Determine the non-zero real numbers $x$ such that $x+\frac{1}{x}=2$.
We could just multiply by $x$ (non-zero) to obtain $(x-1)^{2}=0$ and that's it, but that's not fun, we need to use Viète!

Let's take a polynomial whose roots are $x$ and $\frac{1}{x}$. Their sum is 2 and their product is 1, so they are roots of $X^{2}-2 X+1$, hence $x=1$.