Ask Uncle Colin: Why is it called “completing the square”?

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

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.