1. ## Algebra 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

2. ## Re: Algebra modulus problem

You don't know how to compare the real and imaginary parts of the two sides?

3. ## Re: 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

4. ## Re: 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?

5. ## Re: Algebra modulus problem

Originally Posted by Vistus
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?
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*}.

6. ## Re: 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

7. ## Re: Algebra modulus problem

Originally Posted by Petrus
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
Surely if bi = -2i, then dividing by i gives b = -2.

Also, if you're studying complex numbers, surely you know how to solve a quadratic equation (such as factorising and using NFL, quadratic formula, completing the square...)

8. ## Re: Algebra modulus problem

Originally Posted by Prove It
Surely if bi = -2i, then dividing by i gives b = -2.

Also, if you're studying complex numbers, surely you know how to solve a quadratic equation (such as factorising and using NFL, quadratic formula, completing the square...)
Ehmm are u saying that i did it wrong? U confused me

9. ## Re: Algebra modulus problem

Originally Posted by Petrus
Ehmm are u saying that i did it wrong? U confused me
You said you didn't understand why b = -2. I was explaining why.

10. ## Re: Algebra modulus problem

Originally Posted by Vistus
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?
two complex numbers:

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

so if a2+b2+ a + bi = 10 - 2i, then:

a2 + b2 + a = 10, and b = -2.

since b = -2, a2 + b2 + a = a2 + 4 + a.

since we know this already equals 10, we get:

a2 + a - 6 = 0

and a2 + 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).

11. ## Re: Algebra modulus problem

Originally Posted by Prove It
You said you didn't understand why b = -2. I was explaining why.
Ohh no i did understand was just Teling him if he did not get it sometimes My iPhone Change when i say "u" to "i" i was just making it easy to understand but maybe waste, My fault