You can save some headaches by calculating the squares of constants before putting them in your patch. For example if 12200 is constant and always squared you can just put 148840000 and avoid calculating the square each time you need to process a new value.
Not sure how to do this with [expr]. You can always patch it using several math objects for each calculation or if you use IRCAM's FTM libraries ( http://ftm.ircam.fr/index.php/Main_Page ) you can do it like this:
For a lot of reasons I prefer to write f^2 in an expr as f*f: [expr $f1*$f1]. The general-purpose exponentiation of $f1^$f2 is [expr pow($f1,$f2)]. But you seem to have worked that out, so I'm not sure what you need help with.
The thing is that pow() can be a fairly expensive function call, whereas multiplication is just a multiplication. I'm not sure if the implementation of pow() used by Max in expr optimizes for simple integer exponents, but I wouldn't rely on it.
Do I understand well that the original question is that you want to isolate the term (f to the power of 2) so that you can find the inverse relationship? It's too long ago I worked with the math to do such things. Keep in mind however that the inverse is not a function, as the function you describe is parabolic; you would need to define the inverse with two functions.