Thank you for finding the solution but I have already found it.
I must proof that this equation has only one decision. (I can't proof it using graph because it is enough difficult to construct y=(4+sqrt(15))^X + (4-sqrt(15))^X without, for example, WolframAlpha).
Let $\displaystyle a = 4 + \sqrt{15}$ and $\displaystyle b = 4 - \sqrt{15}$
so that
$\displaystyle a^{x/2} + b^{x/2} = 8^{x/2}$
Multiply this equation by $\displaystyle a^{x/2}$
$\displaystyle a^x + 1 = \left(\sqrt{8a}\right)^x$ since $\displaystyle a b=1$
I don't know how to solve this equation either and I don't see how letting $\displaystyle a^x = t^2$ will help.
On the other hand if you graph the function
$\displaystyle f(x) = a^x+1 - \left(\sqrt{8a}\right)^x$ with $\displaystyle x > 0$
you will notice that it is decreasing and that $\displaystyle f(2) = 0$
This shows that $\displaystyle x = 2$ is the only solution
To prove that it is the only solution you would have to show $\displaystyle f ' (x) < 0$