Let [S = x:x R,and,( 3 + 2 )^x^2 - 4 + ( 3 - 2 )^x^2 - 4 = 10 ].

Question

Let $$S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3  + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3  - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}$$. Then $n(S)$ is equal to

Accepted Answer

Let $(\sqrt{3}+\sqrt{2})^{x^{2}-4}=t$

$$
\begin{aligned}
& t+\frac{1}{t}=10 \\
\Rightarrow & t^{2}-10 t+1=0 \\
\Rightarrow & t=\frac{10 \pm \sqrt{100-4}}{2}=5 \pm 2 \sqrt{6}
\end{aligned}
$$

Case-I

$$
\begin{aligned}
& t=5+2 \sqrt{6} = (\sqrt{3}+\sqrt{2})^{2} \\
\Rightarrow & (\sqrt{3}+\sqrt{2})^{x^{2}-4}=(\sqrt{3}+\sqrt{2})^{2} \\
\Rightarrow & x^{2}-4=2 \Rightarrow x^{2}=6 \Rightarrow x=\pm \sqrt{6}
\end{aligned}
$$

Case-II :

$t=5-2 \sqrt{6}$ = $(\sqrt{3}-\sqrt{2})^{2}$

$(\sqrt{3}+\sqrt{2})^{x^{2}-4}=(\sqrt{3}-\sqrt{2})^{2}$

$\Rightarrow\left((\sqrt{3}-\sqrt{2})^{-1}\right)^{x^{2}-4}=(\sqrt{3}-\sqrt{2})^{2}$

$\Rightarrow 4-x^{2}=2$

$\Rightarrow x^{2}=2$

$\Rightarrow x=\pm \sqrt{2}$

Solutions · 1