The machine should still boot with the power adapter plugged in even though the battery is bad, although, if the battery has been allowed to drain to nothing you may have problems. It will not boot with the battery out and the power supply plugged in.

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I have the same machine here in the house and have twice had to resolve these symptoms. It won't boot with a dead battery and it boots just fine with the battery removed. Once running, if you pop the dead battery in, it won't wake from sleep. Stick a good battery in and all is well.