<s>[INST] Extraneous solution when solving $x^2+x+1=0$ by getting $x^2=1/x$ Let's assume that we have$$x^2+x+1=0.\tag1$$
Substituting $x=0$, we get $1=0$, so $0$ is not a root for the quadratic equation and thus, $x\neq0$. Therefore, there exists $\frac{1}{x}$, which we'll multiply by both sides of $(1)$, giving us $$x+1+\frac{1}{x}=0.$$
We will, then, move $\frac{1}{x}$ to the other side and get $$x+1=-\frac{1}{x}.$$ If we add $x^2$ to both sides and note that $x^2+x+1=0$, we will have $x^2-\frac{1}{x}=0$. The real root of this equation is $x=1$, which is not a root of $(1)$.
I was wondering at which step did I do something that was incorrect and resulted in this supposed root.
 [/INST] The first step when we got an extraneous root is the equation $x^2 - \frac{1}{x} = 0$. It is a corollary of equation(1). But it's only corollary: it's not equivalent  to it.
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