BSOD ntoskrnl.exe Even after resetting to factory settings

[Blurberry]

SF Diagnostic File attached

I'm not a professional at using computers so I really need help with this.
I share this computer with my family, which also consists of some kids who play this game called League of Legends.
However, one day after I got on, the computer blue screened after a few minutes of surfing (around 20 minutes).

I tried System restore and Memory Diagnostic and it said I had something wrong with my hardware.

I was worried but because I had a back-up file, I went ahead and reformatted the computer, returning it to factory settings. However, after that the computer still had problems and I need help figuring out if it can be fixed without buying new hardware. Otherwise, I'll just buy a new computer.

I don't know if these will help but:

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[Blurberry]

The Bsod has been appearing even while no one is on the computer.
If you require extra info, the affected computer is a Acer Aspire with a Intel 8960 Processor.
It runs a Windows 7 Family Premium OS and is being affected by the error code 0x0000001A

 

[koolkat77]

Running Multiple antiviruses can cause things like BSOD's. Please uninstall Avast! and McAfee with their removal tools from the link below:
Uninstallers (removal tools) for common antivirus software - ESET Knowledgebase

Microsoft Security Essentials is recommended from a strict BSOD perspective.
Microsoft Security Essentials, Free antivirus for windows
Malwarebytes Anti-Malware Free
Good and Free system security combination

   Warning

Do not start the free trial of Malware Bytes; remember to deselect that option when prompted.

Run a full scan with both (separately) once downloaded, installed and updated.

Take memtest. Run for 8 passes and test each stick in a know good slot for an additional 6 passes.
http://www.sevenforums.com/tutorials/105647-ram-test-memtest86.html

   Note

Pay close attention to part 3 of the tutorial in order to rule out the faulty stick.

   Information

Errors are sometimes found after 8 passes. It can even take 20 passes for one error to show up.

   Tip

Do this test overnight, before going to bed.

Let us know the outcome.

 

[Blurberry]

Sorry for the late update. I've tried all the recommendations and the problem seems to be with the ram. There were over 400 errors over the 9 passes.

I still have to run the selective ram tests to see which stick is faulty.

However, now the problem is just opening the laptop and finding where the ram is.

Edit: I've also attached an updated rar file of the SF Diagnostic file if it would help with any further problems.

 

[koolkat77]

Bad ram can cause all sorts of problem.

Replace the faulty stick and if you continue to have BSOD's after that we shall troubleshoot further.

 

[koolkat77]

Have you replaced the bad RAM stick?
Any new BSOD's?