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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

Respuesta :

A power [tex]a^b[/tex] gives 1 as a result in two cases

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

So, we need either

[tex]x^2-4x+5=1[/tex]

or

[tex]x^2+x^{-30}=0[/tex]

The first happens if

[tex]x^2-4x+5=1 \iff x^2-4x+4=0 \iff (x-2)^2=0 \iff x=2[/tex]

The second happens if

[tex]x^2+x^{-30}=0 \iff \dfrac{1+x^{32}}{x^{30}}=0[/tex]

which is impossible.

So, the only solution is [tex]x=2[/tex]

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