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

The answer to (z-1)/(1+z)=i+1 is z=-1+2i

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

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

z = - 1 + 2 i

Solution steps

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

Substitute

z = x + y i

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

Multiply both sides by

1 + x + i y

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

Expand

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

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

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

[x - 1 = - y + 1 + x y = 1 + x + y] : x = - 1, y = 2

Substitute back

z = x + y i

z = - 1 + 2 i

x + i y = ∣ x - y i ∣

(1 + i) z^{2} = - 1 + 7

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

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

x - y i = 4 + 7 i

What is the answer to (z-1)/(1+z)=i+1 ?
The answer to (z-1)/(1+z)=i+1 is z=-1+2i