Expert: Steve Holleran - 8/12/2008

Question
Completely simplify the following rational expressions:

x-3   -   1     =    x^2-2
____     ____      ________

x + 2    x-2       x^2-4

Answer
Hi Amanda,

What you've written here is a rational equation, so I'm assuming that when you say "completely simplify", you really want to solve it.

To solve any rational equation, you need to factor all the denominators that are not already factored, then find the LCD of all the denominators, then multiply each term in the equation by this LCD.

The denominator of the fraction on the right side will factor
as (x+2)(x-2), so the equation looks like:

    (x-3) / (x+2) - 1 / (x-2) = (x^2-2) / [(x+2)(x-2)]

Now, the LCD for all the denominators is (x+2)(x-2).

So, multiply each term by this LCD:

(x-3)/(x+2) * (x+2)(x-2) - 1/(x-2) * (x+2)(x-2) =

                           (x^2-2)/[(x+2)(x-2)] * (x+2)(x-2)

After canceling, you should have:

   (x-3)(x-2) - 1(x-2) = x^2-2

   x^2 - 5x + 6 - x - 2 = x^2 - 2

       -6x + 6 =  0   so x = 1.


Hope this is what you wanted.
Steve