Question

For what values of x does 25 Superscript x Baseline = 5 Superscript x squared minus 3?

Answer 1

For this case we have the following equation:

$25^{x} = 5^{ x^2-3}$

Rewriting we have:

$5^{2 (x)} = 5^{ x^ 2-3}\\5^{ 2x} = 5^{ x^ 2-3}$

For expressions to be equal, exponents must be equal. So:

$2x = x ^ 2-3\\x ^ 2-2x-3 = 0$

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.

$(x-3) (x + 1) = 0$

Thus, the roots are:

$x_ {1} = 3\\x_ {2} = - 1$

Answer:

$x_ {1} = 3\\x_ {2} = - 1$