Find $x$ such that $2^x+3^x-4^x+6^x-9^x=1$

If the sum of a finite number of non-negative expressions is $0$, each of them has to be zero.

In other words, when $a,b,c\geq0$

$a+b+c=0 \implies a=b=c=0$

You have done the hard part by showing that it can be written as the sum of three squares. This means that $a=b$, $a=1$ and $b=1$. What does that tell you about $x$?

A sum of squares equals zero if and only if each of the squares equals zero. So you get $a=b=1$.