<s>[INST] Which step is wrong in this proof 
Proof: Consider the quadratic equation $x^2+x+1=0$. Then, we can see that $x^2=−x−1$. Assuming that $x$ is not zero (which it clearly isn't, from the equation) we can divide by $x$ to give
  $$x=−1−\frac{1}{x}$$
  Substitute this back into the $x$ term in the middle of the original equation, so
  $$x^2+(−1−\frac{1}{x})+1=0$$
  This reduces to
  $$x^2=\frac{1}{x}$$
  So, $x^3=1$, so $x=1$ is the solution. Substituting back into the equation for $x$ gives
  $1^2+1+1=0$
  Therefore, $3=0$. 

What happened?
 [/INST] if $x^2+x+1=0$ then $(x-1)(x^2+x+1)=0$ thus $x^3-1=0$, thus: $x^3=1$.
if  ...  then ....  is  not a logical equivalence but only  a logical implication.
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