At some point in studying Algebra, we learn about the Quadratic Equation: given a polynomial equation of the form

$ax_{2}+bx+c=0,$mathematicians derived a **closed-form** solution, instantly giving us the two solutions:

In Medieval Italy, mathematicians further developed a **cubic equation**, giving us the three solutions to

and then the **quartic** equation, giving us the four solutions to

But then they seemed stumped after this conquest of **fourth-degree** polynomial equations. There seemed to be something qualitatively different about **fifth-degree**, or **quintic** polynomial equations.

Long story short, a 20-year-old named Evariste Galois frantically wrote down a **proof** while languishing in prison for his participation in the July Revolution,