Samsung Galaxy A3 (2017) User Manual

I'm sorry but damages like that are not covered by warranty, you will have to pay if you want it repaired. Is the crack really bad? If not, I suggest just continue using it and use a case. I have friends who used phones with cracked screens, it's all cool. They still can use it normally except for watching videos, which then sucks because of the crack. But yours in the back, so it should be ok.

Maybe you should try and remove the back off it, where the battery is, then take out the battery, dust it a bit then put it back in. 

Try and turn back on your phone.

If not, go see a phone technician, the battery could be damaged/broken, but where did you buy the phone? If it was offline (Ebay, Amazon etc.) I would see the phone store its running with. (Like Tesco Mobile, EE, Three, Orange, Virgin etc.)