Power up,(Balance Laptop)

Chances are that your battery doesn't / wouldn't hold a charge anymore. The laptop runs off battery power. When it is plugged in, the AC power "goes right through the battery", to support the laptop's operations. When the power cord is unplugged, and the battery doesn't hold a charge, the laptop will shut down, as your battery is (too) low to continue supporting the laptop's operations.

The easiest way around this problem is to get a new battery for the laptop.
If your battery is (quite) new (and a Lithium battery), you may wish to see if you can "revive" the battery's "life" ... first by depleting the battery (completely), then charging it (until it shows it is full, plus a couple more hours), and repeating this process a few times. If you're lucky, the battery will slowly/gradually start to hold some charge, but, don't expect it to be very long.
The (preferred) way to use re-chargeable batteries is to charge it fully, unplug it from the external power source, use it until it is depleted, then plug in the external power source to charge it up (fully) again .... and NOT to keep it on a power source all the time.

Unplug the battery and plug in the AC power plug into the laptop;

Does the laptop start up and stay on?

If yes,

Then put the battery in WHILE IT IS ON; See if it starts charging.

If it does not; You have a bad battery and need to replace it.

If it does start charging here's what happend;

Some laptop batteries if they are fully depleted on charge will not let you power up the laptop EVEN IF the power cable is plugged in. In some cases the PC won't turn on at all.

The reason for this is because the battery must have a certain % of charge in order for you to start the PC even IF the power cable is plugged in.

By removing the battery we eliminated the battery as being a cause for this issue; Once the PC is up and running in windows.. LET THE BATTERY CHARGE! You'll want at least 10% charge minimum for the PC to start up.

If however the battery replaced still doesn't hold a charge it's going to be the AC Adapter power cable that's causing the issue.

Best advice start off with the power cable replacement first; This is the cheapest to fix; Otherwise the battery can cost in excess of $50-$150 or more. But universal AC power adapters can be cheap $70 or less which varies depending on laptop.

POWER SUPPLY & BATTERY PROBLEM
If the laptop battery is more than 3 - 4 years old then the battery could be worn out. Rechargeable batteries have a finite number of charge and discharge cycles and will lose their charge capacity over time, i.e. won't charge to 100% and gradually the charge reduces until the battery won't charge up at all. OR The battery shows a 100% charge but when the adapter is disconnected the battery drops off to zero capacity in a very short time. If the battery drops to an unacceptable charge level then the battery needs to be replaced. When you connect the power adapter into the laptop, does the power LED light up? If not then - The battery adapter or the adapter's power lead or the laptop's power socket could be faulty. Check power adapter's DC plug that connects into the laptop power socket with a multi-meter, the voltage should be slightly higher than the voltage that is printed on the label on the bottom of the adapter. If the voltage is zero or way below the voltage that is printed on the label then the power adapter is faulty and needs to be replaced. If the adapter is OK then connect it to the laptop and power it up, then look at the power/battery charging LED light and wriggle the power plug in the laptop's power socket. If the power/battery LED light flashes and in a certain position this LED light stays firm, then the power socket is either faulty or it has a dry solder joint where it is connected to the motherboard. This can be repaired but it requires the laptop to be completely dismantled to get at this power socket and repair. If you cannot DIY then I suggest you get a quote first.

POWER SUPPLY & BATTERY PROBLEM
If thelaptop battery is more than 3 - 4 years old then the battery could be worn out.Rechargeable batteries have a finite number of charge and discharge cycles andwill lose their charge capacity over time, i.e. won't charge to 100% andgradually the charge reduces until the battery won't charge up at all. OR Thebattery shows a 100% charge but when the adapter is disconnected the batterydrops off to zero capacity in a very short time. If the battery drops to anunacceptable charge level then the battery needs to be replaced.Whenyou connect the power adapter into the laptop, does the power LED lightup? If not then -The battery adapter or the adapter's power lead or the laptop's powersocket could be faulty.Check power adapter's DCplug that connects into the laptop power socket with a multi-meter, the voltageshould be slightly higher than the voltage that is printed on the label on thebottom of the adapter. If the voltage is zero or way below the voltage that isprinted on the label then the power adapter is faulty and needs to be replaced.If the adapter is OK thenconnect it to the laptop and power it up, then look at the power/batterycharging LED light and wriggle the power plug in the laptop's powersocket. If the power/battery LED light flashes and in a certain position thisLED light stays firm, then the power socket is either faulty or it has a drysolder joint where it is connected to the motherboard.This can be repaired but itrequires the laptop to be completely dismantled to get at this power socket andrepair. If you cannot DIY then I suggest you get a quote first.

Though your observations are explained well, I just want to clarify once again.
If you leave the adaptor plugged on for charging does the battery get charged up to display or show on display in the power options when switched on that the battery is fully charged.
Now if the adapter is removed does the laptop last much on the battery or does it drain off easily? If so the battery is bad and must be replaced.
Also if you are able to check the voltage on the battery terminals of the laptop and it shows a good reading then the battery is a suspect.
please check the link below for some tips on the power adaptor.:

http://www.fixya.com/support/r5164850-laptop_power_supply

First of all, you need to make sure your power adapter is providing enough power to both run the laptop & charge the battery. To to do this is to compare your laptop voltage & amp requirements (written on a sticker on the bottom of the laptop) with the voltage & amp ratings of the power adapter.

Voltages should be the same, or the power adapter's voltage about up to 0.5 volts more than the laptop's voltage rating. Amps of the power adapter should be at least equal to or greater than the laptop's rating. It should never be less than the laptop's rating, or else the power adapter will be providing insufficient amperes to charge the battery. Of course, the power adapter's nominal amperage might be ok, but it degraded on usage. You can verify this by testing using a good or new power adapter.

If the battery does not charge using the new power adapter, take the battery off the laptop, & using a dry tooth brush, scrub in between the fins of the battery connector on the motherboard & on the battery it self, then put the battery back into the laptop & try again.

If this still did not resolve the problem, then the battery itself could be bad. Try it on another laptop, or try a good or new battery in your laptop.

If this did not work, there might be a possibility that the charging circuitry on the laptop's motherboard is defective & needs servicing.

In order to do this, the laptop needs to be disassembled & the motherboard checked for defective/burned out components in the area near the DC jack. Any defective/burned out components found should be replaced, or you might need to replace the laptop's motherboard altogether.

Shows a clear indication of the battery to be faulty. So in the first place remove the battery and checck the voltage, now keeping the battery off the laptop, plug the adaptor to the mains, Now checck the voltage in the battery terminals which is in the battery compartment. If voltage is present then the battery is bad as it is not charging.
However if there is no voltage then the main board has a fault in the charging section which must be looked into.

Sounds like you have a faulty ac adapter or jack on the laptop and its not charging the battery fully. More like a trickle charge. I would recommend leaving the laptop on default power settings or the best power settings and see if their is a differance in life or either shutdown the laptop and let it charge until the light shows fully charged.

Always start with the simplest solution first so start by cleaning all the contacts, AC adapter laptop plug, laptop socket, laptop battery contacts and battery contacts. How have you confirmed power output from the AC adapter? have you used a voltmeter to ensure you are getting power at the laptop end? Have you used a voltmeter to test the battery? Assuming you have tried all this don't forget that almost any component can interrupt power on some laptops, however you're 2 main culprits with this kind of problem are usually power controller (voltage regulator) and in some cases, on a laptop of this vintage, the charge controller built in to the battery. Try it with a different battery if you have one.

Ive had a look at the manufacturer web site but ther doesn't seem to be any driver for your model. You should be able to turn the battery warning alarm off in the management settings