What is the answer to |z|^2+z-i=11 ?

The answer to |z|^2+z-i=11 is z=(-1+sqrt(41))/2+i,z=(-1-sqrt(41))/2+i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

|z|^2+z-i=11

∣ z ∣^{2} + z - i = 11

z = \frac{- 1 + 41}{2} + i, z = \frac{- 1 - 41}{2} + i

Solution steps

∣ z ∣^{2} + z - i = 11

Substitute

z = x + y i

∣ (x + y i) ∣^{2} + x + y i - i = 11

Simplify

∣ x + y i ∣^{2} + x + y i - i : (x^{2} + y^{2} + x) + i (- 1 + y)

(x^{2} + y^{2} + x) + i (- 1 + y) = 11

Rewrite

11

in standard complex form:

11 + 0 i

(x^{2} + y^{2} + x) + i (- 1 + y) = 11 + 0 i

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

[x^{2} + y^{2} + x = 11 - 1 + y = 0]

[x^{2} + y^{2} + x = 11 - 1 + y = 0] : (x = \frac{- 1 + 41}{2}, x = \frac{- 1 - 41}{2}, y = 1 y = 1)

Substitute back

z = x + y i

z = \frac{- 1 + 41}{2} + i, z = \frac{- 1 - 41}{2} + i

(x + i y)^{2} = 5 - 12 i

- x^{2} + 125.6637 x i + 1184 = 0

x^{2} - 9 - 40 i = 0

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

8 x^{2} + 9 x + i = 0

What is the answer to |z|^2+z-i=11 ?
The answer to |z|^2+z-i=11 is z=(-1+sqrt(41))/2+i,z=(-1-sqrt(41))/2+i