Do I need to reinforce the floor?

[Myka]

You're welcome.

You have space between the joists and the dropped ceiling? I'm just wondering if you have space to hide the beam(s).

I don't think 2x4's are going to be enough, and I don't think you need to add another joist (that's a real pain in @$$ to do). You want to transfer some of the weight from the 2 loaded joists onto other joists (whether you are using jack posts or a beam or WHY). If you use hangers to mount a small steel beam(s) perpendicular to the joists and attach them to neighbouring joists you can transfer the weight by preventing flex. Need good hangers and good hanger nails. i bet you would only need 4" steel beams.