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Question Number 136303 by SOMEDAVONG last updated on 20/Mar/21

$$\mathrm{If}{\mathrm{x}}^3-\frac{1}{{\mathrm{x}}^3}=1.\mathrm{compute}(2+\sqrt{5}){\mathrm{x}}^6-\frac{1}{(2+\sqrt{5}){\mathrm{x}}^6}=?$$

Answered by mr W last updated on 20/Mar/21

$$x^6+\frac{1}{x^6}-3=0$$$$x^{12}-3x^6+1=0$$$$x^6=\frac{3\pm \sqrt{5}}{2}$$$$\frac{1}{x^6}=3-\frac{3\pm \sqrt{5}}{2}=\frac{3\mp \sqrt{5}}{2}$$$$$$$$(2+\sqrt{5}){\mathrm{x}}^6-\frac{1}{(2+\sqrt{5}){\mathrm{x}}^6}$$$$=(\sqrt{5}+2){\mathrm{x}}^6-\frac{\sqrt{5}-2}{{\mathrm{x}}^6}$$$$=(\sqrt{5}+2)\times \frac{3\pm \sqrt{5}}{2}-(\sqrt{5}-2)\times \frac{3\mp \sqrt{5}}{2}$$$$=6\pm 5$$$$=11or1$$

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