SOLUTION: Algebraically determine the EXACT roots in simplest form: (2x/x+2)=(3/x-2)-(12/x^2-4)

Question 551700: Algebraically determine the EXACT roots in simplest form:
(2x/x+2)=(3/x-2)-(12/x^2-4)
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I think you meant
= - .
I would start by multiplying both sides by a common denominator type of expression to eliminate denominators. (It's sort of the same as going to a common denominator and then working only with the numerators. However, you have to be aware that you may be introducing extraneous solutions that are solutions of the new equation, but make some of the original denominators zero.
Multiplying both sides by we get

And then we re-arrange and simplify

Performing the indicated operations we go from
to and to
Then we add to both sides
to get the quadratic equation
which we can solve by factoring
--> or
Or we could apply the quadratic formula to find the same solutions to the final equation.
However, is an extraneous solution. It's not a solution of the original equation, because it would make the denominators and zero.
THE SOLUTION
The only solution is , which we verify:
For , = = =
and - = - = - = - =
NOTES:
Parentheses are tricky.
(2x/x+2)=2x/x+2=, but I think you meant 2x/(x+2)=
(3/x-2)-(12/x^2-4)=3/x-2-(12/x^2-4)= - = - + = - + ,
but I think you meant - .
The horizontal lines include invisible parentheses enclosing the expressions above and below them.
=(1+2+3+4)/(5+6+7+8)
If you can only draw the slanted lines, you have to include the parentheses.

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