What is the answer to z-1/z =3+5i ?

The answer to z-1/z =3+5i is z=-0.09338…+0.14651…i,z=3.09338…+4.85348…i

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z-1/z =3+5i

z - \frac{1}{z} = 3 + 5 i

z = - 0.09338 \dots + 0.14651 \dots i, z = 3.09338 \dots + 4.85348 \dots i

Solution steps

z - \frac{1}{z} = 3 + 5 i

Substitute

z = x + y i

x + y i - \frac{1}{x + y i} = 3 + 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) = 3 (x + y i) + 5 i (x + y i)

Expand

(x^{2} - y^{2} - 1) + 2 i x y = (3 x - 5 y) + i (3 y + 5 x)

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

[x^{2} - y^{2} - 1 = 3 x - 5 y 2 x y = 3 y + 5 x]

[x^{2} - y^{2} - 1 = 3 x - 5 y 2 x y = 3 y + 5 x] : (x = - 0.09338 \dots, x = 3.09338 \dots, y = 0.14651 \dots y = 4.85348 \dots)

Substitute back

z = x + y i

z = - 0.09338 \dots + 0.14651 \dots i, z = 3.09338 \dots + 4.85348 \dots i

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

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

x, cs in y + x^{2} y = c

(x + 2 y + 3) + i (3 x - y - 1) = 0

4 z^{2} + (- 13 + 18 i) z - 5 - 10 i = 0

What is the answer to z-1/z =3+5i ?
The answer to z-1/z =3+5i is z=-0.09338…+0.14651…i,z=3.09338…+4.85348…i