This section introduces the technique of completing the square.

### Completing the Square

The principle idea of completing the square is to rewrite a quadratic form as a binomial term squared plus/minus a constant. Let’s see an example of what we mean.

*add zero cleverly*by adding and subtracting 1, we could get the following: So we have rewritten the polynomial as a binomial squared (ie ) minus . This process of producing a perfect square out of the and terms, by adding and subtracting some constant value (in the above case, we added and subtracted ) is the technique of completing the square.

In our previous example we were given what the binomial square was at the start but it might not be as obvious what you should choose as your target perfect square just by looking at your polynomial. Luckily there is a way to determine what the target perfect square should be based on your polynomial.

Let’s say we have the polynomial . Our goal is to end up with something of the form: . As is often the case in mathematics, we will work backwards, expanding our goal to see how it compares to our original form. Thus expanding we get Since is the coefficient of on the left, and is the coefficient of on the right, we can deduce that ; meaning that we want half of the term on the left to be the value that we will be added to in the completing the square form. Moreover it also shows us that we need as a constant value on the left hand side of the equality to mirror the that is on the right hand side of the equality.

That may seem a bit dense, but the take away is the following: For our ;

- (a)
- Divide the term in half; getting .
- (b)
- Square and then add
**and**subtract it from the expression, this yields: - (c)
- The left three terms above factor perfectly into which means we can simplify our polynomial as follows:
- (d)
- The far right side of the equality above is our Completing the Square end result.

Ok, the above is a bit of alphabet soup, so it will help to see another concrete example following these steps.

- (a)
- Divide the term in half. So we get .
- (b)
- Square and then add
**and**subtract it from the expression. So we get: thus we now get: Notice that we*don’t*want to simplify the above; that would defeat the point. - (c)
- The left three terms above factor perfectly. So we get:
- (d)
- The far right side of the equality above is our Completing the Square end result.

So; completing the square on the expression yields .