There’s the quadratic formula and the one for 3 and 4 factor ones (apologies if I’m wording this wrong), but I just heard that apparently there isn’t anything like a quintic(?) formula and so on. Why is this?? Googling gives me a bunch of confusing terminology that’s difficult to parse.
In: 75
The quadratic formula is a few short operations. The [cubic](https://math.vanderbilt.edu/schectex/courses/cubic/) formula is a few lines. The [quartic](https://en.wikipedia.org/wiki/Quartic_equation#The_general_case) formula takes a couple of pages to describe. The quintic equation is unknown, but presumably much much worse.
At which point the question is, “who needs it?” Any scientist or engineer faced with a quintic equation can use an iterative numerical method, which repeats a short easy bit of math over and over to find an approximate answer that can be as accurate as you need. A mathematician cares about proving the *existence* and *properties* of the solution, but doesn’t care what the solution actually *is*.
The [properties of the solutions](https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots) of *all* polynomial equations of any degree are well understood, so the mathematician’s work is done, and there are plenty of great [root-finding algorithms](https://en.wikipedia.org/wiki/Root-finding_algorithms) for the scientists and engineers to use, so the quintic formula, if it existed, would be just a useless curiosity.
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