Q:

How do I set up two monitors on two very different graphics cards?

I have two video cards, one running each of two identical monitors. My main card is an Nvidia GeForce 750 Ti, and the secondary is a Radeon HD5500. When I boot from live USB, the monitor attached to the HD5500 displays the desktop, and the other displays a gray screen with a flashing cursor (but doesn't do anything else). After installing, the desktop shows up on the GeForce, and the Radeon monitor is blank. 
I followed this guide to attempt to install proprietary Nvidia drivers, but it didn't work. So I tried downloading the package straight from Nvidia. I think that worked, but I'm not sure (I'm completely new to this). I then tried to select a different driver for the Radeon from the list on the Additional Drivers settings tab. Now when I boot Ubuntu, the Radeon monitor gets no signal, and the GeForce monitor displays black. 
What did I do wrong, and how do I fix it?
Ubuntu 14.04 dual boot with Windows 8.1 U1

A:

You didn't do anything wrong.  What you're trying to do is "impossible".  Both graphics cards have their own discrete memory and you need a shared frame buffer between the two to make this work...
The BumbleBee project is doing this for Intel and NVidia cards, but they don't support AMD/ATI cards, so unless you start programming this yourself for AMD, it's "impossible"... (for a certain definition of impossible.  Look up Charles De Gaulle)
Sorry to be the harbinger of bad news...

