Find the non-zero real numbers $x$ such that $x+\frac{1}{x}=2$.
Let $x$ be a real number such that $x+\frac{1}{x}=2$. It suffices to consider the polynomial

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

to deduce that $x=1$. Conversely, $x=1$ indeed works.