I'd go with a clean installation of OS X on a pre-owned Mac, anyway. You never know what the previous owner has messed up, anyway, and this certainly looks to be the case. So: Boot from the installation medium (probably came with a DVD for that case) by holding down the "c" key at boot time (the DVD in the drive, of course). You'll have to reinstall all 3rd party applications, of course, as well as bring back any data you'll still need from a backup. (But since this wasn't your computer before, I guess you didn't have any data on it, anyway...) - In the installer, choose to erase the drive and install. Cleanest way to bring the Mac back to life.
If the problem persists, though, it looks like a hardware problem. Right now we can't really decide that. (Until you've tried reinstalling the system software.)