What is the answer to z-1/z =4+4i ?

The answer to z-1/z =4+4i is z=-0.12864…+0.12087…i,z=4.12864…+3.87912…i

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z-1/z =4+4i

z - \frac{1}{z} = 4 + 4 i

z = - 0.12864 \dots + 0.12087 \dots i, z = 4.12864 \dots + 3.87912 \dots i

Solution steps

z - \frac{1}{z} = 4 + 4 i

Substitute

z = x + y i

x + y i - \frac{1}{x + y i} = 4 + 4 i

Multiply both sides by

x + y i

x (x + y i) + y i (x + y i) - \frac{1}{x + y i} (x + y i) = 4 (x + y i) + 4 i (x + y i)

Expand

(x^{2} - y^{2} - 1) + 2 i x y = (4 x - 4 y) + i (4 x + 4 y)

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

[x^{2} - y^{2} - 1 = 4 x - 4 y 2 x y = 4 x + 4 y]

[x^{2} - y^{2} - 1 = 4 x - 4 y 2 x y = 4 x + 4 y] : (x = - 0.12864 \dots, x = 4.12864 \dots, y = 0.12087 \dots y = 3.87912 \dots)

Substitute back

z = x + y i

z = - 0.12864 \dots + 0.12087 \dots i, z = 4.12864 \dots + 3.87912 \dots i

x^{2} + 2 i x - 1 - i = 0

z^{2} - (5 + 4 i) z - (3 - 5 i) = 0

s i f (x) = x^{2} - x, f (x + 1) = f (- x)

(5 x + 4) + 7 i = (3 y + 1) i + 9

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

What is the answer to z-1/z =4+4i ?
The answer to z-1/z =4+4i is z=-0.12864…+0.12087…i,z=4.12864…+3.87912…i