The two issues aren't related - if you have a script that processes the captcha on your page, then it'll work.
Brad is right, however, to point out that you (anyone) shouldn't be leaving the action blank. It works in most cases, but it's not really supposed to work, so you (anyone) shouldn't rely on it.
It's also a bad idea, generally speaking, to blindly copy-and-paste code of of other people's sites onto yours.