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A shorter introduction to working with Root objects is in the below answer.

or read another related post. Using Solve you could include this option InverseFunctions -> True to avoid any messages generated :

Root includes a pure function and an integer number pointing out explicitly a given root (here e.g. Root[1 - 4 #1 + 8 #1^2 - 4 #1^3 + 24 #1^5 - 24 #1^6 - 16 #1^7 + 16 #1^8 &, 1]) or (since ver.7) a list including a pure function and numerical approximation where we can find a root in case of a transcendental equation. This post may be helpful as well. Regardless of the form of representation Root can be exactly determined with an arbitrary accuracy, whatever one needs, let's take the fourth solution in s e.g. :

A shorter introduction to working with Root objects is in the below answer.

or read another related post. Using Solve you could include this option InverseFunctions -> True to avoid any messages generated :

Root includes a pure function and an integer number pointing out explicitly a given root (here e.g. Root[1 - 4 #1 + 8 #1^2 - 4 #1^3 + 24 #1^5 - 24 #1^6 - 16 #1^7 + 16 #1^8 &, 1]) or (since ver.7) a list including a pure function and numerical approximation where we can find a root in case of a transcendental equation. This post may be helpful as well. Regardless of the form of representation Root can be exactly determined with an arbitrary accuracy, whatever one needs, let's take the fourth solution in s e.g. :

A shorter introduction to working with Root objects is in the below answer.

Solutions to algebraic or transcendental equations are expressed in terms of Root objects whenever it is impossible to find explicit solutions. When polynomial Root object does have an expression in terms of radicals, it can be expanded using ToRadicals. The reverse operation is RootReduce.

In general, there is no way express roots of 5-th (or higher) order polynomials in terms of radicals. However even higher order algebraic equations can be solved explicitly if an associated Galois group is solvable. On the other hand Solve and Reduce behave differently by default, e.g. evaluate Reduce[x^4 + 3 x + 1 == 0, x] and Solve[x^4 + 3 x + 1 == 0, x], this justifies apparently different outputs :

Solutions to algebraic or transcendental equations are expressed in terms of Root objects whenever it is impossible to find explicit solutions. In general there is no way express roots of 5-th (or higher) order polynomials in terms of radicals. However even higher order algebraic equations can be solved explicitly if an associated Galois group is solvable. On the other hand Solve and Reduce behave differently by default, e.g. evaluate Reduce[x^4 + 3 x + 1 == 0, x] and Solve[x^4 + 3 x + 1 == 0, x], this justifies apparently different outputs :