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If-x-3-1-x-3-1-compute-2-5-x-6-1-2-5-x-6-




Question Number 136303 by SOMEDAVONG last updated on 20/Mar/21
If x^3 − (1/x^3 ) = 1 .compute (2+(√5))x^6 − (1/((2+(√5))x^6 )) = ?
$$\mathrm{If}\:\mathrm{x}^{\mathrm{3}} −\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }\:=\:\mathrm{1}\:.\mathrm{compute}\:\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)\mathrm{x}^{\mathrm{6}} −\:\frac{\mathrm{1}}{\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)\mathrm{x}^{\mathrm{6}} }\:=\:? \\ $$
Answered by mr W last updated on 20/Mar/21
x^6 +(1/x^6 )−3=0  x^(12) −3x^6 +1=0  x^6 =((3±(√5))/2)  (1/x^6 )=3−((3±(√5))/2)=((3∓(√5))/2)    (2+(√5))x^6 − (1/((2+(√5))x^6 ))  =((√5)+2)x^6 − (((√5)−2)/x^6 )  =((√5)+2)×((3±(√5))/2)−((√5)−2) ×((3∓(√5))/2)  =6±5  =11 or 1
$${x}^{\mathrm{6}} +\frac{\mathrm{1}}{{x}^{\mathrm{6}} }−\mathrm{3}=\mathrm{0} \\ $$$${x}^{\mathrm{12}} −\mathrm{3}{x}^{\mathrm{6}} +\mathrm{1}=\mathrm{0} \\ $$$${x}^{\mathrm{6}} =\frac{\mathrm{3}\pm\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{{x}^{\mathrm{6}} }=\mathrm{3}−\frac{\mathrm{3}\pm\sqrt{\mathrm{5}}}{\mathrm{2}}=\frac{\mathrm{3}\mp\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$ \\ $$$$\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)\mathrm{x}^{\mathrm{6}} −\:\frac{\mathrm{1}}{\left(\mathrm{2}+\sqrt{\mathrm{5}}\right)\mathrm{x}^{\mathrm{6}} } \\ $$$$=\left(\sqrt{\mathrm{5}}+\mathrm{2}\right)\mathrm{x}^{\mathrm{6}} −\:\frac{\sqrt{\mathrm{5}}−\mathrm{2}}{\mathrm{x}^{\mathrm{6}} } \\ $$$$=\left(\sqrt{\mathrm{5}}+\mathrm{2}\right)×\frac{\mathrm{3}\pm\sqrt{\mathrm{5}}}{\mathrm{2}}−\left(\sqrt{\mathrm{5}}−\mathrm{2}\right)\:×\frac{\mathrm{3}\mp\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$=\mathrm{6}\pm\mathrm{5} \\ $$$$=\mathrm{11}\:{or}\:\mathrm{1} \\ $$

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