I can make a few suggestions.

(1) Avoid use of capitalized symbols, expecially those tht might already be built-in symbols. D is an example.

(2) You might rename x to xx wherever it appears with the big variable (I'll call it bigD), and expand as a series in xx at the origin. Then substitute xx->x. I think the expansion would be to order zero, else total degree of resulting Taylor polynomials will exceed 4.

(3) Notice that this gives a biquadratic in x so the solution will be done by iterating solutions for quadratics. This gives those complicated formulas. One can get a more concise form via Root objects (prametrized in terms of the variables other than x). To do this, add a "junk" linear term to avoid the biquadratic, and set Quartics->False in Solve to avoid the Cardano-Tartaglia form of result. Then remove that junk term. If expr is that big quartic, then the code to do this would be:

Solve[expr+junk*x==0,x,Quartics->False]/.junk->0

Hi Daniel,

I think I might solve it if I get rid of square roots in it. I attempted to expand the f5 function in order to get rid of square roots because for my situation Det[f5]==0. I attempted by using Eliminate command as well as GroebnerBasis command. For some reasons it does not work...it takes more than 15 minutes.

eli=ReplaceAll[Sqrt[f5,D^2 - (x - c)^2]->u]
Eliminate[{eli==0,u^2==D^2 - (x - c)^2},u]
GroebnerBasis[{eli,u^2-D^2 + (x - c)^2},x,u]

Am I right with the code?

(1) It would help to have actual code provided in the question.

(2) Might want to look at ref guide page for Series.

Hey Adam, have you tried FullSimplify ?