What is the answer to |z|-z=2+i ?

The answer to |z|-z=2+i is z=-3/4-i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

|z|-z=2+i

∣ z ∣ - z = 2 + i

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

Solution steps

∣ z ∣ - z = 2 + i

Substitute

z = x + y i

∣ (x + y i) ∣ - (x + y i) = 2 + i

Expand

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

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

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

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

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

Substitute back

z = x + y i

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

x, x^{2} y^{2} + x sin (y) = 4

z^{2} - z - z i = 0

z^{2} - (1 + 4 i) z - (5 + i) = 0

z^{2} - 2 i = 0

x^{2} - (3 + 2 i) x + 1 + 3 i = 0