Finding the Difference of the Cubes of Two Numbers

Date: 9/29/95 at 21:57:3
From: Anonymous
Subject: Solve for x^3 - y^3

Here's the problem :

The difference of two numbers is 1.
The product of the two numbers is also 1.
What is the difference of the cubes of the numbers? 

Date: 9/30/95 at 1:52:37
From: Doctor Andrew
Subject: Re: Solve for x^3 - y^3

Let x be the bigger number and y be the smaller.  Then we know that 
x - y = 1.

We also know that x * y = 1

So we know two things:

  x - y = 1

  x * y = 1

If you want to solve this algebraically, pick one of these 
equations and use it to solve for either x or y in terms of the 
other variable (y or x, respectively).  Then substitute that 
expression for x or y into the equation you didn't use.  

You will get a quadratic expression (that means you'll have a term 
raised to the second power) so you can use the quadratic formula to 
solve it.  If you're not familiar with the quadratic formula,
you can read about it here:

  http://mathforum.org/library/drmath/view/53198.html

Once you've found values for x and y, you can cube them and subtract
to get the difference. 

-Doctor Andrew,  The Geometry Forum

Date: 9/30/95 at 14:54:47
From: Doctor Ken
Subject: Re: Solve for x^3 - y^3

Hello!

Let me offer another way you could approach this problem.  I'll 
give you an example of a different but similar problem:

     x - y = 2

       xy  = 7

What is x^2 + y^2?

Well, since x-y = 2, squaring both sides gives 

   x^2 - 2xy + y^2 = 4.

Since xy = 7, we can add 2xy to both sides (it will get rid of the 
middle term on the left, and add 14 on the right).  That gives us   

     x^2 + y^2 = 18.

Now do you see how you could do your problem?

-Doctor Ken,  The Geometry Forum