I have seen this many times, the battery is shot
It is a matter of a new battery, there are plenty out there available online that are not full priced. For some models of computers I have seen them as low as $20 USD. I do not go for the cheapest one I see. I shop prices yes, but I also look at the reviews, which is most important.
Sometimes if you buy cheap, cheap is what you get. There is a long life battery on Amazon with great reviews, just confirm your model # of the BATTERY before you purchase.
In my opinion I would get that one, but be aware that HOW you the battery you have now may have led top it's early demise.
If you let the battery drain down completely, say less than 5% charge, that will kill the battery. If you leave it plugged in 100% of the time, that will kill the battery. You need to let the battery cycle, that is let it charge fully, then unplug it for a while and run it, let it go down then plug it back in. Lithium batteries do not have a memory as the nickel batteries do. they can be plugged back in at anytime of the discharge cycle if need be.
I keep my laptop plugged in all the time, and about once a month I unplug it to let it run on the battery. I watch the icon as I use my laptop, and plug it back in at about 10% charge left. There may be an alarm already set on your laptop for this margin, maybe higher, that's ok too.
Mine is a an older Dell Core2 Duo I have had for years, and was used when I got it, still have the original battery in it. Yours is a nice rig, the small investment of a new battery is a wise move. Get a spare and keep it charged if you find yourself in places that you cannot recharge at.