If your machine is using the nVida 8600 GPU it was know to fail and had class action law suits on it and Apple had a replacement program for a while. Unfortunately there is little you can do to repair it. I have a stack of these machine all with the same issue.
'''UPDATE'''
Yes, Your machine does have the nVidia GeForce 8600M GT Video Card
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The cost of replacing the logic board and probably ending up with the same problem would not be wise. I would suggest going to either the 13” or 15” 2012 non-retina model.
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I got one last week for $250. It is repairable, upgradable and will even take the latest OS X Catalina.
If your machine is using the nVida 8600 GPU it was know to fail and had class action law suits on it and Apple had a replacement program for a while. Unfortunately there is little you can do to repair it. I have a stack of these machine all with the same issue.
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'''UPDATE'''
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+
Yes, Your machine does have the nVidia GeForce 8600M GT Video Card
If your machine is using the nVida 8600 GPU it was know to fail and had class action law suits on it and Apple had a replacement program for a while. Unfortunately there is little you can do to repair it. I have a stack of these machine all with the same issue.