On a problem of Peth\H{o}

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.