What is the answer to z^2-4zi+4i=7 ?

The answer to z^2-4zi+4i=7 is z=2+i,z=-2+3i

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z^2-4zi+4i=7

z^{2} - 4 z i + 4 i = 7

z = 2 + i, z = - 2 + 3 i

Solution steps

z^{2} - 4 z i + 4 i = 7

Substitute

z = x + y i

(x + y i)^{2} - 4 (x + y i) i + 4 i = 7

Expand

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

(x^{2} - y^{2} + 4 y) + i (4 - 4 x + 2 x y) = 7

Rewrite

7

in standard complex form:

7 + 0 i

(x^{2} - y^{2} + 4 y) + i (4 - 4 x + 2 x y) = 7 + 0 i

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

[x^{2} - y^{2} + 4 y = 7 4 - 4 x + 2 x y = 0]

[x^{2} - y^{2} + 4 y = 7 4 - 4 x + 2 x y = 0] : (x = 2, x = - 2, y = 1 y = 3)

Substitute back

z = x + y i

z = 2 + i, z = - 2 + 3 i

(1 + i) x - i y = 2 + i

z^{2} + (i 4 - 1) z + i - 5 = 0

z^{2} - (7 - i) z + (14 - 5 i) = 0

2 x + 2 y = 2 x + 2 y i + 5

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

What is the answer to z^2-4zi+4i=7 ?
The answer to z^2-4zi+4i=7 is z=2+i,z=-2+3i