Question Number 143731 by haladu last updated on 17/Jun/21
Answered by MJS_new last updated on 17/Jun/21
$$\mathrm{if}\:\mathrm{we}\:\mathrm{find}\:{x}>\mathrm{0}\:\mathrm{with}\:\frac{\mathrm{2}{x}+\mathrm{1}}{{x}−\mathrm{1}}={x}^{\mathrm{4}} \:\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{real} \\ $$$$\mathrm{solution} \\ $$$$\Rightarrow \\ $$$${x}^{\mathrm{5}} −{x}^{\mathrm{4}} −\mathrm{2}{x}−\mathrm{1}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} −{x}−\mathrm{1}\right)\left({x}^{\mathrm{3}} +{x}+\mathrm{1}\right)=\mathrm{0} \\ $$$$\Rightarrow\:{x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$