Let $S=\left\{x \in R:(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10\right\}$. Then the number o

Question

Let $S=\left\{x \in R:(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10ight\}$. Then the number of elements in $S$ is\

Professor PC Thomas Classes · Answer

(b): $(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10$ Let $\sqrt{3}+\sqrt{2}=t$ $$ t^{x}+\frac{1}{t^{x}}=10 $$ Let $t^{x}=y$, then $y+\frac{1}{y}=10$ $$ \begin{aligned} \Rightarrow & y^{2}-10 y+1=0 \ \Rightarrow \quad & y=\frac{10 \pm \sqrt{100-4}}{2}=\frac{10 \pm \sqrt{96}}{2} \ & =\frac{10 \pm 4 \sqrt{6}}{2}=5 \pm 2 \sqrt{6} \ & (\sqrt{3}+\sqrt{2})^{x}=5 \pm 2 \sqrt{6} \end{aligned} $$$$ \begin{aligned} & x \ln (\sqrt{3}+\sqrt{2})=\ln (5+2 \sqrt{6}) 	ext { or } \ln (5-2 \sqrt{6}) \ & x=\frac{\ln (5+2 \sqrt{6})}{\ln (\sqrt{3}+\sqrt{2})} 	ext { or } x=\frac{\ln (5-2 \sqrt{6}}{\ln (\sqrt{3}+\sqrt{2})} \end{aligned} $$ So, two real values of $x$.\ $	herefore \quad$ The number of elements in $S$ is 2 .\