<s>[INST] Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$ Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$
Solutions are $x=2$ or $x=-1$.
But $x=2$ does not satisfy $\sqrt{x+2}+x=0$.Because $\sqrt{4}+2 \neq0$
So does it mean that they are different ? Why ?
 [/INST] Ofcourse there lies a difference,
Starting with $ x^2-x-2=0 \implies x^2=x+2 $ , Now follow from here that ,
$ x = \pm \sqrt{x+2} $ , When you take the negative value you get the above result as the solutions are $ x=2 $ or $x=-1 $ , so the negative value satisfies when you have a equation created with the negative sign taken.
In particular , $ (x+\sqrt{x+2})(x-\sqrt{x+2}) = x^2-x-2=0 $ here, so they doesn't represent the same equation. 
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