Fibonacci numbers – gcd (Part 1)

Find all ordered pairs   $(m, \; n)$   of positive integers for which there’s an integer   $x$   satisfying the equation:

$gcd \,(F_m, \, F_n) \, x^2 \; - \; (F_m + F_n) \, x \; + \; F_{gcd \,(m, \,n)} \; = \; 0$

Next blog:   The gcd of any two Fibonacci numbers is also a Fibonacci number