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

The answer to z/(1+z)=-1+i is z=-3/5+1/5 i

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z/(1+z)=-1+i

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

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

Solution steps

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

Substitute

z = x + y i

\frac{x + y i}{1 + x + y i} = - 1 + i

Multiply both sides by

1 + x + i y

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

Expand

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

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

[x = - 1 - x - y y = 1 + x - y]

[x = - 1 - x - y y = 1 + x - y] : x = - \frac{3}{5}, y = \frac{1}{5}

Substitute back

z = x + y i

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

x^{2} - 4 i x - 1 = 0

z^{2} + (2 i + 7) z + 6 + 3 i = 0

y, e = \frac{sin ( x )}{x e n y}

\frac{6 - 5 i}{2 + 3 i} = a + bi

x^{2} - y^{2} - 2 y x i = 4 x + 4