The Cubic Formula (Cardano's Method)

Everyone knows the exact solution for a quadratic, but how about the cubic? There is indeed a cubic formula that's not very hard to work with if you are comfortable with nested roots and complex numbers.

The cubic is shifted so that the POI is on the y-axis (depressed cubic), which can then be solved given a certain pairing with the perfect cube expansion, and the shift is undone at the end.

This is one of the biggest achievements in the history of Mathematics, originally created/solved by Tartaglia and Del Ferro, and then further popularized by Cardano. It kickstarted the interest in the nature of polynomials and Galois Theory, as well as the discovery of complex numbers.

Timecodes:
0:00 - Intro and statement
2:45 - Formula derivation
22:37 - Discriminant cases
31:46 - Special calculus shortcut
34:35 - Worked example: D greater than 0
39:42 - Worked example: D equal to 0
45:15 - Worked example: D less than 0

Sources and other tidbits:
Info about the cubic function: https://en.wikipedia.org/wiki/Cubic_f...
Depressed cubic on Geogebra: https://www.geogebra.org/m/CEzzAQNZ
Alternate derivation involves Lagrange Resolvents: https://en.wikipedia.org/wiki/Resolve...)
True "completing the cube" method: https://web.archive.org/web/201810240...

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