<s>[INST] What is the solution to $x^2=y^2$? What is the solution to $x^2=y^2$? I know it seems simple but I seem to have got myself all confused. Is this correct?
$$x^2=y^2$$
$$\pm\sqrt{x^2}=\pm\sqrt{y^2}$$
$$\pm x=\pm y$$
$$\Rightarrow x_{1}=y,\quad x_{2}=-y, \quad -x_{3}=y, \quad -x_{4}=-y$$
$$\iff x_{1}=x_{4}=y,\quad x_{2}=x_{3}=-y $$
 [/INST] $x^2 = y^2 \iff x^2 - y^2 = 0 \iff (x-y)(x+y) = 0$
A product is equal to $0$ if and only if one of its factors is equal to $0$
Thus:
$x^2 = y^2 \iff x - y = 0$ or $x + y =0 \iff x = -y$ or $x = y$
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