What is the best way of removing 1.25" of concrete from slab floor?
Generally you'll not be able to remove 30mm from a concrete slab without cracking up the remainder. In the UK a typical slab might only be 100mm thick. 

However (in the UK at least) it's unlikely you're looking at the floor slab, but rather the floor screed (sharp sand and cement 3:1) used to finish off the floor. The 100mm concrete floor slab is actually a sub-floor slab underneath the finish screed.

Now, the floor screed can be hard, but it's laid fairly dry and is often not too difficult to chop out. Typical thickness ranges from 50mm to 75mm (but can be as little as 25mm...) 

Starting in a corner, try chopping a small exploratory hole with a sharp cold chisel and a club hammer. Check for pipes first, as radiator pipes typically get buried in the screed around the edges of the room.