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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! :)

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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. May 13 '12 at 8:51
up vote 2 down vote accepted

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.

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Sounds intriguing. Could you write out this integral solution of yours, for the benefit of everybody else? – J. M. 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

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