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Question Number 71406 by mind is power last updated on 15/Oct/19

$$\mathrm{Hello}$$$$\mathrm{Solve}\sqrt[3]{{\mathrm{x}}^2}-3\sqrt[3]{\mathrm{x}(\mathrm{x}-1)}+2\sqrt[3]{\mathrm{x}-1}=0$$

Commented by Prithwish sen last updated on 15/Oct/19

$${\mathrm{x}}^2+(\mathrm{x}-1)-27\mathrm{x}(\mathrm{x}-1)+18\mathrm{x}{(\mathrm{x}-1)}^{\frac{2}{3}}=0$$$$\mathbf{putting}\mathbf{x}={\mathbf{a}}^3\mathbf{and}\mathbf{x}-1={\mathbf{b}}^3\Rightarrow {\mathbf{a}}^3=1+{\mathbf{b}}^3$$$$\therefore {\mathrm{a}}^6+{8\mathrm{b}}^3-{27\mathrm{a}}^3{\mathrm{b}}^3+{18\mathrm{a}}^3{\mathrm{b}}^2=0$$$$\Rightarrow 26{\mathbf{b}}^6-18{\mathbf{b}}^5+17{\mathbf{b}}^3-18{\mathbf{b}}^2-1=0$$$$\mathbf{and}\mathbf{I}\mathbf{got}\mathbf{a}\mathbf{equation}\mathbf{of}6\mathbf{degrees}.$$$$\mathbf{Sir}\mathbf{please}\mathbf{comment}.$$

Commented by MJS last updated on 15/Oct/19

$$a^{1/3}-b^{1/3}+c^{1/3}=0$$$$a^{1/3}+c^{1/3}=b^{1/3}$$$$a+c+3a^{1/3}{c}^{1/3}(a^{1/3}+c^{1/3})=b$$$$3a^{1/3}{b}^{1/3}{c}^{1/3}=b-a-c$$$$27abc={(b-a-c)}^3$$$$a^3-b^3+c^3+21abc-3(a^{2}b+a^{2}c+{ab}^2+{ac}^2+b^{2}c-{bc}^2)=0$$$$a=x^2$$$$b=27x(x-1)$$$$c=8(x-1)$$$$x^6-\frac{19203}{4394}{x}^5+\frac{61719}{8788}{x}^4-\frac{92387}{17576}{x}^3+\frac{4299}{2197}{x}^2-\frac{840}{2197}x+\frac{64}{2197}=0$$$$\mathrm{I}\mathrm{found}\mathrm{no}\mathrm{exact}\mathrm{solution}$$$$x_1\approx .172484$$$$x_2\approx 1.72890$$$$x_{3,4}\approx .230494\pm .209231\mathrm{i}$$$$x_{5,6}\approx 1.00395\pm .0115209\mathrm{i}$$$$\mathrm{if}\mathrm{we}\mathrm{generally}\mathrm{use}\sqrt[3]{z}=\mathrm{abs}{\left(z\right)}^{1/3}\times {\mathrm{e}}^{\mathrm{i}\frac{\arg \left(z\right)}{3}}$$$$\mathrm{only}{x}_2\mathrm{is}\mathrm{a}\mathrm{solution}$$$$\mathrm{if}\mathrm{we}\mathrm{use}\mathrm{the}\mathrm{common}\mathrm{exception}\sqrt[3]{-r}=-\sqrt[3]{r}$$$$\mathrm{also}{x}_1\mathrm{is}\mathrm{a}\mathrm{solution}$$$$\mathrm{the}\mathrm{complex}\mathrm{solutions}\mathrm{are}\mathrm{false}$$

Commented by Prithwish sen last updated on 15/Oct/19

$$\mathrm{you}\mathrm{just}\mathrm{turn}\mathrm{the}\mathrm{equation}\mathrm{in}\mathrm{cubic}\mathrm{form}.$$$$\mathrm{Great}\mathrm{sir}.$$

Commented by mind is power last updated on 15/Oct/19

$$\mathrm{thank}\mathrm{you}!,\mathrm{i}\mathrm{have}\mathrm{no}\mathrm{better}\mathrm{solution}\mathrm{my}\mathrm{bee}\mathrm{we}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{find}\mathrm{exacte}\mathrm{root}{}^{\prime }$$$$\mathrm{or}\mathrm{its}\mathrm{hard}\mathrm{we}\mathrm{need}\mathrm{find}\mathrm{function}\mathrm{g}\left(\mathrm{t}\right)\mathrm{such}\mathrm{that}\mathrm{if}\mathrm{we}\mathrm{put}\mathrm{x}=\mathrm{g}\left(\mathrm{t}\right)$$$$\mathrm{equation}\mathrm{withe}\mathrm{t}\mathrm{is}\mathrm{easier}\mathrm{i}\mathrm{try}\mathrm{but}\mathrm{no}\mathrm{result}$$

Commented by MJS last updated on 15/Oct/19

$$\mathrm{your}\mathrm{equation}\mathrm{is}\mathrm{equivalent}\mathrm{to}\mathrm{mine}$$

Commented by mind is power last updated on 15/Oct/19

$$\mathrm{i}\mathrm{have}\mathrm{did}\mathrm{like}\mathrm{you}\mathrm{sir}!$$

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