# What properties make this equation difficult to symbolically solve?

I tried to solve the following equation with Mathematica's solve:

Solve[K*(2*Tan[L/2*Sqrt[P/(EI)]]-L*Sqrt[P/(EI)])+4*P*Sqrt[P/(EI)] == 0, P]


It gave the following error:

Solve::nsmet: This system cannot be solved with the methods available to
Solve.


I do not see any strange functions being called here, and I am curious about what properties make this expression difficult for Mathematica.

Your equation is one of the many transcendental equations (as opposed to being an algebraic equation that can be solved with methods for solving polynomials) that one cannot solve with the usual ways taught in undergraduate algebra/precalculus courses. In particular, the difficulty in solving this transcendental equation is that your P is inside a square root, outside a square root, and inside a square root inside a tangent (and the tangent is what makes this equation transcendental).
Thus, you might have to content yourself with using FindRoot[] for a numerical solution. You might wish to plot the left hand side of your equation with specific values of your other parameters to assist in searching for this equation's roots.
As an aside, K is a reserved System` symbol; consider using a different name for that parameter.