RAM slot not working

Re-soldering a memory slot can be costly. It takes specialized equipment to solder this component and it has to be done right. If you tested the same stick in the two different slots and it fails to boot when the stick is in that one slot, then you have correctly identified a bad memory slot. To be cost efficient, it is propbably beter to change out the main board. There are however companies out there that will repair motherboards regardless of the problem, prices range widely for the standard repair plus parts plus shipping. E-Bay has several companies that advertise these types of repairs.

Assuming the header reads correctly, HP-Compaq 6720, there are only 2 RAM slots and it's possible you've overloaded the memory slots. Although a computer may technically accommodate a certain amount of memory, what many people don't know is that the maximum memory allotment is based on 64bit systems. Since most machines (well over 90% of personal use computers) are 32bit systems, they will never be able to MAX out their RAM memory. Your machine may show up to 2GB MAX of RAM but without running a 64bit system, 1GB may well be the most you can run. If you've got 2 1GB sticks of memory, it makes sense that only 1 slot is being read while the other is being ignored. Hope this clarifies your predicament. Best of luck.
Greg

I am having the EXACT same issue on 3 Compaq/HP 6720s. Both machines have been reimaged to WIN 7 32bit Pro. Both machines will only accept 3GB RAM.
2GB bottom slot, original 1GB top slot.
With 4GB you get the BSOD.