There should actually be no visible degradation of the display when using an adapter. It all depends on the LCD that you are using, if it only has HDMI for input then most definitely it can support HD resolutions. If it is native 720p then set your resolution to 1366 X 768 and if it native 1080i then set it to 2078 X 1080. Windows would actually show you the recommended resolution set that you need to use as well.
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There should actually be no visible degradation of the display when using an adapter. It all depends on the LCD that you are using, if it only has HDMI for input then most definitely it can support HD resolutions. If it is native 720p then set your resolution to 1366 X 768 and if it native 1080i then set it to 2078 X 1080. Windows would actually show you the recommended resolution set that you need to use as well.