Are you able to run the pre boot Dell Diagnostics to see if the hard drive or other hardware has failed? The light sequence says it is that very unhelpful Other Failure. 

http://www.dell.com/support/manuals/us/en/04/optiplex-9020-desktop/opt9020mtom-v2/power-led-diagnostics?guid=guid-52106768-c0ec-411e-a583-375a950df672&lang=en-us

3, 7 flashes indicate a possible memory error:

http://www.dell.com/support/article/us/en/19/sln284978/a-reference-guide-to-the-dell-optiplex-diagnostic-indicators?lang=en#2012_to_Present

Do you have other memory sticks that you can try in the system? If not, try them one at a time in each slot.

You can also test the LCD:

With the system plugged with a known good power source, hold the menu button located on the right side of the chassis.
While holding the menu button, press the power button to invoke the LCD BIST. At this point, you will see a series of colours flashing on-screen. Once LCD BIST is completed, the system should display the Dell logo and proceed to boot.

Let us know how you get on.

By the way, I appreciate the responses. Still no working solution, but having the community available to help is very much appreciated. 

Hi ChurchIT,

Were you able to test the memory one at a time or with some different memory from another system that is working? 

Thanks, 

Mick.

Mick,

I was unable to do so, because systems like it are all in use. However, I did check each memory stick individually in each memory bank leaning on the improbability that both sticks failed at the same time. The system still failed to post regardless of which bank each memory stick was installed in. 

I'd agree its pretty improbable that all of the memory failed at the same time, sounds more like a motherboard issue then if anything. If you can PM (envelope in the top right) over your service tag I can have a further look into this for you.

Thanks.