Question

The number of solutions of $(5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10$ are

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

The correct option is C $4$

$(5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10$
Let $t=(5+2\sqrt{6}{)}^{x^2-3}$, then $(5-2\sqrt{6}{)}^{x^2-3}=\frac{1}{t}$
$\Rightarrow t+\frac{1}{t}=10$
$\Rightarrow t^2-10t+1=0$
$\Rightarrow t=\frac{10+\sqrt{96}}{2}$ or $t=\frac{10-\sqrt{96}}{2}$
$\Rightarrow t=5+2\sqrt{6}$ or $t=5-2\sqrt{6}$
$\Rightarrow x^2-3=1$ or $x^2-3=-1$
Hence, $x$ can take four real values.