Inspiron 5150

The battery for my Inspiron 5150 will not recharge and the computer stayed on battery power even though it was plugged into A/C.  The computer will run on A/C if the battery is removed. Dell Support told me the motherboard needs to be replaced.  My computer is 5 years old so replacing the motherboard is not a sensible decision.  My question is whether the problem could be with the 5-year old battery rather than the motherboard.