  Recently, Girolami and Adesso have demonstrated that the calculation of
quantum discord for two-qubit case can be viewed as to solve a pair of
transcendental equation (Phys. Rev. A, {\bf 83}, 052108(2011)). In present
work, we introduce the generalized Choi-Jamiolkowski isomorphism and apply it
as a convenient tool for constructing transcendental equations. For the general
two-qubit case, we show that the transcendental equations always have a finite
set of universal solutions, this result can be viewed as a generalization of
the one get by Ali, Rau, and Alber (Phys. Rev. A, {\bf 81}, 042105 (2010)). For
a subclass of $X$ state, we find the analytical solutions by solving the
transcendental equations.
