Question

I have to Solve x^2 - (4-i)x + (5-5i) = 0 and
Prove/Verify/Derive de Moivre's Theorem

Answer 1

quadratic formula for the first

Answer 2

for demoive's, you can prove by induction.

Answer 3

ok i will try right now thx for pointing me in the right direction

Answer 4

you're welcome

Answer 5

Bump having trouble with quadratic formula..

I am stuck at (4-i) +- sqrt(-4+12i+i^2)

Answer 6

* (4-i) +- sqrt(-4+12i+i^2)/(2)

Answer 7

any ideas?

Answer 8

i got to (4-i) +- 2i+i sqrt(12i)/(2) but I feel i made a mistake somewhere..

Answer 9

sqrt(-4+12i+i^2)= sqrt(-5 + 12 i)
if we write this in polar form, we have
sqrt ( 13 * exp( i * A) ) = (13 * (exp(i*A)) ^ (1/2)
where A = arctan(12/-5)

or
\[ \sqrt{13} e^{ i \frac{A}{2}}= \sqrt{13}(\cos A/2 + i \sin A/2) \]

cos A= -5/12, sin A= 12/13

cos A/2 = sqrt( (1/2) (1 + cos A))
sin A/2 = sqrt ( (1/2) (1 - cos A))

it's a bit complicated, but eventually you can get to an answer

Answer 10

Do I have to write it in polar form? I can't pick A to be the coefficent of x^2 and so on for B and C. So A = 1, B= -(4-i), and C=(5-5i) Then plug into quadratic formula: x=-b +- sqrt(b^2-4ac)/(2)

Answer 11

First, my A is just the angle, not related to your A,B,C coefficients

I am just trying to find the square root of -5 + 12i
Normally I use polar coordinates. You could use this approach
http://www.1728.org/compnum2.htm (scroll down to square root)

Answer 12

im lost I thought you just plugged in the coefficients into the formula. My teacher didnt talk about anything else like that

Answer 13

@phi it's too complicated to me, too. XD

Answer 14

lol @Loser66

Answer 15

As I said, the easiest way to take the square root of a complex number is write it in polar form, and then raise to the 1/2 power.

Answer 16

sorry if this is a stupid question but how is sqrt(-4+12i+i^2) = sqrt(-5 + 12 i)

Answer 17

i= sqrt(-1), and i*i = sqrt(-1)*sqrt(-1) = -1

Answer 18

@phi, so why don't we convert \(\sqrt{-5+12i}\) to \((2+3i)^2\) to take off the squaroot?

Answer 19

that is good. How did you do that ?

Answer 20

to be honest, I didn't know and I got stuck like what the asker did until I put it into worfram. To me, it's not good at aaaaall. Because I didn't know how to do. :)

Answer 21

loser i put it into wolfram also but...idk what they did. I thought they just plugged in

Answer 22

@Andysebb if I go backward, its answer makes sense to me. the problem is I don't know how to go forward hehehehe, shame on me

Answer 23

Lol how did u go backwards o.o

Answer 24

@Andysebb expand (2+3i)^2 , it leads to \(\sqrt{-5+12i}\)