For example, if two students in a class are having frequent confrontations that bring learning in the class to a halt, and attempts by teachers and counselors to address their conflict directly haven't been effective, the expedient solution might be to place them in separate classes. The "right thing" would be to address the problem on the social and emotional level, but if continued efforts to do so is likely to result in continued disruption to the students' education, it might be better to separate them. "Expedient" acknowledges the trade-off, while emphasizing the positive outcome.
Often a course of action is described as "expedient" when it seems to dodge an issue of morality or virtue. For example, if we solve climate change with geoengineering instead of by addressing thoughtless consumerism, corporate impunity, and lack of international accountability, many people would feel frustrated or let down by the solution because it would solve the problem without addressing the moral shortcomings that led to the problem. The word expedient stresses the positive side of this, the effectiveness and practicality of the solution, while acknowledging that it leaves other, perhaps deeper issues unaddressed.