It does have an x. Has become an adjective of course-taking on an impossible and pointless challenge is described as Quixotic. And the book also gives us "tilting at windmills" to describe the same thing.
So, people on TTF read proper books, eh? Who knew? I was listening to an old ‘In Our Time’ podcast (It’s a BBC radio broadcast, for our non-UK friends) about the book the other day and the learned people speaking with Melvyn insisted it was pronounced ‘kwiks-oat’. My girlfriend, a fluent Spanish speaker, is having none of it.