As I understand it, a quorum is the minimum necessary for business to get done in a meeting. My understanding is that a quorum merely prevents business from taking place without meeting the required number of members present. It is not an grant of authority for things to take place when you do happen to meet the minimum threshold.
For example, the quorum may be set at a majority of members. But without a rule stipulating that a majority vote wins, a quorum is not sufficient to tell you whether or not a majority can carry a motion. To go further, the quorum certainly does not override any requirement that a supermajority carry any motion.
In sum, my understanding is that a quorum shouldn't be read to explain what number is necessary to carry a motion. My understanding is that a quorum merely denotes the number required for any business to take place at all.