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Question Number 63383 by minh2001 last updated on 03/Jul/19

$$solvethisequationinall$$$$partofcomplexnumber:$$$$\sqrt{(x^9-3x^2+1)(x-6)+4}=(x^9-3x^2+1)(x-6)-16$$

Commented by MJS last updated on 04/Jul/19

$$\mathrm{waiting}\mathrm{to}\mathrm{see}\mathrm{your}\mathrm{solution}$$

Answered by MJS last updated on 03/Jul/19

$$\sqrt{t+4}=t-16$$$$t+4={(t-16)}^2$$$$t^2-33t+252=0$$$$\Rightarrow t=12\vee t=21$$$$\mathrm{but}t=12\mathrm{is}\mathrm{no}\mathrm{solution}\mathrm{of}\sqrt{t+4}=t-16$$$$\Rightarrow t=21$$$$(x^9-3x^2+1)(x-6)=21$$$$(x+1)(x^9-7x^8+7x^7-7x^6+7x^5-7x^4+7x^3-10x^2+28x-27)=0$$$$x_1=-1$$$$\mathrm{we}\mathrm{cannot}\mathrm{exactly}\mathrm{solve}\mathrm{the}\mathrm{rest}$$$$x_2\approx 6.00000208$$$$\mathrm{the}\mathrm{other}8\mathrm{roots}\mathrm{are}\mathrm{complex}\left(4\mathrm{pairs}\mathrm{of}\mathrm{conjugated}\mathrm{complex}\mathrm{roots}\right)$$$$\mathrm{if}{\mathrm{it}}^{\prime }\mathrm{s}$$$$\sqrt{(x^3-3x^2+1)(x-6)+4}=(x^3-3x^2+1)(x-6)-16$$$$\mathrm{we}\mathrm{can}:$$$$(x^3-3x^2+1)(x-6)=21$$$$(x+1)(x^3-10x^2+28x-27)=0$$$$x_1=-1$$$$x_2=\frac{10}{3}+\sqrt[3]{\frac{209}{54}-\frac{\sqrt{337}}{6}}+\sqrt[3]{\frac{209}{54}+\frac{\sqrt{337}}{6}}$$$$x_3=\frac{10}{3}+(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})\sqrt[3]{\frac{209}{54}-\frac{\sqrt{337}}{6}}+(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})\sqrt[3]{\frac{209}{54}+\frac{\sqrt{337}}{6}}$$$$x_4=\frac{10}{3}+(-\frac{1}{2}+\frac{\sqrt{3}}{2}\mathrm{i})\sqrt[3]{\frac{209}{54}-\frac{\sqrt{337}}{6}}+(-\frac{1}{2}-\frac{\sqrt{3}}{2}\mathrm{i})\sqrt[3]{\frac{209}{54}+\frac{\sqrt{337}}{6}}$$

Commented by minh2001 last updated on 04/Jul/19

$$No,youwerewrong.Why$$$$youset{x}^{3}butnot{x}^9.I^{\prime }ve$$$$foundother8solutionsas$$$$yourmean,thankyou$$

Commented by MJS last updated on 04/Jul/19

$$\mathrm{so}\mathrm{show}\mathrm{us}\mathrm{how}\mathrm{you}\mathrm{found}\mathrm{them}$$$$\mathrm{I}\mathrm{can}\mathrm{approximate}\mathrm{them}\mathrm{all}$$

Commented by MJS last updated on 04/Jul/19

$$\mathrm{with}{x}^9$$$$x_1=-1$$$$x_2\approx 6.00000208383$$$$x_{3,4}\approx -.994927692081\pm .650751638656\mathrm{i}$$$$x_{5,6}\approx -.262576070729\pm 1.25073913893\mathrm{i}$$$$x_{7,8}\approx .684103814572\pm 1.04916880346\mathrm{i}$$$$x_{9,10}\approx 1.07339890633\pm .300735402858\mathrm{i}$$

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