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

The answer to z-1/z =1+4i is z=-0.07024…+0.24637…i,z=1.07024…+3.75362…i

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z-1/z =1+4i

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

z = - 0.07024 \dots + 0.24637 \dots i, z = 1.07024 \dots + 3.75362 \dots i

Solution steps

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

Substitute

z = x + y i

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

Expand

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

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

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

[x^{2} - y^{2} - 1 = x - 4 y 2 x y = y + 4 x] : (x = - 0.07024 \dots, x = 1.07024 \dots, y = 0.24637 \dots y = 3.75362 \dots)

Substitute back

z = x + y i

z = - 0.07024 \dots + 0.24637 \dots i, z = 1.07024 \dots + 3.75362 \dots i

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

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

z^{2} + i z = 2 i^{2}

a + ib = (4 + 3 i) (5 - 2 i)

\frac{- 1 + 2 i}{8 i} = a + bi

What is the answer to z-1/z =1+4i ?
The answer to z-1/z =1+4i is z=-0.07024…+0.24637…i,z=1.07024…+3.75362…i