What is the answer to z-1/z =3+4i ?

The answer to z-1/z =3+4i is z=-0.12737…+0.15653…i,z=3.12737…+3.84346…i

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z-1/z =3+4i

z - \frac{1}{z} = 3 + 4 i

z = - 0.12737 \dots + 0.15653 \dots i, z = 3.12737 \dots + 3.84346 \dots i

Solution steps

z - \frac{1}{z} = 3 + 4 i

Substitute

z = x + y i

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

Expand

(x^{2} - y^{2} - 1) + 2 i x y = (3 x - 4 y) + i (3 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 = 3 x - 4 y 2 x y = 3 y + 4 x]

[x^{2} - y^{2} - 1 = 3 x - 4 y 2 x y = 3 y + 4 x] : (x = - 0.12737 \dots, x = 3.12737 \dots, y = 0.15653 \dots y = 3.84346 \dots)

Substitute back

z = x + y i

z = - 0.12737 \dots + 0.15653 \dots i, z = 3.12737 \dots + 3.84346 \dots i

i (x) = ∣ 2 y + 5 ∣

(a + bi)^{2} + 2 (a - bi) + 1 = 0

m, y = (x - 3)^{2} minim o

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

2 - i x - y = 0

What is the answer to z-1/z =3+4i ?
The answer to z-1/z =3+4i is z=-0.12737…+0.15653…i,z=3.12737…+3.84346…i