When the battery dies it should run on AC power. Yes those batteries do recharge, but they do need to be replaced from time to time. If it won't run on AC power, you may have a bad cord or internal connection problems. I am not an expert in laptops and their power supply units.
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If your equation requires either a complex combination of variables to get the desired result OR you can't seem to get the result regardless of the variables....it ain't your variables, it's your constant.