Given that ` 1 + 2|z|^(2) = |z^(2) + 1|^(2) + 2 | z + 1 | ^(2)`, then the value of `|z(z + 1 )|` is ______.

Answer

1

Explanation

`1+2|z|^(2) = |z^(2) +1|^(2)+ 2|z+1|^(2)`
`rArr 1+2barzz = (barz^(2)+1)+2(barz +1)`
`rArr (zbarz)^(2)+ 2(z +barz) +z^(2) + barz^(2) + 2=0`
`rArr (zbarz -1)^(2) + (z+barz +1)^(2) = 0`
`rArr zbarz =1 rArr barz = (1)/(2)`
and `z+barz+1=0`
or `z+(1)/(2)+1=0`
`rArr z^(2) + z =1`
`rArr |z^(2)+z| =1`