Completing the Square

Another method of solving a quadratic is to complete the square. This changes the form of the quadratic from

(1)  

Into the form

(2)  

We can use the square of a binomial:

(3)  

Using this we can see that for any equation of the form , in other words, where we can form a square:

(4)  

So:

(5)  

Where is:

(6)  

So this gives us:

(7)  

We can expand this to form a general case where

(8)  

into the form

(9)  

So the general case is:

(10)  

Where

(11)  

and

   

The completed square form is useful as the values of and give us the Cartesian coordinates of the stationary point of the parabola, so we have a local minimum or maximum at .

So for example, take the quadratic:

So completing the square with , and , we have:

   

and

   

So the local minimum in this case is at , and the completed square form is:

(12)