Math simplify problem

I have a problem where I got the final answer:

Code:
       1              1
-  ---------   -  --------
   (u - 1)^2       (u + 1)^2

which is correct, but the book further simplifies it to:

Code:
     2(1 + u^2)
-  -------------
    (u^2 - 1)^2
I tried and tried, but couldn't figure out how to simplify it to that result. Can someone please explain the process to me?

Thanks!
 

HallsofIvy

Science Advisor
41,626
821
Do you remember adding fractions from fourth grade arithmetic?

Get a common denominator and add.

The denominator of one fraction is (u-1)2= (u-1)(u-1) and the denominator of the other is (u+1)2[/sup= (u+1)(u+1). The "least common denominator" is (u-1)(u-1)(u+1)(u+1)= (u-1)(u+1)(u-1)(u+1)= (u2-1)(u2-1)= (u2-1)2.
[tex]-\frac{1}{(u-1)^2}- \frac{1}{(u+1)^2}= -\frac{(u+1)^2}{(u-1)^2(u+1)^2}-\frac{(u-1)^2}{(u-1)^2(u+1)^2}[/tex]
[tex]= -\frac{u^2+ 2u+ 1}{(u^2-1)^2}-\frac{u^2-2u+1}{(u^2-1)^2}[/tex]
[tex]= -\frac{2u^2+ 2}{(u^2-1)^2}= -\frac{2(u^2+1)}{(u^2-1)^2}[/tex]
 

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