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One never knows when one might need a particular bit of mathematics. I wrote a mathematical model of DNA transcription events inside E. coli, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5425810/. For this, I needed the solution to a cubic equation. Thankfully, Cardano had published the solution in the 15th century, and Wang had shown which was the relevant root in more recent times, https://febs.onlinelibrary.wiley.com/doi/epdf/10.1016/0014-5...

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Why not use Newton’s method? Your computer is using numerical approximations for computing those roots anyway. The reason the cubic formula isn’t used is because it’s very convoluted and it doesn’t solve any problems you couldn’t solve with Newton’s method, unless you’re doing algebra and you need an exact solution.

exact analytical solutions are nice as a check against any numerical solutions imo. (checking the same calculations in multiple ways is handy for error catching ime)

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