Complex Analysis - Solutions of the Equation

I need to find all the solutions of the equation

$\displaystyle z^2-4iz-4-2i=0$.

I know that I can use the quadratic equation and then use a partial result of that to solve for the roots. However, when I use the quadratic equation, I come up with an answer that just doesn't make sense to me, i.e. $\displaystyle 4i +- \sqrt{2i}$. I am not sure where to go from here. I have gone over my calculations several times and this is just what I get... I will appreciate your help. (Sorry, I can't remember how to do the plus/minus sign). Thanks.

Re: Complex Analysis - Solutions of the Equation

Quote:

Originally Posted by

**tarheelborn** I need to find all the solutions of the equation

$\displaystyle z^2-4iz-4-2i=0$.

I know that I can use the quadratic equation and then use a partial result of that to solve for the roots. However, when I use the quadratic equation, I come up with an answer that just doesn't make sense to me, i.e. $\displaystyle 4i +- \sqrt{2i}$. I am not sure where to go from here. I have gone over my calculations several times and this is just what I get... I will appreciate your help. (Sorry, I can't remember how to do the plus/minus sign). Thanks.

I think it will be $\displaystyle 2i \pm \sqrt{2i}$. You can use DeMoivre's Theorem to get the two square roots of i in polar form and then convert to cartesian form.

Re: Complex Analysis - Solutions of the Equation

Quote:

Originally Posted by

**tarheelborn** I need to find all the solutions of the equation

$\displaystyle z^2-4iz-4-2i=0$.

I know that I can use the quadratic equation and then use a partial result of that to solve for the roots. However, when I use the quadratic equation, I come up with an answer that just doesn't make sense to me, i.e. $\displaystyle 4i +- \sqrt{2i}$. I am not sure where to go from here. I have gone over my calculations several times and this is just what I get... I will appreciate your help. (Sorry, I can't remember how to do the plus/minus sign). Thanks.

Thanks so much, but I really just need to know how to do the quadratic equation part. I think I can do the de Moivre's part if I can get that far. Silly, but that's the way it is. Sorry I posted in the wrong forum; I am really tired.

Re: Complex Analysis - Solutions of the Equation

Quote:

Originally Posted by

**tarheelborn** Thanks so much, but I really just need to know how to do the quadratic equation part. I think I can do the de Moivre's part if I can get that far. Silly, but that's the way it is. Sorry I posted in the wrong forum; I am really tired.

I'd assumed that you could do the 'quadratic equation part' and had just made a careless error. Please post all your working and say where you get stuck.

Re: Complex Analysis - Solutions of the Equation

Quote:

Originally Posted by

**tarheelborn** I need to find all the solutions of the equation

$\displaystyle z^2-4iz-4-2i=0$.

I know that I can use the quadratic equation and then use a partial result of that to solve for the roots. However, when I use the quadratic equation, I come up with an answer that just doesn't make sense to me, i.e. $\displaystyle 4i +- \sqrt{2i}$. I am not sure where to go from here. I have gone over my calculations several times and this is just what I get... I will appreciate your help. (Sorry, I can't remember how to do the plus/minus sign). Thanks.

Either complete the square or use the Quadratic Formula.

Re: Complex Analysis - Solutions of the Equation

Actually, you did NOT make an error, you simply have not finished the problem. You want to write the solutions in the "standard" form "a+ bi".

Use the quadratic **formula** (not "quadratic equation") $\displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$ with a= 1, b= 4i, and c= -4- 2i.

$\displaystyle \frac{-4i\pm\sqrt{(-4i)^2- (4)(1)(-4- 2i)}}{2(1)}= \frac{-4i\pm\sqrt{-16+ 16+ 8i}}{2}= \frac{-4i\pm\sqrt{8i}}{2}$

$\displaystyle = -2i+ \sqrt{2i}$ (which is what you got) where I have dropped the "$\displaystyle \pm$" since I will be using the two roots of 2i.

"2i" is at (0, 2) in the complex plane, 90 degrees from the real axis, so its square roots are at 45 and -45 degrees, still distance 2 from the origin. That is, they are $\displaystyle \sqrt{2}(1+ i)$ and $\displaystyle \sqrt{2}(1- i)$. The solutions to the equation are

$\displaystyle -2i+ \sqrt{2}(1+ i)= \sqrt{2}+ (\sqrt{2}- 2)i$ and $\displaystyle -2i+ \sqrt{2}(1- i)= \sqrt{2}- (\sqrt{2}+ 2)i$.

Re: Complex Analysis - Solutions of the Equation

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Originally Posted by

**HallsofIvy** Actually, you did NOT make an error, you simply have not finished the problem. You want to write the solutions in the "standard" form "a+ bi".

Use the quadratic **formula** (not "quadratic equation") $\displaystyle \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$ with a= 1, b= 4i, and c= -4- 2i.

$\displaystyle \frac{-4i\pm\sqrt{(-4i)^2- (4)(1)(-4- 2i)}}{2(1)}= \frac{-4i\pm\sqrt{-16+ 16+ 8i}}{2}= \frac{-4i\pm\sqrt{8i}}{2}$

$\displaystyle = -2i+ \sqrt{2i}$ (which is what you got) where I have dropped the "$\displaystyle \pm$" since I will be using the two roots of 2i.

"2i" is at (0, 2) in the complex plane, 90 degrees from the real axis, so its square roots are at 45 and -45 degrees, still distance 2 from the origin. That is, they are $\displaystyle \sqrt{2}(1+ i)$ and $\displaystyle \sqrt{2}(1- i)$. The solutions to the equation are

$\displaystyle -2i+ \sqrt{2}(1+ i)= \sqrt{2}+ (\sqrt{2}- 2)i$ and $\displaystyle -2i+ \sqrt{2}(1- i)= \sqrt{2}- (\sqrt{2}+ 2)i$.

The OP had 4i. (You have -2i !?). If the OP had read my post properly, s/he would have seen that it's 2i. As I said over a day ago, a careless error made by the OP.

However, the OP has marked the thread as solved and Thanked HoI and generally seems perfectly happy to swap one careless error for another, so who am I to rock the boat ....

Re: Complex Analysis - Solutions of the Equation

Re: Complex Analysis - Solutions of the Equation

Quote:

Originally Posted by

**mr fantastic** The OP had 4i. (You have -2i !?). If the OP had read my post properly, s/he would have seen that it's 2i. As I said over a day ago, a careless error made by the OP.

However, the OP has marked the thread as solved and Thanked HoI and generally seems perfectly happy to swap one careless error for another, so who am I to rock the boat ....

What is an OP?

Re: Complex Analysis - Solutions of the Equation

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Originally Posted by

**tarheelborn** What is an OP?

Original Poster or Original Post .

Re: Complex Analysis - Solutions of the Equation

Thank you. I feared it was something negative.