Just acqured a used Alienware Desktop Computer and it won't power up

[Phil]

You either have a dead power supply, perhaps the one you tried was not compatible, or the motherboard has a problem. The fact you aren't even getting the POST screen rules out memory, disk drive, etc. I've had an alienware laptop for 15 years and it still works. Huge, heavy with metal cabinet, extra cooling and so forth as has been mentioned. I would imagine the desktop versions are similar.
Bad video card would keep you from seeing the POST screen but you would still be seeing fans, lights come on and so forth.