Question

Find the sum of solutions of equation $(5+2\sqrt{6}{)}^{x^2-3}+(5-2\sqrt{6}{)}^{x^2-3}=10.$

• A. $0$
• B. $2\sqrt{2}$
• C. $-2\sqrt{2}$
• D. $2$

Answer 1

The correct option is A $0$

Substitute $t=(5+2\sqrt{6}{)}^{x^2-3}$
Hence, $\frac{1}{t}=(5-2\sqrt{6}{)}^{x^2-3}$
$\Rightarrow t+\frac{1}{t}=10$
$\Rightarrow t=5\pm 2\sqrt{6}$
Hence, $x^2-3=\pm 1$
Hence, $x=2,-2,\sqrt{2},-\sqrt{2}$
Hence, the sum of the roots is 0