$(5 + (24)^{\frac{1}{2}})^x + (5 - (24)^{\frac{1}{2}})^x = 10$ , solve for $x$

Question

I have been stuck to this question lately

$(5 + \sqrt{24})^x + (5 - \sqrt{24})^x = 10$ , solve for $x$

Answer 1

HINT:

$$(5+\sqrt{24})(5-\sqrt{24})=1$$

Let $(5+\sqrt{24})^x=y\iff(5-\sqrt{24})^x=\dfrac1{(5+\sqrt{24})^x}=?$

Now solve for $y$

Now if $\displaystyle u^m=u^n,$

either $\displaystyle m-n=0,u\ne0; $

or $\displaystyle u=1$

or $\displaystyle u=-1,m-n$ is even

Answer 2

HINT

Well, the answer is $x= \pm 1$.

Realize that the function on the left is increasing for $x \ge 0$ and decreasing for $x \le 0$.