complex number (1 Viewer)

[Hehe]

3-2i is one root of x^2+bx+c=0 where b and c are real. Find b and c. Do we use the sum and the products of the roots to do it ?
Thanx

 

[Heinz]

The other root is 3 + 2i (if the coefficients are real, the roots are conjugates of eachother). with this knowledge you can easily figure out the rest, i.e. the sum of the roots to find b and the product of the roots to find c.

x^2 - (sum of the roots)x + (product of the roots)

Edit: it had a typo

 

[Grey Council]

simply:
complex roots occur in complex conjugates. in other words, if you know that one root is a complex number, the other root is the conjugate of the root you know.

In this case 3-2i is one root. Therefore 3+2i is the other root. The use basic 2 unit polynomial theory to solve.

heh, im sure you understood Heinz's explanation, this is just a simpler explanation. heck, i didn't know that complex roots occur in conjugates, thats why i'm posting this. Maybe someone else doesn't know either.

 

[AGB]

can someone explain why complex roots occur in complex conjugates??

 

[Heinz]

I think its because of the quadractic equation which is used to find the roots. The +- component of the equation generates two roots which are equal and opposite in value. If you have the 4u arnold text, theres a brief explanation at the top of page 29.

 

[J0n]

Originally posted by AGB
can someone explain why complex roots occur in complex conjugates??

If a + ib is a root of P(z), so is a - ib

Proof
Since P(a + ib) = 0
.'. conjugate[P(a + ib)] = conjugate[0]
.'. P(conjugate[a + ib]) = 0

 

[KeypadSDM]

Originally posted by Hehe
3-2i is one root of x^2+bx+c=0 where b and c are real. Find b and c. Do we use the sum and the products of the roots to do it ?
Thanx

Don't you just plug in the root into the equation and then solve?
(3-2i)^2 + b(3-2i) + c = 0
9 - 4 - 12i + 3b -2bi + c = 0
:. -12i -2bi = 0
b = -6
9 - 4 + 3b + c = 0
c = 13

(I hope ... I haven't checked it)

 

[Grey Council]

why woiuld that give the other root? Sorry, can you please explain?

 

[Heinz]

I dont think it does give you the other root. just solves b and c which is what the question is asking for.

 

[Grey Council]

thought not.

 

[Supra]

yeh thats jus to find the equation, but mayb u can find the otha roots by solving that equation...
so by quadratic formula u solve...
x^2-6x+13=0

but i prefer sum and product of roots neway