Mathematical Modeling Question, need help

This question deals with a 1-parameter equation:

fc(x)= (4/x)+(x/2)-c for 0≤c≤ 2.3

a.) Use algebra to find the positive fixed point p(c) of fc(x) (in terms of c) and identify its exact interval of existence

a.) Ok so for the fixed point I got p(c)= -c+√(c2+8) I'm not sure if this is correct though. I used the quadratic formula to find this after setting the initial equation equal to x and solving in terms of c. Now for the interval of existence I need to take the first derivative of the initial equation which I have as -4x-2+(1/2). Is this right? And I think I'm supposed to plug in the fixed point i have into this first derivative to get the interval of stability for c by setting an inequality up like: -1<c<1. This would be the interval of existence for the parameter c right? Please correct me if i'm making any mistakes. I can't do this last step because i end up getting ridiculous numbers.