Screen black iMac 27inch

I have an:

Apple iMac 27 inch All-In-One Desktop PC (Intel Core i5 3.1GHz Quad-Core Processor, 2X2GB RAM, 1TB HDD, AMD Radeon HD 6970M with 1GB graphics)

I purchased the machine in August 2011. Recently the screen went black but you could still hear the chimes and the computer working.

I took it to the local Apple store and a Genius said that the LCD screen need replacing, we agreed a price and the work was due to take place last week.

Then a couple of days later I had a strange phone call saying that there was a problem with the logic board and it needed to be replaced doubling the original quote for the repair.

I have asked what is wrong with the logic board and they have said "The logic board is the main brain of the computer. Our technicians cannot isolate the problem to a specific component of the board but we are as certain as we can be that this is the component which is causing you the issue based on the tests we have run against your machine."

Can anyone explain what this means, they have also stated that the computer will work fine if I connect it to an external monitor. I am tempted to buy an external monitor as this is a more cost effective course of action.