SOLUTION: x^2=-1, Find all n complex solutions of the equation of the form x^n=k

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Question 951807: x^2=-1, Find all n complex solutions of the equation of the form x^n=k
Answer by MathLover1(20849) About Me  (Show Source):
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The fundamental theorem of algebra implies that a polynomial equation of degree
n has precisely n solutions in the complex number system.
These solutions can be real or complex and may be repeated.
recall the quadratic formula and its discriminant:

using the discriminant b%5E2-4ac, we are able to determine solutions which are real or complex and may be repeated
if the discriminant is less than zero, equation has two complex solutions
b%5E2-4ac%3C0
like you have here x%5E2=-1=>x%5E2%2B1=0=>a=1, b=0, c=1
using b%5E2-4ac%3C0=>0%5E2-4%2A1%2A1%3C0=>-4%3C0; so, discriminant is negative and we have two complex solutions
these are x%5E2=-1=>x=sqrt%28-1%29=>x=ior x=-i

now, find all n complex solutions of the equation of the form x%5En=k:
x%5En-k=0+=> a=1, b=0, c=-k
0%5E2-4%2A1%28-k%29%3C0
4k%3C0
k%3C0%2F4
k%3C0+
for any k%3C0 there will be two complex solutions of the equation x%5En=k
and k can be any number from:
(-infinity%3Ck%3C0+)