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The equation is : $$ -x^3-2x^2-4x-5.4+K=0 $$ and I want my solutions in terms of $K$.

How do I simplify this polynomial?

Blue
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  • For $K=5.4$ you can simplify to a quadratic equation, otherwise you can solve it as a cubic equation as usual. – Dietrich Burde Jan 13 '21 at 12:07
  • Also see: https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_formula – VIVID Jan 13 '21 at 12:09
  • I do not want to set K=5.4. I the roots x of this polynomial in terms of K. I do not know how to solve this. –  Jan 13 '21 at 12:18

1 Answers1

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You face a cubic equation in $x$. So, just follow the steps given in the Wikipedia page.

The first is to compute $$\Delta=-27 k^2+\frac{898 k}{5}-\frac{9363}{25}$$ which is always ngative since this quadratic in $k$ does not show real root.

So, use the hyperbolic method for one real root and get $$x=-\frac{2}{3} \left(1+2 \sqrt{2} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{449-135 k}{160 \sqrt{2}}\right)\right)\right)$$

Claude Leibovici
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