Understanding how to find the fundamental fequency when given the beat frequency?

[lunchtime_browser]

1) You choose F1 and F2 so that you get the correct answer. What on earth is a negative frequency? You know before you start that you are looking for an answer which is positive.

2) The tuning fork frequency is 256 Hz and the beat frequency is 6 Hz. So there are TWO possible values for the frequency of the string:

Either
(i) f = (256 - 6) = 250 Hz
In which case FB = (256 - 250) = 6 Hz

Or
(ii) f = (256 + 6) = 262 Hz
In which case FB = (262 - 256) = 6 Hz

Up to this point, you have no way of knowing which it is. To find out, you load the prongs of the fork.
What will that do? It will lower the frequency of the tuning fork slightly, to something a bit less than 256 Hz.

Now you can see that it can't possibly be option (ii).
262 Hz - [something a bit less than 256 Hz] would make FB bigger than 6 Hz and that is not what happens here. FB reduces to 4 Hz.

So we are dealing with option (i) here and the wire frequency f = 250 Hz
(loaded fork frequency - 250) = 4 Hz
loaded fork frequency = (250 + 4) = 254 Hz

The bottom line is that this is all about solving logic problems; not just plugging values into a remembered formula.