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Question Number 122360 by benjo_mathlover last updated on 16/Nov/20

Answered by $@y@m last updated on 16/Nov/20

$$Putt={\left(\sqrt{5-2\sqrt{6}}\right)}^x$$$$t+\frac{1}{t}=10$$$$t^2-10t+1=0$$$$t=\frac{10\pm \sqrt{100-4}}{2}$$$$t=\frac{10\pm 4\sqrt{6}}{2}=5\pm 2\sqrt{6}$$$$\Rightarrow x=2,-2$$

Answered by nimnim last updated on 16/Nov/20

$$byinspectionx=2$$

Answered by liberty last updated on 16/Nov/20

$$\mathrm{consider}:5-2\sqrt{6}=\frac{(5-2\sqrt{6}).(5+2\sqrt{6})}{5+2\sqrt{6}}=\frac{1}{5+2\sqrt{6}}$$$$\mathrm{or}5-2\sqrt{6}={(5+2\sqrt{6})}^{-1}$$$$\mathrm{now}\mathrm{let}{\left(\sqrt{5-2\sqrt{6}}\right)}^{\mathrm{x}}=\mathrm{p},\mathrm{gives}$$$$\Rightarrow \mathrm{p}+{\mathrm{p}}^{-1}=10\mathrm{or}\frac{{\mathrm{p}}^2+1}{\mathrm{p}}=10$$$$\Rightarrow {\mathrm{p}}^2-10\mathrm{p}+1=0;{\mathrm{p}}_{1,2}=\frac{10\pm \sqrt{100-4}}{2}$$$$\Rightarrow {\mathrm{p}}_{1,2}=5\pm 2\sqrt{6}$$$$\mathrm{for}{\mathrm{p}}_1=5+2\sqrt{6}=\sqrt{{(5-2\sqrt{6})}^{\mathrm{x}}}$$$$\to 5+2\sqrt{6}=\sqrt{{(5+2\sqrt{6})}^{-\mathrm{x}}},\mathrm{we}\mathrm{get}\mathrm{x}=-2$$$$\mathrm{similarly}\mathrm{for}{\mathrm{p}}_2=5-2\sqrt{6}\mathrm{we}\mathrm{get}\mathrm{x}=2$$$$\mathrm{thus}\mathrm{the}\mathrm{solutions}\mathrm{is}\mathrm{x}=\pm 2.$$

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