Functions – How to Solve 5th Degree Polynomials

Question

I looked on Wikipedia for a formula for roots of a 5th degree polynomial, but it said that by Abel's theorem it isn't possible. The Abel's theorem states that you can't solve specific polynomials of the 5th degree using basic operations and root extractions.

Can you find the roots of a specific quintic with only real irrational roots (e.g. $f(x)=x^5+x+2$) using other methods (such as logarithms, trigonometry, or convergent sums of infinite series, etc.)?

Basically, how can the exact values of the roots of such functions be expressed other than a radical (since we know that for some functions it is not a radical)?

If no, is numerical solving/graphing the only way to solve such polynomials?

Edit: I found a link here that explains all the ways that the above mentioned functions could be solved.

Answer 1

Here is a summary of posts that should address related questions:

1. Reducing the general quintic to Bring-Jerrard form.
2. Solving (1) using elliptic functions.
3. Reducing the general quintic to Brioschi form.
4. Solving (3) using trigonometric and special functions.
5. Solving the Brioschi form using $R(q)$ (or the Rogers-Ramanujan continued fraction).