Hey! So, I've been working on this problem for awhile but I'm currently a bit dumbfounded on how to solve it. I think this question has been answered before but I still don't quite understand how it works or how to explain it.
The problem is as follows:
|z|^2 -2(complex conjugate of z) + iz= 2i
It'd really love some help with this problem. So far, I'm lost, but I don't know why that is the case or how I would be able to solve this beyond guessing and checking. The question is looking for complex answers, and I tried substituting z for (a + bi), but that wasn't really very helpful, either due to my lack of understanding or getting lost somewhere.
Any and all help would be greatly appreciated!
Let z = x+yi
Then we have
x^2+y^2-2x+2yi+xi-y=2i
That means we have
x^2+y^2-2x-y = 0
2y+x-2 = 0
Thus, x = 2-2y
and so
4-8y+4y^2+y^2-4+4y-y = 0
5y^2-5y = 0
5y(y-1) = 0
so,
y=0 x=2
or
y=1 x=0
Or,
z=2 or z=i
I'll let you verify that both of those z values satisfy the original equation.
Thanks for the response Steve, but this doesn't quite clarify things for me. The step I keep getting lost on is this one:
"x^2+y^2-2x-y = 0
2y+x-2 = 0"
How did you get to this point? What happened to the 2yi + xi?
I'd be happy to help you with this problem! It seems like you're on the right track in substituting z as (a + bi). Let's break down the problem and go step by step:
The equation you have is |z|^2 - 2*(complex conjugate of z) + iz = 2i.
Step 1: Substitute z as (a + bi), where a and b are real numbers.
With this substitution, the equation becomes:
|(a + bi)|^2 - 2*(a - bi) + i(a + bi) = 2i.
Step 2: Simplify each term.
Let's simplify each term:
|(a + bi)|^2 can be written as (a + bi)(a + bi), which simplifies to (a^2 + 2abi - b^2).
-2*(a - bi) simplifies to (-2a + 2bi).
i(a + bi) becomes (ai - b).
Now, the equation becomes:
(a^2 + 2abi - b^2) - 2a + 2bi +ai - b = 2i.
Step 3: Group the real and imaginary terms.
Now, let's group the real and imaginary terms:
(a^2 - b^2 - 2a - b) + (2ab + 2b + a)i = 2i.
The real part of the equation is: (a^2 - b^2 - 2a - b).
The imaginary part of the equation is: (2ab + 2b + a).
Step 4: Equate the real and imaginary parts.
Since the equation is true for all values of a and b, the real and imaginary parts of the equation should be separately equal to zero.
Setting the real part equal to zero:
(a^2 - b^2 - 2a - b) = 0.
Setting the imaginary part equal to zero:
(2ab + 2b + a) = 0.
Step 5: Solve the system of equations.
Now, you have a system of equations with two variables (a and b). You can solve this system of equations to find the values of a and b.
Once you find the values of a and b, you can substitute them back into the expression for z (z = a + bi) to find the complex solutions to the equation.
I hope this explanation helps you understand how to approach and solve the problem! Remember to carefully simplify each term, group the real and imaginary parts, and solve the resulting system of equations.