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Question Number 63095 by aliesam last updated on 28/Jun/19

Answered by MJS last updated on 29/Jun/19

$$\sqrt[3]{4x^4-40x^2+100}-\sqrt[3]{2x^2-10}=20-2x^2$$$$x^2=s$$$$\sqrt[3]{4s^2-40s+100}-\sqrt[3]{2s-10}=20-2s$$$$\sqrt[3]{4{(s-5)}^2}-\sqrt[3]{2(s-5)}=10-2(s-5)$$$$t=2(s-5)\iff s=\frac{t}{2}+5$$$$\sqrt[3]{t^2}-\sqrt[3]{t}=10-t$$$$u=\sqrt[3]{t}\iff t=u^3$$$$u^2-u=10-u^3$$$$(u-2)(u^2+3u+5)=0$$$$u_1=2\Rightarrow t_1=8\Rightarrow s_1=9\Rightarrow x=\pm 3$$$$u_2=-\frac{3}{2}-\frac{\sqrt{11}}{2}\mathrm{i}\Rightarrow t_2=9-2\sqrt{11}\mathrm{i}\mathrm{not}\mathrm{valid}$$$$u_3=-\frac{3}{2}+\frac{\sqrt{11}}{2}\mathrm{i}\Rightarrow t_3=9+2\sqrt{11}\mathrm{i}\mathrm{not}\mathrm{valid}$$$$t_2\mathrm{and}{t}_3\mathrm{do}\mathrm{not}\mathrm{solve}\sqrt[3]{t^2}-\sqrt[3]{t}=10-t$$

Commented by MJS last updated on 29/Jun/19

$$\mathrm{if}\mathrm{we}\mathrm{write}{t}_2\mathrm{and}{t}_3\mathrm{as}r{\mathrm{e}}^{\pm \mathrm{i}\theta }\mathrm{and}\mathrm{add}\pm 2\pi$$$$\mathrm{then}r{\mathrm{e}}^{\pm \mathrm{i}(\theta +2\pi )}\mathrm{solve}\mathrm{the}\mathrm{equation}\mathrm{but}\mathrm{I}\mathrm{doubt}$$$$\mathrm{if}\mathrm{these}\mathrm{solution}\mathrm{are}\mathrm{legit}...$$

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