# Analytical method for root finding

Is there an analytical method to find the roots of the following equation?

$$y = -\frac{1}{2}{x}^{2}-\cos(x)+1.1$$

I'm sorry for the trivial question, I'm new at math! :)

• A transcendental equation like that, where the independent variable is inside and outside a transcendental function, more often than not doesn't admit neat closed-form expressions for its roots. – J. M. isn't a mathematician May 13 '12 at 8:51

## 1 Answer

This equation does not admit an analytic solution, i.e., there is no formula in terms of "elementary functions" (giving the solution in terms of additions, substractions, multiplications, divisions, $n$-th roots, exponentials and logarithms).

Edit: I noticed that my "integral approach" won't work. Proving that this formula admits no analytic solution requires techniques that go way beyond what can be done with basic algebra.

• Sounds intriguing. Could you write out this integral solution of yours, for the benefit of everybody else? – J. M. isn't a mathematician May 13 '12 at 9:08
• I do agree, could you please write out a proof for your statement? – Jashin_212 May 13 '12 at 9:19