Dear Uncle Colin,

Why is it called “completing the square”? To me, it looks like you’re taking something away from a square.

- Some Quadratics, Understandably, Are Requiring Explanation

Hi, SQUARE, and thanks for your message!

Completing the square involves taking a quadratic such as $x^2 + 6x + 5$ and writing it in a specific form - here, it would be $(x+3)^2 - 4$. And you’re absolutely correct, it looks very much like you’re breaking down a square rather than completing it.

The difference, though, is just the side of the equation you’re working on.

A change of perspective

When given a quadratic like $x^2 + 6x + 5$, completing the square asks “what would I need to add to “this for it to be a perfect square?”.

Geometrically speaking, you can think of $x^2 + 6x + 5$ as an $x$-by-$x$ square, two 3-by-$x$ rectangles (one on top of the square and one to the side) and five extra units in the corner gap. The corner gap was originally 9 units (3-by-3), so we’d need to add four on to make a perfect square.

Algebraically, that translates to $(x^2 + 6x + 5) + 4 = (x+3)^2$. The four extra units are needed to ‘complete the square’. Taking them away from either side puts everything in the form you want, $(x+3)^2 - 4$.

Hope that helps!

- Uncle Colin