Macbook pro issues, screen gets pixelated, freezes, shuts down, powers back on

I have a 13" Macbook Pro a1278 that I bought on January 2013. A couple months ago I had problems with it. It would shut down randomly sometimes. Then, on certain occasions, my computer would freeze, the screen would get pixelated and started moving, then it would proceed to shut down and power back on by itself. In addition to this, it would give me the "bad RAM" 3 beeps issue. In order to resolve this, I called apple. I followed their instructions for software problems, my computer would turn back on but hours or days later it would go back to the same issues. I decided to buy a new RAM. I did, with the same specifications as my old ones. I erased all my old stuff just in case. It worked fine for a month, and now, for the last two or three days, it has turned off and on randomly on two occasions as I turned it on. And just a few hours ago, the whole pixelated screen, freezing, turning off and on by itself issue, came back.
If anyone has any idea what the issue might be, please tell. I would love to get this fixed, I need it for school badly. I attached a picture of the screen when it gets pixelated.

Welcome to the Mac-Forums.

Normally I would suspect a graphics problem with your MacBook Pro but your model uses an Intel HD 4000 chipset which is integrated within the CPU. That is usually stable. You changed the memory for new modules and also reinstalled clean so we can probably eliminate bad memory and software. Intermittent problems are some of the most difficult to resolve especially the type of symptoms you've been experiencing. Is there an Apple store nearby where you live? If there is, try making an appointment to take your MacBook Pro to them and have them run diagnostics on it. There is no charge for diagnostics if done at an Apple store. However, this time of year appointments are hard to come by so don't hesitate to call. Hopefully you can get the problem resolved over the Xmas Winter break before you have to return to school.

Bear with me for a bit..... Have you ever dropped your MacBook Pro or spilled a liquid on it? Carried it together in a back pack along with a load of books? (I think you understand why we ask.) Anyway, let us know if you take it to Apple and what they say.

Based upon what you have already tried it's likely you have a logic board failing, possibly with a bad memory slot. If you have 8GB of memory or more installed (If 4GB running with just 2GB installed for any time would be very painful) you could remove the memory from one slot, then test. If it then fails move the the memory into the other slot. If the fault goes away with only one memory module installed in either the upper or lower slot you will have a bad or intermittent slot. You could then have the logic board replaced, or run with either 4GB or better 8GB in the good slot. Using only one slot isn't optimal for max performance, but would get you going.

If you test as above, yet the fault comes back again, then you'll likely need the logic board replacing. Do you have an ARS or AASP in your country?

I haven't spilled anything on it or dropped it, but I do carry it around on my backpack (on the back part that is intended for laptops) or purse. Unfortunately in Puerto Rico there are no official apple stores, so I would have to take it to a local Best Buy store (in which I bought my laptop) for any service and I would get charged $100 and 1 week to see what the problem is. But, I guess the problem is out of my hands and as soon as the holidays pass I'll take it. I'll let you guys know how that goes.

*I do need the whole 8gb in order to run the programs I need for school. I'm actually doing my portfolio to get into architecture graduate school. And I did run it for a bit (about a week) with only one of the cards (4gb),it worked fine, but it only worked for opening Microsoft word and things like that, but nothing like AutoCAD or Adobe programs like Photoshop or Illustrator, which is the whole reason I bought it.

Thanks for replying!