The Daria Episode Guide:
Quinn the Brain

Can renegade surgeons transplant your brain while you sleep? The frightening truth, next on Sick, Sad World.

Quinn is told by Mr. O'Neill that she has to get an A on the next paper to pass "but please take this in a constructive spirit." Quinn doesn't really care until Daria comments that she'd end up being the oldest Freshman, which lights a fire under Quinn.
Quinn, of course, immediately goes to Daria and offers her money to write Quinn's paper. Daria and Quinn agree on a price, then Quinn makes a comment about Daria having something to do on Friday night, so Daria decides not to do it. Quinn has to blow off a date (and takes two hours to get ready to do so) to actually write the paper.
She ends up doing well on her paper (by her and Mr. O'Neill's standards, anyway), and Mr. O'Neill reads it aloud to the class. Suddenly, "smart" is popular (at least, the shallow, pseudo-intellectual bull Quinn goes after). Quinn starts wearing all black because "that's what really deep people do" and soon other girls are wearing it too, which doesn't please the Fashion Club.
Meanwhile, Daria is getting really depressed about people thinking Quinn's "writing" is good, and even suffers a heart-to-heart talk with her father. Soon the Fashion Club attempt to approach Daria to see if she can get the old Quinn back, but they all freak. Soon after that Joey, Jeffy, and Jamie follow, and Daria guesses their intent before they manage to get away.
She dresses up like Quinn and convinces Joey, Jeffy, and Jamie to ask her on a date, which gets Quinn back to normal real quick.

Quinn: Don't worry, all I have to do is get like an A on my next essay and I'll be fine.
Daria: I'll take the Vegas odds on that one.

"I'll bet you do, you... you high-school Casablanca!" -- Brittany ("Quinn the Brain")

"Don't worry. Today its brains, tomorrow pierced tongues, and the next day pierced brains." -- Jane ("Quinn the Brain")

Jane: Well, you know, condition people to expect nothing and the least little something gets them all excited. Ask Pavlov.
Quinn: The Custodian?
Jane: Woh... Nevermind. Return to your world, and I'll return to mine.

Wraith's Ramblings:
Although this episode started off well enough, it rather fell apart after the first commercial break. Just about everyone goes out of character at some point in this show, and Mr. O'Neill simply should not have survived asking Daria to take tutoring from Quinn. This is where second season starts to go downhill fast. It has retained quite a bit of popularity in some groups simply because it has Daria looking cute in it.
Also see my Rambling of 3-3-98 and 4-12-98.

Sick, Sad World (the site) was created by Wraith