Question

lf $(5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10$, then $x=$

• A. $\pm 2,\pm \sqrt{2}$
• B. $\pm 1,\pm \sqrt{2}$
• C. $\pm 4,\pm \sqrt{2}$
• D. $\pm \sqrt{5},\pm \sqrt{2}$

Answer 1

The correct option is A $\pm 2,\pm \sqrt{2}$

We can see the first two terms are conjugate of each other so when the power
of the terms is equal to $\pm \left(1\right)$ we will get the desired solution.
$x^2-3=1\Rightarrow x^2=4\Rightarrow x=\pm 2$
$x^2-3=-1\Rightarrow x^2=2\Rightarrow x=\pm \sqrt{2}$
Hence, option A is correct.