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Question Number 183202 by liuxinnan last updated on 23/Dec/22

$${ax}^3+{bx}^2+c=0$$$$x_1=$$$$x_2=$$$$x_3=$$

Answered by Frix last updated on 23/Dec/22

$${ax}^3+{bx}^2+c=0$$$$m=\frac{b}{a}\wedge n=\frac{c}{a}$$$$x^3+{mx}^2+n=0$$$$x=t-\frac{m}{3}$$$$t^3+\frac{m^2}{3}t+\frac{2m^3}{27}+n=0$$$$p=\frac{m^2}{3}\wedge q=\frac{2m^3}{27}+n$$$$t^3+pt+q=0$$$$\mathrm{Depending}\mathrm{on}D=\frac{p^3}{27}+\frac{q^2}{4}\underset{<}{\stackrel{>}{=}}0\mathrm{you}\mathrm{can}\mathrm{use}$$$${\mathrm{Cartano}}^{\prime }\mathrm{s}\mathrm{Formula}\mathrm{or}\mathrm{the}\mathrm{Trigonometric}$$$$\mathrm{Formula}:$$$$D>0\Rightarrow$$$$u=\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{p^3}{27}+\frac{q^2}{4}}}\wedge v=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{p^3}{27}+\frac{q^2}{4}}}$$$$[\mathrm{you}\mathrm{must}\mathrm{take}\mathrm{the}\mathrm{real}\mathrm{roots}:\sqrt[3]{-r}=-\sqrt[3]{r}]$$$$t_1=u+v$$$$t_2=(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})u+(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})v$$$$t_3=(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})u+(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})v$$$$D=0\wedge p\ne 0\wedge q\ne 0\iff \frac{p^3}{27}=-\frac{q^2}{4}\Rightarrow$$$$u=v=\sqrt[3]{-\frac{q}{2}}$$$$t_1=2\sqrt[3]{-\frac{q}{2}}=\frac{3q}{p}$$$$t_2=t_3=-\sqrt[3]{-\frac{q}{2}}=-\frac{3q}{2p}$$$$D<0\Rightarrow$$$$t_k=\sqrt[3]{-\frac{4p}{3}}\times \cos \left(\frac{2\pi k+{\cos }^{-1}(-\frac{q}{2}\times \sqrt{-\frac{27}{p^3}})}{3}\right)\mathrm{with}k=1,2,3$$$$\mathrm{You}\mathrm{must}\mathrm{insert}\mathrm{backwards}$$$$p=\frac{m^2}{3}\wedge q=\frac{2m^3}{27}+n$$$$x_k=t_k-\frac{m}{3}$$$$m=\frac{b}{a}\wedge n=\frac{c}{a}$$$$\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{demanded}\mathrm{formula}$$

Answered by mr W last updated on 23/Dec/22

$$wecanalsosolvefor\frac{1}{x}insteadof$$$$forx.$$$$\frac{1}{x^3}+\frac{b}{cx}+\frac{a}{c}=0$$$$\mathrm{\Delta }=\frac{1}{4}{\left(\frac{a}{c}\right)}^2+\frac{1}{27}{\left(\frac{b}{c}\right)}^3$$$$\underset{―}{if\mathrm{\Delta }>0:}$$$$\frac{1}{x_1}=\sqrt[3]{\mathrm{\Delta }-\frac{a}{2c}}-\sqrt[3]{\mathrm{\Delta }+\frac{a}{2c}}$$$$\frac{1}{x_{2,3}}=-\frac{1}{2}(\sqrt[3]{\mathrm{\Delta }-\frac{a}{2c}}-\sqrt[3]{\mathrm{\Delta }+\frac{a}{2c}})\pm (\sqrt[3]{\mathrm{\Delta }-\frac{a}{2c}}+\sqrt[3]{\mathrm{\Delta }+\frac{a}{2c}})i$$$$\underset{―}{if\mathrm{\Delta }=0:}$$$$\frac{1}{x_1}=-2\sqrt[3]{\frac{a}{2c}}$$$$\frac{1}{x_2}=\frac{1}{x_3}=\sqrt[3]{\frac{a}{2c}}$$$$\underset{―}{if\mathrm{\Delta }<0:}$$$$\frac{1}{x_{1,2,3}}=2\sqrt{-\frac{b}{3c}}\sin [\frac{1}{3}{\sin }^{-1}\left(\frac{3a}{-2b}\sqrt{-\frac{3c}{b}}\right)+\frac{2k\pi }{3}](k=0,1,2)$$

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