Why won't my ps4 start?

I doubt that the popping noise came from inserting a PS3 game in your PS4. Since you didn’t open up the Playstation it’s just a wild guess, but it could be that the power supply has blown. I would recommend to open the playstation and first do a visual inspection. The PSU is enclosed in plastic so you won’t see much, but it will have a part number/code on it, if you google that part number you’ll find stores that sell them. I’ve seen people around here buy them from Amazon with great success :-)