Finding Solutions for z in z^2 = a + bi

[Bubblegum]

Homework Statement

z^2 = a + bi

a = real number
b = real number

find all the solutions for z

Homework Equations

The Attempt at a Solution

(x+y)^2 = a + bi ?

 

[Mentallic]

Homework Statement

z^2 = a + bi

a = real number
b = real number

find all the solutions for z

Homework Equations

The Attempt at a Solution

(x+y)^2 = a + bi ?

Let z = x+iy

Then

(x+iy)^2 = a+ib.

Expand the LHS and then compare the real parts and imaginary parts. Then what you'll have is a system of two equations in the two unknown x and y.

 

[Bubblegum]

I am stuck at:

0= y^4 + ay^2 - b^2/4

 

[Mentallic]

I am stuck at:

0= y^4 + ay^2 - b^2/4

It's a quadratic in y^2. If you're unsure what that means, let some other variable such as m=y² and then solve for m, and then convert back to y² and solve for y.