Factoring Problems

Date: 8/14/96 at 13:55:0
From: Anonymous
Subject: Word Problems Involving Factorization

Dear Dr. Math, 

I am an English teacher trying to teach Algebra 1. I hope you can help 
me with the following problem. I have been struggling with the chapter 
on factoring. I have drawn several pictures to try and figure out this 
problem, but I'm getting nowhere.

A photograph is 7 inches long and 5 inches wide. It is surrounded 
by a border of uniform width. If the area of the border alone is 
64 sq. inches, what is the width of the border?

Thanks a lot for all your help! 

Laura Sturman

Date: 8/14/96 at 22:50:5
From: Doctor Robert
Subject:Word Problems Involving Factorization

Draw yourself a diagram of a rectangle 7 inches by 5 inches. Now add a 
uniform border all around this rectangle. Call the width of this 
border x. Now we need to find the area of just the border. One way to 
do it is to draw a square in each corner. The area of one square is 
x^2, and since there are four of them, these areas add to 4x^2.  Now 
there are four rectangles left. Two of them have the dimension 5 by x 
and two of them 7 by x. Adding these areas we get 24x. So we have the 
following equation:

   4x^2 + 24x = 64

Subtracting 64 from both sides and dividing all the way through by 4 
we get:

   x^2 + 6x -16 = 0

This factors into (x+8)(x-2) = 0

so that x = -8 or x = 2. We reject the negative solution. Our answer 
is x = 2. Therefore, the width of the border is 2 inches.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/