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CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ Amazon)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: Gigabyte GA-970A-UD3 ATX AM3+ Motherboard ($100.64 @ Amazon)
Memory: G.Skill Sniper Series 8GB (2 x 4GB) DDR3-1866 Memory ($68.00 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Video Card: Gigabyte Radeon R9 270X 2GB Video Card ($204.98 @ SuperBiiz)
Case: Antec One ATX Mid Tower Case ($29.99 @ NCIX US)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($59.99 @ NCIX US)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($16.98 @ OutletPC)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($88.98 @ OutletPC)
Total: $769.51
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-19 14:31 EST-0500)
Here is a good foundation to start with:
At this budget, AMD is a better option in my opinion. This CPU is great because you can overclock it well and really increase your performance.
This is a solid CPU cooler that will handle light/medium overclocks. If you don't have any plans to overclock you can leave it off.
Decent motherboard from a reputable vendor, I would wager you aren't going to be going for any significant overclocks, in which case you don't need to spend a lot on a motherboard.
8GB of RAM is the standard, AMD CPUs benefit from a little faster RAM, so 1866 is a good compromise between speed and cost.
1TB storage drive, not much to say here, buy as much space as you need. Eventually I would consider a SSD for your operating system and applications, it will make a big difference in performance. Put your media and games (assuming you will be gaming) on the storage drive.
This is a solid GPU, should do well in most games at 1080P, and Gigabyte is a good choice for aftermarket coolers.
Cases are subjective, this Antec One is a good beginner case, nothing too flashy, but will hold what you need and has some cable management. You may want to browse Newegg and see what all is out there. I would take a look at offerings from Corsair, Antec, Silverstone, Fractal Design, NZXT, and CoolerMaster.
This XFX has all the power you will need for this GPU, and any additional HDDs or overclocks you might do. It is made by SeaSonic which is one of, if not the top PSU vendors. You can pay a little extra to get a modular or semi-modular PSU, which will help reduce wire clutter, but this is more of a luxury item.
Optical drives are all the same, generally just go with the cheapest one.
Hope this helps explain why I chose the parts I did, and you can use this as a basis and modify depending on your needs.