Asus laptop shuts down when starting the machine with no charger plugged in
Solution 1:
Either the battery very likely needs replacing; or, possibly the AC adapter is not charging well. Based on your post, it seems like battery.
Startup takes extra current (power) which likely explains why it is shutting down as it does.
Try resetting the battery.
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Allow the battery to fully charge.
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Unplug and let the batter fall to shut down in normal use (may take a while).
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Shut down or let the machine remain shut down.
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Plug in and allow to charge to full.
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Restart and use normally.
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If you get the same battery failure when starting up unplugged, then replace the battery with an OEM new battery.
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If during these tests, you perceive charging is deficient because of the AC adapter, replace it or ask a local shop to test it.