Recently I saw the following proof:
\begin{align*}(y+2)^2 &= y^2\\ y^2 + 4y + 4 &= y^2\\ 4y &= -4\\ y &= -1\\ \end{align*}
Geometrically I see the flaw in this solution. Solving for $y + 2 = y$ is basically asking where the lines $f_1 = y+2$ and $f_2 = y$ intersect? Since these are parallel lines, they never intersect and there is no solution for $y$. Squaring sides changes the equation itself to the intersection of two parabolas. So geometrically squaring both sides is obviously a flaw.
But why is this wrong algebraically? Squaring both sides seems to make sense. Furthermore, could someone give me a list of tips to easily identify the problem with a lot of these false proofs? I seem to be seeing a lot of them.
$a = b$ implies $a^2 = b^2$. The converse is not true. Consider $a = 1, b = -1$.
More technically, squaring is not an injective function: It maps distinct numbers to the same number. Thus it cannot be "undone" or "reversed" in general.