I just bought an Alienware x51 gaming PC and it should be arriving tomorrow via FedEx and I didn't buy any extended warranty on the system. Could I still buy extended warranty on the PC I've purchased even though it's already shipped?
You really need to buy the extended warranty when you configure it. That is the cheapest way. Now you will pay a lot extra to extend it. Most computer hardware that fails does so during the first months or first year. I would skip it, but you decide. Desktops are easy to work on to replace parts yourself.
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As Osprey has said, you can definitely extend the warranty. Please refer to the article on Extend Your Warranty by Dell Support.
Hope this helps!