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Question Number 128457 by SLVR last updated on 07/Jan/21

Commented by BHOOPENDRA last updated on 07/Jan/21

$$x^5=\frac{133x-78}{133-78x}$$$$78x^6-133x^5+133-78=0$$$$(x^2-1)(78x^4-133x^3+78x^2+78-133x$$$$\Rightarrow letx+\frac{1}{x}=t,then{x}^2+\frac{1}{x^2}=t^2-2$$$$78(x^2+\frac{1}{x^2})-133(x+\frac{1}{x})+78=0$$$$=78(t^2-2)-133\left(t\right)+78=0$$$$=78t^2-133t-78=0$$$$\Rightarrow t=\frac{13}{6}and-\frac{6}{13}$$$$whenx+\frac{1}{x}=\frac{13}{6}$$$$x=\frac{2}{3},\frac{3}{2}$$$$rootsoftheequationare1,-1,\frac{2}{3},\frac{3}{2}$$

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