What is the answer to Z/(1+Z)=3+i4 ?

The answer to Z/(1+Z)=3+i4 is Z=-11/10+1/5 i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

Z/(1+Z)=3+i4

\frac{Z}{1 + Z} = 3 + i 4

Z = - \frac{11}{10} + \frac{1}{5} i

Solution steps

\frac{Z}{1 + Z} = 3 + i 4

Substitute

Z = x + y i

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

Multiply both sides by

1 + x + i y

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

Expand

x + i y = (3 + 3 x - 4 y) + i (4 + 3 y + 4 x)

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

[x = 3 + 3 x - 4 y y = 4 + 3 y + 4 x]

[x = 3 + 3 x - 4 y y = 4 + 3 y + 4 x] : x = - \frac{11}{10}, y = \frac{1}{5}

Substitute back

Z = x + y i

Z = - \frac{11}{10} + \frac{1}{5} i

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

\frac{( 4 + x i )}{2 + i} = y + 2 i

2 x + 7 + (3 - y) i = - 4 + 6 i

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

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

What is the answer to Z/(1+Z)=3+i4 ?
The answer to Z/(1+Z)=3+i4 is Z=-11/10+1/5 i