What is the answer to z-1/z =2+3i ?

The answer to z-1/z =2+3i is z=-0.17481…+0.22320…i,z=2.17481…+2.77679…i

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z-1/z =2+3i

z - \frac{1}{z} = 2 + 3 i

z = - 0.17481 \dots + 0.22320 \dots i, z = 2.17481 \dots + 2.77679 \dots i

Solution steps

z - \frac{1}{z} = 2 + 3 i

Substitute

z = x + y i

x + y i - \frac{1}{x + y i} = 2 + 3 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) = 2 (x + y i) + 3 i (x + y i)

Expand

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

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

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

[x^{2} - y^{2} - 1 = 2 x - 3 y 2 x y = 2 y + 3 x] : (x = - 0.17481 \dots, x = 2.17481 \dots, y = 0.22320 \dots y = 2.77679 \dots)

Substitute back

z = x + y i

z = - 0.17481 \dots + 0.22320 \dots i, z = 2.17481 \dots + 2.77679 \dots i

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

x + (2 x - y) i = - 7 i - 2

z^{2} - (3 + 3 i) z + (1 + 3 i) = 0

(2 x - y) + (x + y) i = - 4 - 5 i

2 + x i = y - 7 i

What is the answer to z-1/z =2+3i ?
The answer to z-1/z =2+3i is z=-0.17481…+0.22320…i,z=2.17481…+2.77679…i