Question

Z^2+z+1=0,where z is a complex number then value of (z+1/z)^2+(z^2+1/z)^2.

Answer 1

Answer:

-1 ± 2√3 i

Step-by-step explanation:

Divide z² + z + 1 = 0 by z to get z + 1 + 1/z = 0.  So z + 1/z = -1.  The first term is then (-1)² = 1.

From z² + z + 1 = 0, we have z² = - z - 1.  From z + 1/z = -1, also 1/z = - z - 1.  So the second term is:

( z² + 1/z )² = ( - z - 1 - z - 1 )² = ( -2z - 2 )² = 4 ( - z - 1 )² = 4 ( z² + 2z + 1 ) = 4 ( z² + z + 1 + z ) = 4z.

So the whole expression is 1 + 4z.

Using the quadratic formula to solve for z:

z = ( -1 ± √(1 - 4) ) / 2 = ( -1 ± √3 i ) / 2

so

4z = -2 ± 2√3 i

The value of the whole expression is then

1 + 4z = -1 ± 2√3 i

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Is there a mistake in the question though?  Is it possible that you actually wanted

( z + 1/z)² + ( z² + 1/z² ) ?

If so, divide z² + z + 1 = 0 by z² to get 1 + 1/z + 1/z² = 0.  So

1/z² = - ( 1 + 1/z) = z

=> z² + 1/z² = z² + z = -1

All together then, the whole expression is

(-1)² + (-1) = 1 - 1 = 0.