This is part of solving the problem posed in this YouTube video.
I can numerically determine that the real root is about 2.27, but I am wondering how I can find an exact representation of the root.
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1You can try Cardan's method to find roots of a cubic polynomial. – Mr.Gandalf Sauron Sep 13 '21 at 11:06
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The only analytical approach that I know of for the generic cubic is discussed in this article. – user2661923 Sep 13 '21 at 11:06
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https://www.wolframalpha.com/input/?i=x%5E3%2B3x%5E2%E2%88%925x%E2%88%9216%3D0 – jimjim Sep 13 '21 at 11:07
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try Newton-Raphson method – mark Sep 13 '21 at 11:14
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3See the answer at: Is there really analytic solution to cubic equation? – user21820 Sep 18 '21 at 09:09
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If you follow the steps described in the Wikipedia page, using the hyperbolic method for one real root, you should find $$x=4 \sqrt{\frac{2}{3}} \cosh \left(\frac{1}{3} \cosh ^{-1}\left(\frac{27 }{32}\sqrt{\frac{3}{2}}\right)\right)-1$$
Claude Leibovici
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