If Memtest86 is finding errors, then you can be pretty sure have a hardware fault somewhere (and I'm pretty sure it's a memory fault, but it's not guaranteed to be).
I would start with Memtest-ing each memory stick in a different, known-working system. If you're still getting memory failures, then you're in luck - just go replace the memory.
If the memory tests clean in a working system, then the real fun begins. Factory-reset the BIOS and strip the system down to bare essentials - One stick of memory, no DVD drive, just a hard drive to load the system from, no graphics card if the motherboard can do onboard graphics, CPU, no expansion cards, no USB devices. If this fixes things, then you know it's one of the components you removed causing the issue. Test those one-by-one in another, working system.
If you still have errors, then it's probably the motherboard or CPU or harddrive. Test the CPu and harddrive one at a time in another known-working system. If they check out fine, then you know it's the motherboard.