Dell Vostro 1510, starts up accidentally

Hello,
I have a Dell Vostro 1510 laptop, it was working fine until last week that it just got crashed in middle of a game (note: no one was playing at that moment but the game was running). The screen turned to black with some white lines in screen, and the system froze. I restarted the laptop but it didn't start up again; I only had 3 LEDs when I tried to run the computer, one was my power led, and the other one was the Bluetooth LED, and HDD LED for a short time (3seconds) and I couldn't even able to see the BIOS page. After 2 days, when I was trying to starts up the computer on battery, it started to beeping (a beeps in 2 minutes or so). I gave it to a repair shop, and they couldn't find out what was wrong with it, but they told me it is probably the Motherboard's problem.
After I got my laptop from them, I tried everything else I could (changing RAM places, removing one RAM and trying to boot the system) but they didn't do any good.
It was all until today, I power on the computer with AC power and put it on its own for about 3 hours, after 3 hours I checked it again, and I saw there is a strait line in middle of screen (that was a great improvement, since before that the monitor didn't starts at all) and when restart the laptop, it starts functioning. I ran the many Diagnostic tests (in middle of one of them the system froze again and it restarted automatically but after repeating I found out it isn't because of a special part of hardware) and all of them returned nothing fail, every peace of hardware was fine, CPU, Ram, and most of the harddrive (since I couldn't finnish testing the whole HDD) but when I was in middle of a test the screen crashed and it restarted, then it didn't started again. But after some tries I ran the system again.
Does anyone know what's wrong with it?