I put this here as a sticky for convenience of reference to anyone with the infamous "Dell laptop battery quits within weeks of warranty expiration" issue.
They do indeed quit as soon as the warranty is out. They are told to by a routine in BIOS via the 'smart chip' in the battery. That's right, Dell laptops are programmed to sabotage themselves in order to force the sale of expensive batteries.
Toward the end of the linked thread is a link to software download that resets the chip in the battery to 'all-new' state. Like all software, use at your own risk. We're not endorsing it as safe and effective, we won't know that until someone tries it and reports back.
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