The Quartic Formula (Ferrari's Method)

After finding the cubic formula using Cardano's method, the next step is to find the quartic formula using Ferrari's method. It is quite similar and builds on the ideas and concepts of Cardano's method with an extra step to form two perfect square quadratics.

Our journey for polynomial solution formulas in radicals ends here though since there is no formula for the quintic and higher degrees using elementary functions, but the methods to solve the quadratic, cubic, and quartic equation were definitely ingenious and worth the ride.

Timecodes:
0:00 - Intro and statement
0:35 - Formula derivation (choose λ s.t. p + 2λ ≠ 0 in the rare case)**
21:13 - Worked example
32:32 - Alternative steps in solving
34:26 - Extra on the discriminant
42:29 - Closing and unsolvability of the quintic

Sources and other tidbits:
Info on the quartic function: https://en.wikipedia.org/wiki/Quartic...
Alternate derivation involves Lagrange Resolvents: https://en.wikipedia.org/wiki/Resolve...)
Vieta's formulas: https://en.wikipedia.org/wiki/Vieta%2...
General discriminant: https://artofproblemsolving.com/wiki/...
Abel-Ruffini Theorem and the unsolvability of the quintic (in elementary functions): https://en.wikipedia.org/wiki/Abel%E2...

Twitter -   / michaelmaths_  
Instagram -   / michaelmaths_