Question

a similar question to one before...
find all complex solutions for
4x^4+16x^2+4>0

Answer 1

use the same method I outlined in your previous question.

Answer 2

that will give you the 4 complex roots of:\[4x^4+16x^2+4=0\]
are you sure that's a > sign there?

Answer 3

if it is, then first solve the quadratic (using \(y=x^2\)) to get the range of values that y must satisfy.
then work out the range that x must satisfy from those.

Answer 4

For the solution to be able to compare to zero in an order relation, the solution would have to be a real number. That means your suitable candidates for the solution are either real numbers, or strictly imaginary numbers of the form 0 +bi.

Answer 5

ookay thank you both

Answer 6

Edit of my previous input:

For the function to be able to compare to zero in an order relation, the output would have to be a real number. That means your suitable candidates for the solution (values of x) are either real numbers, or strictly imaginary numbers of the form 0 +bi.

Answer 7

After looking at this problem a little more, I have a bit more insight. As I mentioned before, the output of the function must be a real number to compare to zero. That means the imaginary part of the function output must be zero. There are three cases to consider:

i) x=a+0i
ii) x=0+bi
iii) x=a+bi

In case i), the solution is all real values of a, i.e., any real number.

In case ii), we need to solve b^4-4b^2+1>0. By my best estimate, you will get three intervals on the imaginary line.

In case iii), you need to express the function in terms of a + bi. Separate the real and imaginary parts. Set the imaginary part of the function to zero, and get a in terms of b (when I did it, I ended up with a relation of a^2 as a function of b^2). Now substitute this relationship into the inequality for the real part of the problem, and you will have a polynomial inequality in b that might have solutions. I haven't worked the whole thing out, but think the answer (if it exists) will be parts of a hyperbola on the complex plane.

I hope this gives you some ideas. Thanks for posting an interesting problem.

Answer 8

Further update. We confirmed that the second case has three intervals on the imaginary line, and that the third case has no solution. Thanks again for an interesting problem.