Completing the Square Using a Diagram

Date: 11/09/1999 at 00:08:15
From: Andy Ko
Subject: Completing the square with diagram

My question involves completing the square, but instead of the usual 
form of brackets and formula, I have to do it something like this:

     x^2 + 6x + ...


     +----------+-----+
     |          |     |
     |          |     |
     |   x^2    | 3x  |
     |          |     |
     |          |     |
     +----------+------
     |          |
     |    3x    |
     +----------+


I don't know how to draw a diagram for completing the square for:

     x^2 - 6x + ...


The answer is x^2 - 6x + 9, but I can't draw the diagram. My teacher 
told me the only hint we get is that it overlaps. I can't seem to 
figure it out. Can you help?

Thanks.

Date: 11/09/1999 at 11:00:45
From: Doctor Rob
Subject: Re: Completing the square with diagram

Thanks for writing to Ask Dr. Math.

Try this diagram:

                 x
     o------------------o-----o
     |        x-3       |  3  |
     |                  |     |
     |                  |     |
     |x-3               |     |
     |                  |     |
     |                  |     | x
     |                  |     |
     |                  |     |
     o------------------o-----o
     |                  |     |
     |3                 |     |
     o------------------o-----o
                 x

The lower right has area 3^2. The entire lower edge rectangle (small 
rectangle and square together making a longer rectangle) has area 3*x, 
and likewise the entire right edge rectangle has area 3*x. They 
overlap in the 3-by-3 square in the lower right corner.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/