Maxine.
There are hundreds, possibly thousands, of faults that could prevent the engine from starting, so the chances of diagnosing the problem from the tiny amount of information you supplied the forum so far are negligible.
I shall assume the RAC mechanic checked the obvious things like fuel level, fuel pump, relays, coils, EMC etc. Do you have a full service history for the car? What mileage has the car done? Has the car displayed any other unusual symptoms before it failed such as smoke, high oil consumption, low power, high fuel consumption, coolant loss? if you can answer those questions we might, and I stress might, be able to home in on the fault. For a quicker result I suggest you get it diagnosed at a specialist VAG garage or, failing that, an Audi dealer though I must admit I don't have a high opinion of them.
Did the trader you bought it from guarantee it in any way? Was that trader commercial or private? Your legal rights depend very much on precisely what he guaranteed and for how long. Some guarantees only last three months and cover so little that they are not worth the paper they are printed on.