Actually, there might be a way. This would only work if you actually made saved at some point. It doesn't if you didn't, sorry
Open up your essay file again in TextEdit. Hover on the file name at the top of the title bar and you will see a little triangle. Click on it and then click on Browse All Versions. There is a chance that TextEdit saved a previous version of your essay. Just go back to the version you want and click restore.
Hope that works for you. Here's an image of what I'm talking about:
I suppose you can also check in ~/Library/AutoSave Information folder to see if the file is in there but I don't think that works when you deliberately quit the app.
I also find that if I open a blank textedit document and type in something and then quit and reopen Textedit, it resumes that file without any text loss. So maybe you can reopen TextEdit and get that essay file back. If you did have text missing, try browsing all versions on that file TextEdit resumed and see if you can go back a version.
p.s. I have no idea why you use textedit to write essays

If you can't afford Word or Pages, there's a great app called OpenOffice which you might want to try.
