well your motherboard has a pcie x16 slot so you are good on that front....the only thing is I cannot find the psu specs for that machine on hp website. can you open your case and find out what wattage your psu is? and a 550ti would be a nice upgrade for that rig.
yea for sure you are right there is no guarantee...but the chances are slim...really there is no guarantee that any computer components will work together perfectly. there is always small isolated instances of certain combinations of parts not working together