Developed-the-formula-for-the-cubic-equation-when-it-has-three-real-roots-and-Cardano-is-just-not-suitable-as-discriminant-gets-negative-Given-x-3-bx-c-0-with-b-gt-0-c-gt-0-t-b-1-15

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Question Number 114388 by ajfour last updated on 18/Sep/20

$$Developedtheformulaforthecubic$$$$equationwhenithasthreerealroots$$$$andCardanoisjustnotsuitableas$$$$discriminantgetsnegative..$$$$Given{\mathit{x}}^3-\mathit{b}\mathit{x}-\mathit{c}=0$$$$with\mathit{b}>0,\mathit{c}>0$$$$\frac{\mathit{t}}{\mathit{b}}=1+\frac{15}{2}\left(\frac{{\mathit{c}}^2}{{\mathit{b}}^3}\right)+\sqrt{\left(\frac{{\mathit{c}}^2}{{\mathit{b}}^3}\right)[49-\frac{75}{4}\left(\frac{{\mathit{c}}^2}{{\mathit{b}}^3}\right)]}$$$$\mathit{x}=\sqrt{\mathit{t}+\frac{4{\mathit{c}}^2}{{\mathit{b}}^2}}-\frac{3\mathit{c}}{\mathit{b}}★$$$$mrWSir,MjSSirpleasecheck!$$

Commented by MJS_new last updated on 18/Sep/20

$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{not}\mathrm{exact}$$$$x^3-\frac{244}{3}x-\frac{1456}{27}=0\Rightarrow x\in \{-\frac{26}{3},-\frac{2}{3},\frac{28}{3}\}$$$$\mathrm{but}\mathrm{your}\mathrm{formula}\mathrm{gives}$$$$x\approx 9.33355157\mathrm{instead}\mathrm{of}9.\stackrel{\bullet }{3}$$

Commented by ajfour last updated on 18/Sep/20

$$yessir,ifoundtheerror!thanks.$$

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