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

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We have to solve : 5^x^2 = 15625 / 5^x

5^x^2 = 15625 / 5^x

=> 5^x^2 = 5^6 / 5^x

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

As the base 5 is the same we can equate the exponent

=> x^2 = 6 - x

=> x^2 + x - 6 = 0

=> x^2 + 3x - 2x - 6 = 0

=> x(x + 3) - 2(x + 3) = 0

=> (x - 2)(x + 3) = 0

=> x = 2 and x = -3

**The required values of x are x = 2 and x = -3**

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}.**