problems with battery charging

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For a friend of mine.  Thanks for any help you can provide.

I own a compaq presario X1000 laptop and have been having a lot of
problems with the battery charging. Even when I have the laptop plugged
into the power socket the battery of the laptop keeps discharging. If I
remove the battery and install it back, it start charging.
If I dont remove the battery and install it again it drains completely
and the computer shuts off.

Is there any way the computer can keep running on the power supply even
if the battery drains completely?

What are the possible problems with the battery? Can it be optimized?

Any help would be greatly appreciated.

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