A student divided every term of the equation

$$
x^{2}-b x+1=0
$$

by $x^{2}$, arriving at the equation

$$
\left(\frac{1}{x}\right)^{2}-b \frac{1}{x}+1=0
$$

However, they could have also obtained the same equation by substituting $\frac{1}{x}$ for $x$ in the first equation everywhere. From this, they concluded that $x=\frac{1}{x}$, or $x= \pm 1$. Where is the mistake?
The error simply lies in the fact that the student forgets that a quadratic equation has two roots, and thus from the fact that $\frac{1}{x}$ is also a solution to the equation besides $x$, it does not follow that $x=\frac{1}{x}$, but only that one root is $x_{1}$, and the other root is $x_{2}=\frac{1}{x_{1}}$ (The student is only accidentally half-right when $b=2$, in which case indeed $\left.x_{1}=x_{2}=1.\right)$