A general solution to the cubic equation was long considered impossible, until we gave up the requirement that math reflect reality. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription.

Thanks to Dr Amir Alexander, Dr Alexander Kontorovich, Dr Chris Ferrie, and Dr Adam Becker for the helpful advice and feedback on the earlier versions of the script.

▀▀▀

References:

Some great videos about the cubic:

500 years of not teaching the cubic formula. -- https://youtu.be/N-KXStupwsc

Imaginary Numbers are Real -- https://youtu.be/T647CGsuOVU

Dunham, W. (1990). Journey through genius: The great theorems of mathematics. New York. -- https://ve42.co/Dunham90

Toscano, F. (2020). The Secret Formula. Princeton University Press. -- https://ve42.co/Toscano2020

Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179-205. -- https://ve42.co/Bochner63

Muroi, K. (2019). Cubic equations of Babylonian mathematics. arXiv preprint arXiv:1905.08034. -- https://ve42.co/Murio21

Branson, W. Solving the cubic with Cardano, -- https://ve42.co/Branson2014

Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural. arXiv preprint arXiv:1308.2181. -- https://ve42.co/Rothman

Vali Siadat, M., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations. Math Horizons, 28(1), 12-15. -- https://ve42.co/Siadat21

Merino, O. (2006). A short history of complex numbers. University of Rhode Island. -- https://ve42.co/Merino2006

Cardano, G (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3

Bombelli, R (1579) L’Algebra https://ve42.co/Bombelli

▀▀▀

Special thanks to Patreon supporters: Luis Felipe, Anton Ragin, Paul Peijzel, S S, Benedikt Heinen, Diffbot, Micah Mangione, Juan Benet, Ruslan Khroma, Richard Sundvall, Lee Redden, Sam Lutfi, MJP, Gnare, Nick DiCandilo, Dave Kircher, Edward Larsen, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Ruslan Khroma, Robert Blum, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson,Ron Neal

Written by Derek Muller, Alex Kontorovich, Stephen Welch and Petr Lebedev

Animation by Fabio Albertelli, Jakub Misiek, Iván Tello and Jesús Rascón

Mathematical animations done with Manim -- thanks Grant Sanderson and the Manim community!

SFX by Shaun Clifford

Filmed by Derek Muller and Emily Zhang

Edited by Derek Muller and Petr Lebedev

Additional video supplied by Getty Images

Music from Epidemic Sound

Additional Music By Jonny Hyman

Produced by Derek Muller, Petr Lebedev and Emily Zhang