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

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Homework Help Overview

The discussion revolves around finding solutions for the equation z^2 = a + bi, where a and b are real numbers. Participants are exploring the implications of this equation in the context of complex numbers.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants attempt to express z in terms of its real and imaginary components, suggesting the substitution z = x + iy. There is discussion about expanding the equation and comparing real and imaginary parts to form a system of equations.

Discussion Status

Some participants have reached a point of confusion regarding the resulting equations, particularly a quadratic in y^2. There is a suggestion to introduce a new variable to facilitate solving the equation, indicating a productive line of inquiry.

Contextual Notes

Participants are working under the constraints of the original equation and are exploring various interpretations of the resulting equations without reaching a consensus on the next steps.

Bubblegum
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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 ?
 
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Bubblegum said:

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.
 
I am stuck at:

0= y^4 + ay^2 - b^2/4
 
Bubblegum said:
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.
 

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