What is the answer to z^2=i(|z|^2-4) ?

The answer to z^2=i(|z|^2-4) is z=1-i,z=-1+i

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z^2=i(|z|^2-4)

z^{2} = i (∣ z ∣^{2} - 4)

z = 1 - i, z = - 1 + i

Solution steps

z^{2} = i (∣ z ∣^{2} - 4)

Substitute

z = x + y i

(x + y i)^{2} = i (∣ (x + y i) ∣^{2} - 4)

Expand

(x^{2} - y^{2}) + 2 i x y = i (- 4 + x^{2} + y^{2})

Rewrite

i (- 4 + x^{2} + y^{2})

in standard complex form:

0 + (- 4 + x^{2} + y^{2}) i

(x^{2} - y^{2}) + 2 i x y = 0 + (- 4 + x^{2} + y^{2}) i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[x^{2} - y^{2} = 0 2 x y = - 4 + x^{2} + y^{2}]

[x^{2} - y^{2} = 0 2 x y = - 4 + x^{2} + y^{2}] : (x = 1, x = - 1, y = - 1 y = 1)

Substitute back

z = x + y i

z = 1 - i, z = - 1 + i

(2 - 3 i) x + (3 - 2 i) y = 14 + 8 i

y - in y = x^{2} + 1

(a + b + 3) + (8 + b) i = 2 a + (3 a - b) i

(x + 1) + 6 y i = 4 + 24 i

s i \frac{159}{100} = a, b c

What is the answer to z^2=i(|z|^2-4) ?
The answer to z^2=i(|z|^2-4) is z=1-i,z=-1+i