SOLUTION: {{{x+(10)/(x-2)=(x^2+3x)/(x-2)}}}

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Question 135271: x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29
Found 2 solutions by Fombitz, solver91311:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29
x%28x-2%29%2B10=x%5E2%2B3xMultiply both sides by (x-2).
x%5E2-2x%2B10=x%5E2%2B3xDistribute.
-5x%2B10=0
x=2
This cannot be a solution since (x-2) is a denominator in your original equation.
It would lead to division by zero.
There is no solution.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29

Put everything on the left:
x%2B%2810%29%2F%28x-2%29-%28%28x%5E2%2B3x%29%2F%28x-2%29%29=0

x-2 is the common denominator:
%28x%28x-2%29%29%2F%28x-2%29%2B%2810%29%2F%28x-2%29-%28%28x%5E2%2B3x%29%2F%28x-2%29%29=0

Expand and collect like terms:
%28x%5E2-2x%2B10-x%5E2-3x%29%2F%28x-2%29=0
%28-5x%2B10%29%2F%28x-2%29=0

Note that a%2Fb=0 if and only if a=0 AND b%3C%3E0. So set the numerator equal to 0 and solve:

-5x%2B10=0
-5x=-10
x=2

However, if x=2, then the denominator x-2 equals 0, so x=2 must be excluded as a root of this equation, and there are no other roots.

Therefore the solution set of x%2B%2810%29%2F%28x-2%29=%28x%5E2%2B3x%29%2F%28x-2%29 is the empty set.