Question

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

A:1
B:2
C:3
D:4
E:infinitely many

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\\eq 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 (1+x^(32))/(x^(30))=0`

which is impossible.

So, the only solution is
`x=2`