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Question Number 215992 by MATHEMATICSAM last updated on 24/Jan/25

$${(\sqrt{3}+\sqrt{2})}^x+{(\sqrt{3}-\sqrt{2})}^x=10.\mathrm{Solve}$$$$\mathrm{for}x.$$

Answered by Rasheed.Sindhi last updated on 24/Jan/25

$${(\sqrt{3}+\sqrt{2})}^x+{(\sqrt{3}-\sqrt{2})}^x=10.\mathrm{Solve}\mathrm{for}\mathrm{x}.$$$$$$$$\sqrt{3}+\sqrt{2}=a$$$$\frac{1}{a}=\frac{1}{\sqrt{3}+\sqrt{2}}\cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}=\sqrt{3}-\sqrt{2}$$$$a^x+\frac{1}{a^x}=10$$$$a^{2x}-10a^x+1=0$$$$a^x=\frac{10\pm \sqrt{100-4}}{2}=5\pm 2\sqrt{6}$$$$x\log a=\log (5\pm 2\sqrt{6})$$$$x=\frac{\log (5\pm 2\sqrt{6})}{\log \left(a\right)}$$$$=\frac{\log (5\pm 2\sqrt{6})}{\log (\sqrt{3}+\sqrt{2})}$$$$x={\log }_{(\sqrt{3}+\sqrt{2})}(5\pm 2\sqrt{6})$$$$={\log }_{(\sqrt{3}+\sqrt{2})}{(\sqrt{3}\pm \sqrt{2})}^2$$$$=2,-2$$

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