Your question seems to concern the issue of the computability of solutions of computable functions, and the larger context for such a question is the subject known as [computable analysis](http://en.wikipedia.org/wiki/Computable_analysis). 

Carl Mummert has [a very nice blog post concerning the following theorems](http://m6c.org/w/2012/02/computable-roots/), which I believe lie at the heart of your question.

>>Here are several interesting results from computable analysis:

>><b>Theorem 1.</b> If $f$ is a computable function from $\mathbb{R}$ to $\mathbb{R}$ and $a$ is an isolated root of $f$, then α is computable.

>><b>Corollary 2.</b> If $p(x)$ is a polynomial over $\mathbb{R}$ with computable coefficients, then every root of $p(x)$ is computable.

>><b>Theorem 3.</b> There is a effective closed subset of $\mathbb{R}$ which is nonempty (in fact, uncountable) but which has no computable point.

>><b>Theorem 4.</b> There is a computable function from $\mathbb{R}$ to $\mathbb{R}$ which has uncountably many roots but no computable roots.