2014/077) Solve for x

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

let $y =5 + 2\sqrt{6}\cdots (1)$
so we have $\dfrac{1}{y} =5 - 2\sqrt{6}\cdots (2)$

now let $z= (5 + 2\sqrt{6})^{x^2-3}$

so $\dfrac{1}{z}= (5 - 2\sqrt{6})^{x^2-3}$

hence $z+\dfrac{1}{z}= 10$

or $z^2-10z+z= 0$
hence $z= 5 + 2\sqrt{6}$ or $z= 5 - 2\sqrt{6}$

 $z= 5 + 2\sqrt{6}$
=> $x^2-3 =1$ or $x=\pm 2$

$z= 5 - 2\sqrt{6}$
=> $x^2-3 =-1$ or $x=\pm \sqrt{2}$