I am having problems with the following :

Z +|Z|^2 = 10-2i (the solution must be in the form a+bi)

I have gotten to :

a +bi + a^2 +b^2 = 10 -2i

and cannot think how to solve it:(

Any help would be appreciated.

Jeff

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- Sep 17th 2012, 12:06 PMjholgateAlgebra modulus problem
I am having problems with the following :

Z +|Z|^2 = 10-2i (the solution must be in the form a+bi)

I have gotten to :

a +bi + a^2 +b^2 = 10 -2i

and cannot think how to solve it:(

Any help would be appreciated.

Jeff - Sep 17th 2012, 12:19 PMemakarovRe: Algebra modulus problem
You don't know how to compare the real and imaginary parts of the two sides?

- Sep 18th 2012, 05:43 AMjholgateRe: Algebra modulus problem
Ok solvedit. a^2 and b^2 are real numbers. Don't know why I never seen that. You're reply helped me see that. Thank you emakarov:)

- Sep 19th 2012, 07:30 AMVistusRe: Algebra modulus problem
I'm still having problems with this, I got

a + bi + a^2 +b^2 = 10 -2i

but I don't know how to proceed. Apparently you solved it with the help of previous post, but I don't see how. A little hint, please? :) - Sep 19th 2012, 07:41 AMProve ItRe: Algebra modulus problem
As you were told, compare the real and imaginary parts.

You have $\displaystyle \displaystyle \begin{align*} a + a^2 + b^2 + b\,i = 10 - 2i \end{align*}$, so $\displaystyle \displaystyle \begin{align*} a + a^2 + b^2 = 10 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} b = -2 \end{align*}$. Solve for $\displaystyle \displaystyle \begin{align*} a \end{align*}$. - Sep 20th 2012, 11:09 AMPetrusRe: Algebra modulus problem
With other words a+a^2+4=10

a+a^2=6

a=???

If i did not get it why b=-2 its because bi=-2i - Sep 20th 2012, 07:48 PMProve ItRe: Algebra modulus problem
- Sep 20th 2012, 08:34 PMPetrusRe: Algebra modulus problem
- Sep 20th 2012, 08:42 PMProve ItRe: Algebra modulus problem
- Sep 20th 2012, 10:11 PMDevenoRe: Algebra modulus problem
two complex numbers:

a+bi and c+di are equal if and only if: a = c, and b = d.

so if a^{2}+b^{2}+ a + bi = 10 - 2i, then:

a^{2}+ b^{2}+ a = 10, and b = -2.

since b = -2, a^{2}+ b^{2}+ a = a^{2}+ 4 + a.

since we know this already equals 10, we get:

a^{2}+ a - 6 = 0

and a^{2}+ a - 6 = (a - 2)(a + 3), so a = 2, or -3.

as a final check, we verify that:

z = 2 - 2i

z = -3 - 2i both satisfy:

z + |z|^{2}= 10 - 2i

2 - 2i + |2 - 2i|^{2}= 2 - 2i + 4 + 4 = (2 + 4 + 4) - 2i = 10 - 2i

-3 - 2i + |-3 - 2i|^{2}= -3 - 2i + 9 + 4 = (-3 + 9 + 4) - 2i = 10 - 2i (checked). - Sep 20th 2012, 10:43 PMPetrusRe: Algebra modulus problem