If a part of the tooth is missing and you have pain then the decay must have entered your pulp chamber which will innervate the nerve endings and that is why you have pain. considering your age i would advise you a root canal procedure than an extraction, thereby you can preserve your natural teeth. Yes the problem will be resolved by root canal. The root canal procedure can be done by an specialist (Endodontist) and it would be done in a single appointment. Please do get a crown done after the root canal procedure to preserve your natural tooth
Next Steps
Please visit a dental clinic or hospital as soon as possible
Health Tips
It is good that you had taken a course of antibiotics which will help in any infection. Please take a painkiller as a remedy for pain. Tab Zerodol or any other Analgesic would do