I've read that I should buy a PCI-Express 2.0 card if my motherboard has PCI-Express slot 1.0 or 1.1. However, a radeon card that I want to buy is a PCI-Express 2.1 card. Should I buy some other card in order for my game to work?
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generally speaking all pci-e 1.0x16 slots will work with any pci-e gpu... (yes, even a 3.0x16). the issue is some motherboards aren't supported by their manufacturers much, and without a propper bios update will fail to work with a gpu from time to time.
It's not a 100% certain thing, but generally, yes it will work as long as you have the latest bios in your motherboard. (you can shrink that small chance of failure to close to 0 chance if it's an asus motherboard, they have tremendous legacy support for their old boards)