What is the solution for 5^x^2=15625/5^x ?

We'll write 15625 as a power of 5, to create matching
bases.

5^(x^2) = 5^6/5^x

We'll
apply the quotient rule of the exponentials that have matching
bases:

5^(x^2) = 5^(6-x)

Since
the base are matching, we'll apply one to one property:

x^2
= 6 - x

We'll subtract 6 - x both
sides:

x^2 + x - 6 = 0

We'll
apply quadratic formula:

x1 = [-1 + sqrt(1 +
24)]/2

x1 = (-1 + 5)/2

x1 =
2

x2 = (-1-5)/2

x2 =
-3

The equation will have 2 solutions and
they are: {-3 ; 2}.