Completing the Square: a Diagram

Date: 03/06/2002 at 05:28:21
From: Kelvon Goh
Subject: Completing the Square Using a Diagram

Can you use a diagram to show completing the square? Something 
like x^2+3x.

Thanks.

Date: 03/06/2002 at 09:03:44
From: Doctor Rick
Subject: Re: Completing the Square Using a Diagram

Hi, Kelvon.

Are you talking about a diagram like this for x^2+3x?

    +----------------+-----+
3/2 |     (3/2)x     |     |
    |                |     |
    +----------------+-----+
    |                |     |
    |                |     |
    |                |     |
x   |       x^2      | 3   |
    |                | - x |
    |                | 2   |
    |                |     |
    |                |     |
    +----------------+-----+
            x          3/2

I divided the 3x into two equal rectangles, 3/2 by x, and stuck them 
on the sides of the x-by-x square. To complete a large square, x+3/2 
on a side, we need to add the little square with sides 3/2.

  x^2 + 2(3/2)x + (3/2)^2 = (x + 3/2)^2

Now, how can we make a figure when the coefficient of x is negative? 
We can work backwards - the x-by-x square is the BIG one, and we 
SUBTRACT two rectangles:

    +----------------+-----+
    |3/2/////////////|XXXXX|
    |////////////////|XXXXX|
    +----------------+-----+
    |                |\\\\\|
    |                |\\\\\|
x   |                |\\\\\|
    |x-3/2           |\\\\\|
    |                |\\\\\|
    |                |\\\\\|
    |                |\\\\\|
    |     x-3/2      |\3/2\|
    +----------------+-----+
               x          

We subtract two rectangles, 3/2 by x, from the x-by-x square. In doing 
so, we remove that little square with the X's TWICE; it's part of both 
rectangles. Therefore, we must add it back once, if we want to be left 
with just the (x-3/2) by (x-3/2) square:

  x^2 - 2(3/2)x + (3/2)^2 = (x - 3/2)^2

Is this what you're looking for?

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