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And yes, if you remove the battery (while the computer is shut down!) and then plug in the AC adapter, you can run the laptop without the battery. However, if there is a power outage, or you accidentally unplug the adapter, the system will shut down and you may lose unsaved work.
The system may not be shutting down because of the battery (especially if it happens also when the machine is plugged in). Does the computer run hot or all the fans running at high speed?
You've been helped by a 16 year old.