What is the answer to solvefor z,z-1/z =5+4i ?

The answer to solvefor z,z-1/z =5+4i is z=-0.12343…+0.09410…i,z=5.12343…+3.90589…i

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solvefor z,z-1/z =5+4i

solve for

z, z - \frac{1}{z} = 5 + 4 i

z = - 0.12343 \dots + 0.09410 \dots i, z = 5.12343 \dots + 3.90589 \dots i

Solution steps

z - \frac{1}{z} = 5 + 4 i

Substitute

z = x + y i

x + y i - \frac{1}{x + y i} = 5 + 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) = 5 (x + y i) + 4 i (x + y i)

Expand

(x^{2} - y^{2} - 1) + 2 i x y = (5 x - 4 y) + i (4 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 - 4 y 2 x y = 4 x + 5 y]

[x^{2} - y^{2} - 1 = 5 x - 4 y 2 x y = 4 x + 5 y] : (x = - 0.12343 \dots, x = 5.12343 \dots, y = 0.09410 \dots y = 3.90589 \dots)

Substitute back

z = x + y i

z = - 0.12343 \dots + 0.09410 \dots i, z = 5.12343 \dots + 3.90589 \dots i

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

∣ z - i ∣ = 3

∣ z - i ∣ = 4

(1 + i) x - (1 - 2 i) y = 1 + 4 i

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

What is the answer to solvefor z,z-1/z =5+4i ?
The answer to solvefor z,z-1/z =5+4i is z=-0.12343…+0.09410…i,z=5.12343…+3.90589…i