What is the answer to Z^2=3-4i ?

The answer to Z^2=3-4i is Z=2-i,Z=-2+i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

Z^2=3-4i

Z^{2} = 3 - 4 i

Z = 2 - i, Z = - 2 + i

Solution steps

Z^{2} = 3 - 4 i

Substitute

Z = x + y i

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

Expand

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

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

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

[x^{2} - y^{2} = 3 2 x y = - 4]

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

Substitute back

Z = x + y i

Z = 2 - i, Z = - 2 + i

(a^{2} - b^{2}) + 2 abi = - 8 - 6 i

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

i y (2 - i) + 1 = 4 i - x (1 - i)

x^{2} + (i - 6) x + 10 = 0

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