Complex Variables Q&A: Solving Quadratics and Finding Roots

[kingwinner]

Homework Statement

I'm beginning my studies in complex variables and have some questions...

Q1) We know that x²=9 => x=+/- √9 = +/- 3.
Suppose z^2 = w where z and w are COMPLEX numbers, then is it still true to say that
z = +/- √w ? Why or why not?

Q2) "Let az² + bz + c =0, where a,b,c are COMPLEX numbers, a≠0.
Then the usual quadratic formula still holds."

My concern is with the √(b²-4ac) part. How can we find √(b²-4ac) when b²-4ac is a COMPLEX number?
For example, what does √(-1+4i) mean on its own and how can we find it? I know there is a general procedure(using polar form and angles) to solve for the nth root of a complex number (z^n=w), but I still don't understand what √(-1+4i) means on its own.
Even for real numbers, there is a difference between solving x²=9 and finding √9, right? So is there any difference between finding √(-1+4i) and solving z²=-1+4i for z using polar form and angles?

Homework Equations

Complex variables

The Attempt at a Solution

As shown above.

I hope someone can explain these. Any help is much appreciated!

 

[Office_Shredder]

z = +/- √w loses meaning because there is no distinguished choice for √w. Normally we define √x to be the positive square root of x, but there's no such thing as positive and negative in the complex numbers. However what we CAN say is that if a²=w and b²=w, then a²-b²=0, and factoring (a-b)(a+b)=0. So a=b or a=-b

√(-1+4i) is just going to mean a complex number which, when squared, gives -1+4i. It doesn't specify which one, but for the quadratic formula it doesn't matter which one you pick, because they both work. When writing $$\pm$$ in the quadratic formula for real numbers, what you really mean is that you can use either square root of the number and you will get a root. The same holds for complex numbers as well

 

[kingwinner]

Q2) z= [-b +/- sqrt(b^2 - 4ac)] / 2a

Say b^2 - 4ac turns out to be -1+4i
Let w^2 = -1+4i. Use polar form and angles to solve for w, we get 2 solutions, call them w1 and w2.

And you mean we can pick either one of w1, w2 as √(-1+4i), right?
Say if I pick w1, then the solutions will be
z=[-b +/- w1] / 2a

Alternatively, if I pick w2, then the solutions will be
z=[-b +/- w2] / 2a which will be the SAME as the above, right? Why are they the same?

Thanks for explaining!

 

[ehild]

Q2) z= [-b +/- sqrt(b^2 - 4ac)] / 2a

Say b^2 - 4ac turns out to be -1+4i
Let w^2 = -1+4i. Use polar form and angles to solve for w, we get 2 solutions, call them w1 and w2.

How are w1 and w2 related? Compare their polar forms.

ehild

 

[kingwinner]

I think w1 = -w2, but is this always true? Why or why not?

 

[ehild]

w₁ and w₂ can be written also in polar form:

as w=|w₁|=|w₂|,

and -1=e^iΠ

w₁=w e^îφ and w₂=-w₁=w e^i(φ+Π).

Use w₁ and w₂ in the quadratic form instead of ±.
ehild