Good words of advice from pnyklr3, though there's one detail I have a comment on...
Level of bacterial growth is more dependent on stocking than on tank volume. For instance, if you have three neon tetras in a 55g, you will certainly not have sufficient bacterial growth when you transfer that filter media to your 40g stocked with 3 fantail goldfish. On the other hand, transferring a filter from a 55g with 3 goldfish to a 40g with three neons will be (on average) more than sufficient.
So, it is possible to "instantly cycle" a tank by adding a good quantity of filter media, but it's important to account for varying stocking levels when making the transfer.
Translating "drops of ammonia" to "quantity of fish" is hard to do and may account for why you're getting nitrites. How high were the nitrites?