Quantum discord provides a measure for quantifying quantum correlations beyond entanglement and is very hard to compute even for two-qubit states because of the minimization over all possible measurements. Recently a simple algorithm to evaluate the quantum discord for two-qubit
X states was proposed by Ali, Rau, and Alber [Phys. Rev. A 81, 042105 (2010)] with minimization taken over only a few cases. Here we shall at first identify a class of
X states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of
X states for which their algorithm fails. And then we demonstrate that this special family of
X states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements.