nostr:npub1pt6l3a97fvywrxdlr7j0q8j2klwntng35c40cuhj2xmsxmz696uqfr6mf6 hey mist, I see you're discussing math again. do you perchance have any experience with cubic equations? I'm wondering about the general solution to 1 + x + x² + x³ = z [z∈R], the wiki page on cubic equations almost had me crying and wolfram vomited out this abortion. surely the real root can't be that ugly
nostr:npub175dlmjzxq6fgswh73qym7epwcll0dlrkpvcd479u9fc3yrzgxrqsjdwrtj The general formula for solving cubic equations is extremely ugly, there's no way around it. The one for quartic equations is even worse. Both formulas are obtained by algebraic manipulations which resemble "completing the square" for quadratic equations.
There is no general formula for equations of degree > 4. (Abel's Impossibility Theorem.) So the fact that explicit formulas exist for degrees <= 4 should be considered a miracle, even if those formulas are ugly.
For your specific equation, I don't see any trick which would give a nicer formula. What do you need it for? For many applications, an approximate algorithm like Taylor expansion or Newton's method would be just as good as an explicit formula.
