What is the answer to |z|+z=3+i ?

The answer to |z|+z=3+i is z= 4/3+i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

|z|+z=3+i

∣ z ∣ + z = 3 + i

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

Solution steps

∣ z ∣ + z = 3 + i

Substitute

z = x + y i

∣ (x + y i) ∣ + x + y i = 3 + i

Simplify

∣ x + y i ∣ + x + y i : (x^{2} + y^{2} + x) + i y

(x^{2} + y^{2} + x) + i y = 3 + i

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

[x^{2} + y^{2} + x = 3 y = 1]

[x^{2} + y^{2} + x = 3 y = 1] : (x = \frac{4}{3}, y = 1)

Substitute back

z = x + y i

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

(z^{2} - 4 i z - 1) = 0

∣ 3 z + 2 i ∣ = 6

∣ 3 z + 2 i ∣ = 3

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

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