Solve a linear equation . The equation I'm trying to solve is $x+2=x$.
Here's what I tried:
$(x+2)^2=x^2$
$x^2+4x+4=x^2$
$4x=-4$
so $x=-1$
 A: The statement

$x=-1$ is a solution of $\langle$an equation$\rangle$

means

if $x=-1$ then the equation is true.

But you have proved

if the equation is true then $x=-1$,

which is the converse statement and is logically irrelevant.
A: The error lies in there cannot exist such a number to begin with, the $x+2=x$ is false from the get go because it is also saying that $2=0$ which is evidently false, as we subtract x from both sides. Doing things to it afterward doesn't change anything to correct it.
I should add however by squaring it you are adding extra possible things as $(-1)^2=1^2$
A: The problem is that it's possible to take two unequal values, specifically $-x$ and $x$ (assuming $x \neq 0$), and square them to make an equal value.  If you start out with an equation that is universally false, then you can conclude almost anything, for instance in this case, you concluded that $-1 = 1$.
The problem is that you probably were never taught about what an equation actually means.  If you are given the following:
$$
f(x) = g(x)
$$
Then before you try and solve it, your very first question should be: does a solution, i.e. $x$, exist!
In lower level math, it is generally assumed that if you are given a problem to solve that a solution exists so you never really stop and ask yourself whether or not a solution does exist.  For instance, find a real value that satisfies the following equation: $x^2 + 1 = 0$...it's not possible!
An equation is a boolean statement: $1 = -1$ is a false, $2 = 2$ is true, $0.01 = 1$ is false, etc.  When we say $x + 1 = 0$, we can choose any $x$ we like.  If $x = 1$ then we get $1 + 1 = 2 \neq 0$ so if $x = 1$ then $x + 1 = 0$ is false!  When we solve equations we try to find the value of $x$ which makes the statement true!
If you start with $x + 2 = x$ then we rearrange to find that $0 = 2$ or $0 = -2$ (or $-1 = 1$, or $-.01 = 1.99$, or $1.5 = -0.5$, etc.--there are infinite amount of rearrangements you could do--all lead to inequality and thus a false statement)--all of which are false--unequivocally.  If you try to perform any operations on either side of that equation it will lead to a false result.
p.s. I don't want to confuse you, so I'm going to keep my wording (of my last sentence) the same.  If I'm being very careful, I would say starting with something that is false leads us to an inconclusive result (not necessarily a false conclusion).  Here is a perfect example: if $x + 2 = 4$ then $x = \frac{4}{2} = 2$--we got the correct result, but with faulty reasoning.
A: $x+2=x$
$\to$$2=0$ , a contradiction.
Hence there is no solution to the equation $x+2=x$ . 
Also $-1$ is the solution of the second degree equation $(x+2)^2=x^2$ and not a solution of the first degree equation  $x+2=x$.
