First of all:
DON'T run it from ANY AC adapter, at least before being 100% sure that:
1) you have good (new) rechargeable batteries in the battery pack
AND
2) batteries make good contact (oxide free, etc) with the battery contacts of the calculator.
The chances of permanently damage your calculator (almost with no hope of repair) are VERY high if any of the abovementioned rules are not obeyed.
Suggestions:
a) Clean the contacts;
b) Rebuild the battery pack with new rechargeable AA-size cells;
c) Recharge the cells outside the calculator (i.e.: don't use the calculator as a charger);
d) Always run your calculator from batteries, never from AC power, to avoid the excess voltage the AC adapter may cause (more than 8 volts, which certainly will fry the chips inside; particularly the "continuous memory" RAM chip, even with the calculator turned off);
e) Eventually run the calculator from alkaline batteries. The voltage (3 volt) will be somehow higher than from rechargeables (2.5 volt), but within tolerances (there should be no problems, but this one is at your own risk, no warranties).
You can look here at the MoHPC for more detailed explanations, but first be safe not to damage your HP25C.
Good luck and Welcome!