x=1.18681439028 approx.
Let y=4ˣ+6ˣ-9ˣ, then dy/dx=4ˣln(4)+6ˣln(6)-9ˣln(9).
A graph of y shows an x-intercept near x=1.
Substitute x in:
x-(4ˣ+6ˣ-9ˣ)/(4ˣln(4)+6ˣln(6)-9ˣln(9)).
This gives us another value of x to plug into the expression (Newton’s Method).
Repeat the exercise until the same value of x is repeated (limited by calculator).