I had a similar problem with a 2 year Old G50VT-X5.
There was a burning scent in the air, and the laptop continued to run on battery. When I plugged in the AC adapter, the whole thing turned off , and refused to start if the powerplug was in. The only way to start the comp was to remove the Power plug, remove the battery and reinsert it.
Sent it for repair, turns out the motherboard was completely fried, and they had to replace it. Got it back now and its all good, but be wary, that the replacement Mobo costs around 250 USD + repair costs and shipment. If it is worth it for you you should do it. For all other users ... buy a notebook cooler, as it can save this comp