I need help with solving this complex equation

[zoe_l]

Hi, I need some help solving this complex equation. The question is in dutch, so I'll translate.
"Solve the next equation in

when it's known that it possesses a complex repeated root and two conjugated roots (b and its complex conjugate)."
Not sure if I’ve translated it well, so I'll try to say it like this: the 4 solutions to this equation are (z-c)^2 (z-b) (z-conjugate of b)

I started solving it by factorizing the equation so I could equalize for example (2C+b+conjugate of b) = 6(1+i)
The only part I was able to solve was the imaginary part of C, (ImC=3).

Thank you!

 

[topsquark]

Hi, I need some help solving this complex equation. The question is in dutch, so I'll translate.
"Solve the next equation in

when it's known that it possesses a complex repeated root and two conjugated roots (b and its complex conjugate)."
Not sure if I’ve translated it well, so I'll try to say it like this: the 4 solutions to this equation are (z-c)^2 (z-b) (z-conjugate of b)

I started solving it by factorizing the equation so I could equalize for example (2C+b+conjugate of b) = 6(1+i)
The only part I was able to solve was the imaginary part of C, (ImC=3).

Thank you!

View attachment 34207View attachment 34208

It's going to take a bit of algebra, but one way is to simply take the hint and expand. We know that the equation has the form:
[imath](z - (a + ib))^2(z - (c + id))(z - (c - id))[/imath]

Expand this out and match coefficients. It looks like a monster, but start off with the coefficient of [imath]z^3[/imath]. You can immediately write down a value for b and you can get c in terms of a. The simplest way from there is to plug it back into the original and expand again. Then match coefficients again. The rest won't be easy, but it should be doable.

-Dan

 

[blamocur]

The only part I was able to solve was the imaginary part of C, (ImC=3).

Good job!
Using that result and the equation [imath]c^2 |b|^2 = 24i - 10[/imath] you can get a quadratic equation for the real part of [imath]c[/imath].

 

[zoe_l]

It's going to take a bit of algebra, but one way is to simply take the hint and expand. We know that the equation has the form:
[imath](z - (a + ib))^2(z - (c + id))(z - (c - id))[/imath]

Expand this out and match coefficients. It looks like a monster, but start off with the coefficient of [imath]z^3[/imath]. You can immediately write down a value for b and you can get c in terms of a. The simplest way from there is to plug it back into the original and expand again. Then match coefficients again. The rest won't be easy, but it should be doable.

-Dan

Thank you Dan! But, I'm still not finding an answer.
Im(c)=3, and Re(c)=3-a,

if you plug it in in the coefficient of z^3, they cancel each other out

when plugging it in the constant part it both the real and imaginary part of b are still unknown and I can't find a way to substitute one of them
it turns into a quartic equation and I don't know how to write Re(b) in terms of Im(b)

it's the same with z^2 and z

 

[zoe_l]

Good job!
Using that result and the equation [imath]c^2 |b|^2 = 24i - 10[/imath] you can get a quadratic equation for the real part of [imath]c[/imath].

Thank you! But I think I'm overlooking something because I don't know how to get a quadratic equation for Re(c)
I don't know what to do with Im(b) and if I use c^2 ∣b∣^2=24i−10 I get a quartic equation.

 

[topsquark]

Thank you Dan! But, I'm still not finding an answer.
Im(c)=3, and Re(c)=3-a,

if you plug it in in the coefficient of z^3, they cancel each other out

when plugging it in the constant part it both the real and imaginary part of b are still unknown and I can't find a way to substitute one of them
it turns into a quartic equation and I don't know how to write Re(b) in terms of Im(b)

it's the same with z^2 and z

a, b, c, and d are real numbers!

-Dan

 

[blamocur]

Thank you! But I think I'm overlooking something because I don't know how to get a quadratic equation for Re(c)
I don't know what to do with Im(b) and if I use c^2 ∣b∣^2=24i−10 I get a quartic equation.

Since [imath]|b|^2[/imath] is real you can find the ratio of the real and the imaginary parts of [imath]c[/imath].

 

[zoe_l]

Thank you all!! I was a bit confused and didn't see the obvious. Here's my final answer: