Use Factoring to Find Two Consecutive Integers...

Date: 7/31/96 at 13:52:49
From: Deena Kumar
Subject: Use Factoring to Find Two Consecutive Integers...

I am having a little trouble with using factoring in problem solving. 
The instructions say to solve the problem and reject the solutions 
that do not meet the conditions of the problem.The problem is as 
follows:

Find two consecutive integers whose product is 56.

Here is how I tried to do it:

Let y = the smaller integer. Then let y + 1 = the larger integer. 
                                    2
y(y + 1) = 56 = y * y + y * 1 = 56 = y + y = 56

 2
y + y = 56

 2
y + y - 56 = 0


(y + 8)(y - 7) = 0


y + 8 = 0         y - 7 = 0  

y = -8            y = 7


It is obvious that the answer is 7 and 8, but my dilemma is that; in 
           2
order for y + y - 56 = 0 to be true, the "y" must be a sum of a larger
positive integer (8) and a smaller negative integer (-7). I know my 
method must be wrong, but it is the only way I know. How should I go 
about solving problems such as this? I would truly appreciate your 
help.

D.S.K.

Date: 7/31/96 at 14:12:42
From: Doctor Robert
Subject: Re: Use Factoring to Find Two Consecutive Integers...

You have worked out the problem in the correct fashion.  I think that 
you are not interpreting your results correctly.  There are two values 
of y, namely 7 and -8 that will work.  If you use 7, then 7 and 8 are 
the desired integers (remember you let y and y+1 be your variables).  
If you use -8, then the integers are -8 and -7.  Both these solutions 
are answers to the question. If the question had asked for POSITIVE 
integers, we would have had to reject the -8 solution. I hope that 
this clears things up.

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