The desktop won't come on, just 3 loud beeps
When I turn it on I just hear one long beep followed by 2 short beeps
These are post codes (aka "beep" codes) that the BIOS gives when there's a hardware problem. Unfortunately, there's no universal standard for what means what between board manufacturers (or even between boards by the same!)
I'll need to know the exact model to be able to give you more than that.
Have you recently done anything with the memory or other hardware? This might have something to do with it. Any details you could provide will help. I'll update my answer when I have more info to go on.
In the meantime, here's a link to HP's troubleshooting guide for beep codes. The list at the bottom does not give an explanation of this specific code, but the steps should be followed anyway. I've seen posts about this problem, but nobody had anything useful to say and were all old and closed.
II checked out that document you linked to, saw some more stuff at bottom..
Ive attached the images. The 2nd one I think is correct, and looks like a video issue.
Have you also tried re-seating the ram? What happened prior to the issue starting?