Question Number 213486 by a.lgnaoui last updated on 06/Nov/24
$$\:\:\boldsymbol{\mathrm{x}}^{\mathrm{5}} +\mathrm{5}\boldsymbol{\mathrm{x}}−\frac{\mathrm{6}}{\boldsymbol{\mathrm{x}}}=\mathrm{0}\:\:\:\:\:\:\boldsymbol{\mathrm{x}}? \\ $$
Answered by Frix last updated on 06/Nov/24
![x^6 +5x^2 −6=0 (x^6 −1)+5(x^2 −1)=0 (x^4 +x^2 +1)(x^2 −1)+5(x^2 −1)=0 (x^2 −1)(x^4 +x^2 +6)=0 x=±1 x=±(((√(−1+2(√6)))/2)±((√(1+2(√6)))/2)i) [ _(of signs)^(all 4 combinations) ]](https://www.tinkutara.com/question/Q213490.png)
$${x}^{\mathrm{6}} +\mathrm{5}{x}^{\mathrm{2}} −\mathrm{6}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{6}} −\mathrm{1}\right)+\mathrm{5}\left({x}^{\mathrm{2}} −\mathrm{1}\right)=\mathrm{0} \\ $$$$\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{1}\right)+\mathrm{5}\left({x}^{\mathrm{2}} −\mathrm{1}\right)=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{6}\right)=\mathrm{0} \\ $$$${x}=\pm\mathrm{1} \\ $$$${x}=\pm\left(\frac{\sqrt{−\mathrm{1}+\mathrm{2}\sqrt{\mathrm{6}}}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{6}}}}{\mathrm{2}}\mathrm{i}\right)\:\:\:\:\:\left[\:_{\mathrm{of}\:\mathrm{signs}} ^{\mathrm{all}\:\mathrm{4}\:\mathrm{combinations}} \right] \\ $$
Commented by a.lgnaoui last updated on 06/Nov/24
$$\mathrm{thanks}\:\mathrm{Sir} \\ $$