Just my penny’s worth...
If the laptop is still under warranty and you are unhappy with it in any way, take it to the retailer and get it looked at. If you bought it online, then find out how a replacement battery can be arranged. You've nothing to lose by asking.
Secondly are you sure you're shutting it down. Don't think I'm being condescending, but my wife bought an HP, and thought she was shutting it down. It was going into some sort of hibernation mode, which still used battery power to maintain the RAM.