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### 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/
```
Associated Topics:
High School Basic Algebra
High School Geometry
High School Triangles and Other Polygons

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