How long does it take a bone fragment to come out after a tooth extraction?

i had 2 teeth removed 5 days ago (one had been a root c) i went as i had been having toothache for a few days he said they were rotten and took them out i hoped that was it. the pain never went away in face it got worse and when i took a look the area looked disgusting i went back 2 days later and he gave me a script for metronidazole 400mg 3 times a day, i have been taking them and the infection is definatly clearing up, but i am still in so much pain, the area that was the root c has some white bone sticking out but not out the hole its coming through the side of the gum its very sharp and so painful i am sick to death of the pain now as its been over a week since the toothache started. this shard has been there for 2 days and not moved will it come out itself? does it take long? i have even tried pulling it as the pain is so bad but its stuck fast and i ended up in more pain so i've left it. thanks in advance