For this case we have the following equation:
[tex]25^{x} = 5^{ x^2-3}[/tex]
Rewriting we have:
[tex]5^{2 (x)} = 5^{ x^ 2-3}\\5^{ 2x} = 5^{ x^ 2-3}[/tex]
For expressions to be equal, exponents must be equal. So:
[tex]2x = x ^ 2-3\\x ^ 2-2x-3 = 0[/tex]
To factor, we look for two numbers that, when multiplied, result in -3 and when added, result in -2. These numbers are -3 and 1.
[tex](x-3) (x + 1) = 0[/tex]
Thus, the roots are:
[tex]x_ {1} = 3\\x_ {2} = - 1[/tex]
Answer:
[tex]x_ {1} = 3\\x_ {2} = - 1[/tex]
