What is the answer to z-1/z =5+5i ?

The answer to z-1/z =5+5i is z=-0.10191…+0.09792…i,z=5.10191…+4.90207…i

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z-1/z =5+5i

z - \frac{1}{z} = 5 + 5 i

z = - 0.10191 \dots + 0.09792 \dots i, z = 5.10191 \dots + 4.90207 \dots i

Solution steps

z - \frac{1}{z} = 5 + 5 i

Substitute

z = x + y i

x + y i - \frac{1}{x + y i} = 5 + 5 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) = 5 (x + y i) + 5 i (x + y i)

Expand

(x^{2} - y^{2} - 1) + 2 i x y = (5 x - 5 y) + i (5 x + 5 y)

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

[x^{2} - y^{2} - 1 = 5 x - 5 y 2 x y = 5 x + 5 y]

[x^{2} - y^{2} - 1 = 5 x - 5 y 2 x y = 5 x + 5 y] : (x = - 0.10191 \dots, x = 5.10191 \dots, y = 0.09792 \dots y = 4.90207 \dots)

Substitute back

z = x + y i

z = - 0.10191 \dots + 0.09792 \dots i, z = 5.10191 \dots + 4.90207 \dots i

\frac{x + i y}{1 + x + i y} = 3 + 4 i

4 (3 i + 2) - 2 i = (2 x) i + y

3 x - 2 y i = 6 + 4 i

p (2 i + 6 j) + q (2 i - 4 j) = 6 i - 7 j

(3 + i) x^{2} + 10 x - (9 + 3 i) = 0

What is the answer to z-1/z =5+5i ?
The answer to z-1/z =5+5i is z=-0.10191…+0.09792…i,z=5.10191…+4.90207…i