Question

How many different solutions are there to the equation (x² - 4x + 5 )ˣ²⁺ˣ⁻³⁰=1

Answer 1

A power $a^b$ gives 1 as a result in two cases

1. The base is 1, and every exponent is good: $1^b=1 \forall b$
2. The exponent is 0, and every non-zero base is good: $a^0=1 \forall a\neq 0$

So, we need either

$x^2-4x+5=1$

or

$x^2+x^{-30}=0$

The first happens if

$x^2-4x+5=1 \iff x^2-4x+4=0 \iff (x-2)^2=0 \iff x=2$

The second happens if

$x^2+x^{-30}=0 \iff \dfrac{1+x^{32}}{x^{30}}=0$

which is impossible.

So, the only solution is $x=2$