Please reply fast .?

Express z² in polar form. That is

z² = r(cos(iθ) + isin(θ))

where r = |z| and θ = Arg(z) ( ie - π < θ ≤ π )

Applying De Moivre's Theorem where the nth roots of z^n are of the form

z_k = r^(1/k) [cos(θ/n + 2kπ/n) + isin(θ/n + 2kπ/n)]

(0 ≤ k ≤ n-1)

with z₀ being the principle root,

then the two square roots of z - ie z₀ and z₁ - are

z₀ = r^(1/2) (cos(θ/2) + isin(θ/2))

z₁ = r^(1/2) (cos(θ/2 + π ) + isin(θ/2 + π))

Applying basic trigonometric identities:

cos(α + β) = cos(α)cos(β) - sin(α)sin(β)

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

from which

cos(θ/2 + π ) = -cos(θ/2)

sin(θ/2 + π ) = -sin(θ/2)

we can write

z₁ = -r^(1/2) (cos(θ/2) + isin(θ/2))

The point to notice is that the two square roots of z² are both at angle π apart

I'm not sure what you mean by "+_z" and as for the modulus of z not being a root, the modulus of z - ie |z| - is a real-valued quantity denoting the distance of z from the origin in the complex plane. Is it possible that you might be confusing

|z|² = zz* (where here z* is denoted as the conjugate of z)?