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.
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