Convert following into the form

                                       f (x ) = a ( x - h )²  + k

Solution

We are going to complete the square here.  Though, it is a slightly distinct process than the other times that we've seen it to this point.

       The thing that we've got to keep in mind here is that we should have a coefficient of 1 for the x² term to complete the square. Thus, to get that first we will factor the coefficient of the x² term out of the whole right side as follows.

                                             f ( x ) = 2 ( x² - 6 x + 3 /2)

Note that this will frequently put fractions into the problem that is just something which we'll have to be able to deal with.  Also note that if we're fortunate enough to have a coefficient of 1 on the x² term we won't need to do this step.

Now, it is where the procedure really starts differing from what we've illustrated to this point.  Still we take one-half the coefficient of x & square it. Though, rather than adding this to both of the sides we do the following with it.

 (- 6 /2)² = (-3)² = 9

f ( x ) = 2 ( x² - 6 x + 9 - 9 + 3/2 )

We add & subtract this quantity within the parenthesis as illustrated. Note that all we are actually doing here is adding in zero as 9-9=0! The order listed here is significant. We should add first and then subtract.

The next step is to factor out the first three terms and join the last two as follows.

              f ( x ) = 2 ( ( x - 3)²  - 15 )

As a last step we multiply the 2 back through.

         f ( x ) = 2 ( x - 3)²  -15

And there we go.