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vova2212 [387]
3 years ago
10

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

Mathematics
1 answer:
Anna71 [15]3 years ago
8 0

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

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