Of course, the only way to survive this is with 200 red armor, a power shield (which loses its cells as the BFG fires but still helps a little) and the Auto-Doc tech. Of course, having invulnerability really helps.

See also: rocket jump

When the big green ball hits a solid object, like a wall, another player or in this case the floor it has a large explosion the the immediate vincinity of the big green ball. The explosion happening also triggers the line of sight hit, which means anything that can be seen by the explosion and the player firing the BFG dies (Unless they are at long range, where they take a little damage and run away).

To prevent people just shooting the wall or floor to trigger the ass-kicking line of sight hit, the programmers made the explosion do double damage to the person who fired it. It normally does some ridiculous number in the 32,000 range. With quad its 128,000. Double that. Yes, thats right. 256,000 damage. All to you.

It should be noted that the only thing that gets quadded is the impact of the big green ball, not the death rays or the line of sight hit. Firing the BFG while under the influence of quad damage its not much point, it just makes people mad.

Which is why I like doing it so much.

Anyway, a quick (belated) test of this shows that either quad damage does not affect the BFG ball, or the BFG ball does not do full damage to the player - it's possible to survive the quad BFG jump without even turning on a power shield (red armour is essential, however). It's possible to survive and not travel any further than you would if you'd used the BFG without the quad damage.