quad is short for quadrangle, a rectangular courtyard for people to gather/mingle and which is surrounded on all four sides by buildings. It could be a school playground or University recreation area, or a small area where army personnel train, or any similar area. e.g. "I'll meet you in the quad at lunchtime."
recap is short for recapitulate, to summarise, to re-state briefly
e.g. Teacher to students at end of lesson/lecture: "OK. I'll just recap." means "Finally, I'll just give you a brief summary of the main points of the information we have just been discussing. ..."
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