Complex Variables Q&A: Solving Quadratics and Finding Roots

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In summary, Complex variables can be confusing because there is no distinction between positive and negative numbers in the complex numbers, and √(-1+4i) doesn't specify which one you pick when solving for z. Polar form and angles can be used to solve for z, but it is possible that w1 and w2 are related in a different way than what is shown in the polar form.
  • #1
kingwinner
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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!
 
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  • #2
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 [tex] \pm[/tex] 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
 
  • #3
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!
 
  • #4
kingwinner said:
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
 
  • #5
I think w1 = -w2, but is this always true? Why or why not?
 
  • #6
w1 and w2 can be written also in polar form:

as w=|w1|=|w2|,

and -1=e

w1=w eîφ and w2=-w1=w ei(φ+Π).

Use w1 and w2 in the quadratic form instead of ±.
ehild
 
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FAQ: Complex Variables Q&A: Solving Quadratics and Finding Roots

1. How can complex variables be used to solve quadratics?

Complex variables can be used to solve quadratics by representing the quadratic equation in terms of complex numbers. This allows for complex solutions to be found, which can be graphed on the complex plane.

2. What is the difference between a complex root and a real root?

A complex root is a solution to a quadratic equation that involves the use of complex numbers, while a real root is a solution that only involves real numbers. Complex roots are typically represented as a pair of complex numbers, while real roots are represented as a single real number.

3. How can the quadratic formula be modified to solve for complex roots?

The quadratic formula can be modified to solve for complex roots by using the square root of a negative number, which results in the use of imaginary numbers. This modified formula is known as the complex quadratic formula.

4. Can complex variables be used to solve any type of quadratic equation?

Yes, complex variables can be used to solve any type of quadratic equation, including those with real or complex coefficients. The solutions may involve complex numbers, but they can still be found using the same principles and formulas.

5. How can complex roots be graphed on the complex plane?

Complex roots can be graphed on the complex plane by plotting the real and imaginary parts of the solution as coordinates. The real part is represented on the horizontal axis, while the imaginary part is represented on the vertical axis. This allows for a visual representation of the complex solution.

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