + bx + c = 0 and rearrange it to give (x + b/2)² + c - (b/2)² =0. Sometimes this can help us solve the equation and sometimes it's just a useful result to have. We can explain why this rearranging works using geometry.