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This paper deals with approximation properties of the newly defined$q$-generalization of the Bal\'{a}zs-Szabados operators in the case $q\geq1$.Quantitative estimates of the convergence and Voronovskaja type theorem aregiven. In particular, it is proved that the rate of approximation by the$q$-Bal\'{a}zs-Szabados ($q>1$) is of order $q^{-n}$ versus $1/n$ for theclassical Bal\'{a}zs-Szabados ($q=1$) operators. The results are new even forthe classical case $q=1$.

Approximation properties of the $q$-Bal\'{a}zs-Szabados operators in the case $q\geq1$