After a lot of interview rounds and a lengthy process for a tech job I applied, I got a phone call from the recruiter that they'd like to offer me the role. Over the phone, she asked me they are thinking of offering package of $xk/year and asked me what figure had I in my mind. I promptly said without thinking that I have in my mind between $(x+5)k and $(x+10)k since I need a job. She promptly said that ok then it'll be $(x+5)k. I haven't yet got any formal offer letter and this was a phone conversation.
My question is after the recruiter agreeing to the minimum of my specified range, can I negotiate more? If yes, when should I do it? that is, once I'm emailed the offer letter? Also, I was called in to grab some paperwork and one of the papers was the job application form which has a 'desired salary' field. Should I just mention the amount that I want in that field? Also, can I ask anything beyond $(x+10)k, which is the upper limit of the range that I specified?
EDIT Have another offer now and have two questions related to it which I have posted here. The suggestions for my questions were really helpful so would request the members who answered to have a look and suggest further if possible.