Acer Aspire 5750z Doesn't Start

In most cases it does't even start, don't see anything.
I took out the battery, it works on the AC adapter.
On a few trials, the laptop starts and for a few seconds the windows blue window logon appears and disappears. the resolution, in those few seconds, appears low.
I can't enter BIOS with F2.
It doesn't start from the CD, from a bootable disk.
Only on the few cases, when i see the windows logon, the hard disk indicator light lights

 

If it works fine with the AC adapter alone, but not when the battery is installed, then it seems clear the battery is bad. It likely has developed an internal short as often happens with multi-cell batteries. This is probably the most common cause of car batteries that fail prematurely.

The result is the battery fails to charge fully, fails to maintain a charge, and/or draws too much current from the charger. It sounds like you need a new battery.