complex numbers question! (1 Viewer)

[felixcthecat]

hii.. well umm i was reviewing my complex numbers and i've found that i've completely forgotten the method to solve the following type of question.. can someone do that example so i'd kno? thanx!

z^2 - (1-i)z + 7i-4 = 0

i looked in my notes but the method seems rather stupid cuz it went into z^4.. none-the-less, can someone do it plz? thanx!

 

[Riviet]

Three words: "Use Quadratic Formula"

To find square roots of complex numbers e.g sqrt(1+i), let (a+ib)²=1+i

expand LHS, equate real and imaginary parts from both sides to form two equations in a and b. Simultaneously solve them and substitute the values of a and b into a+ib to get your 2 complex roots.

 

[felixcthecat]

thanx buchanan!!.. do u hav worked solutions for the whole coroneos book or sumthing? =D

haha riviet.. i got to the step where u told me to equate sqrt of the complex number and stuff... but then since i was looking at my notes, i didn't think and tried to find the x and y of (x + iy) haha.. my teacher makes things so difficult when they're so easy...

thanks!~

edit: buchanan.. how'd u rememeber it was coroneos 1C question 9 part 3?!? ...

 

[Rax]

Ok I don't think I have done any complex number questions like that, so can someone please explain the formula bit with the "square root of w" section.
I get the quadratic eqtn bit obviously lol
Thanks

 

[pLuvia]

Ok well to find the square root of 16-30i you let z² = 16-30i (z=x+iy)

z²=16-30i
(x+iy)²=16-30i
Equate Re and Im parts

x²-y²=16 -------------- 1
2xy=-30
y=-15/x ---------------2

Using simulataneous equations solve for x and y

you should end up having
x=5
y=-3

Which you sub into the general equation of a complex number

therefore the square root of 16-30i = 5-3i

Hope that helped

 

[Riviet]

Buchanan- what does the isgn bit mean in the formula?

 

[pLuvia]

It's easier if you use the formula:

Never learnt the formula Only learnt it by equating Re and Im parts and DeMoivre's theorem

 

[Riviet]

The formula is valid, just not in the syllabus. However, use at your own decision and risk.

 

[felixcthecat]

now that i look back at your first answer buchanan.. y'd u need the formula? .. i thought u could do it without the formula exactly the same way u did it?.. or am i just subconciously using that formula ^0^

 

[Rax]

ok Sweet.. I just wasn't too sure as I hadnt seen that formula before. Does anyone know its derivation (Guessing thats out of syllabus for sure) because I would like to find how they got it.
Thanks guys

 

[felixcthecat]

Ok well to find the square root of 16-30i you let z² = 16-30i (z=x+iy)

z²=16-30i
(x+iy)²=16-30i
Equate Re and Im parts

x²-y²=16 -------------- 1
2xy=-30
y=-15/x ---------------2

Using simulataneous equations solve for x and y

you should end up having
x=5
y=-3

Which you sub into the general equation of a complex number

therefore the square root of 16-30i = 5-3i

Hope that helped

i'm also wondering... if u don't use buchanan's equation.. how would u derive x=5 and y=-3 from ..
x²-y²=16
and
2xy=-30

cuz u'd come out with.. x² - 15²/x² = 16

which would come out with x⁴ - 16x² - 15 = 0

how'd u go from there?.. i would hav thought u didn't need to go there and i've missed something realli vital... did i?

 

[felixcthecat]

ohh haha.. stupid me =.=.. the legacy of being lazy all holidays ^^''

 

[pLuvia]

Silly mistakes, just watch out for them

 

[Riviet]

Once you're familiar with the equating real and imaginary parts method, you should be able to go straight to the two equations and solving them.

 

[felixcthecat]

hehe.. i feel like being surrounded by a group of math teachers ><''

riviet... by equating real and im. u get x⁴ - 16x² - 15 = 0

am i rite..? so then u hav to use the quadratic equation AGAIN.. then u'd get the x² answer, and u hav to root that before u get to the answer..

if i'm rite then i think buchanan's equation is so much faster.. but then i've also been told to remember as little equations as u can ^0^.. cuz by the end of the course i'd be too filled with equations?

edit: buchanan..u've already done ur hsc hav u? cuz i'd be soo freaked out if u havn't... =.=

 

[Rax]

LOL Buchanan, nice quote
Yeah when I guess I should have looked into it a bit more then just asking about derivition. Just explaining some stuff I done lately in Ext 2 has been a bit long in its finding. thanks anyhow

 

[Riviet]

so then u hav to use the quadratic equation AGAIN.. then u'd get the x² answer, and u hav to root that before u get to the answer...

There's no need for the quadratic formula, you can simply make a substition for x² and in most cases you should be able to factorise by inspection unless x and y aren't integers.

 

[pLuvia]

Just stick with the equating Re and Im parts, but if you do decide to use it ask your teacher first, or use it at your own risk

 

[STx]

hmm, im thinking of using that formula in my half yearly, so it is indirectly in the syllabus?

 

[Riviet]

Use at your own risk, but if you want a definite answer, ask your maths teacher and/or head of maths and see if they would approve the use of the formula.