Solution of a typical equation with surds power

Question

I was attempting to find a solution for the equation $1 + 12^\sqrt{x} = 9^\sqrt{x} + 10^\sqrt{x}$. By the trial and error, I found a solution $ x = 9$. Is there any method to solve these equation?

Answer 1

Step 1. Set $\,y=\sqrt{x}$, and obtain the equation $$ f(y)=12^y-10^y-9^y+1=0 $$

Step 2. Show that $f(y)<0$, if $y\in(0,3)$, $f(y)>0$, if $y>3$ or $y<0$.