  In this paper we deal with a problem of Peth\H{o} related to existence of
quartic algebraic integer $\alpha$ for which $$
\beta=\frac{4\alpha^4}{\alpha^4-1}-\frac{\alpha}{\alpha-1} $$ is a quadratic
algebraic number. By studying rational solutions of certain Diophantine system
we prove that there are infinitely many $\alpha$'s such that the corresponding
$\beta$ is quadratic. Moreover, we present a description of all quartic numbers
$\alpha$ such that $\beta$ is quadratic real number.
