The prime importance of solving by radicals is actually, that it led to the theory of groups! Groups are used in all sorts of places. (One nice pictorial example is fundamental group of a topological space). Just like Complex numbers arose in trying to solve the cubic. Also, the statement of Fermat's Last Theorem doesn't have any applications but its solution led to lot of interesting theory like how ideals get factorized in rings, elliptic curves, Galois representations...
BTW, the same theory can be extended to differential equations and Differential Galois Theory tells you if you can get a solution by composing basic functions along with exponentials.
Historically, radicals can be motivated by looking at people trying to solve linear, then quadratics, medieval duels about cubics and quartics, the futile search for solving quintics etc. Incidentally, quintics and any degree can have a closed form solution using modular functions.
BTW, the same theory can be extended to differential equations and Differential Galois Theory tells you if you can get a solution by composing basic functions along with exponentials.
Historically, radicals can be motivated by looking at people trying to solve linear, then quadratics, medieval duels about cubics and quartics, the futile search for solving quintics etc. Incidentally, quintics and any degree can have a closed form solution using modular functions.
More discussion on the MathOverflow page https://mathoverflow.net/questions/413468/why-do-we-make-suc...