Find all the real and non real answers to 1/x^2=x

Multiply both side by x^2

 

1= x^3     the only answer is x=1            x=i does not work    x= -1 does not work     .....

Solve for x:
1/x^2 = x

 

Multiply both sides by x^2:
1 = x^3

 

1 = x^3 is equivalent to x^3 = 1:
x^3 = 1

 

Taking cube roots gives 1 times the third roots of unity:
Answer: |  x = 1      or      x = -(-1)^(1/3)       or        x = (-1)^(2/3)