Sum of two square and complex quadratic

[r.a.c.]

I actually have two questions, both are in the complex plane.

1. The problem statement, all variables and given/known data

Q1: Express 962 as a sum of two squares (Hint: 962 = (13)(74)
Q2. Given z,a belong to C (complex). Find a such that the roots of the equation z^2 + az + 1 = 0 have equal absolute values (or modulus).



2. Relevant equations
Well this is about the complex plane so for the first one I think most probably its Euler's Identity. As for the second it could be anything.


3. The attempt at a solution

Q1: It is a sum of squares so the best I can think of is that 962 is a modulus of a vector i.e. z belongs to C, |z| = 962. Because z = a + ib so 962^2 = a^2 + b^2 . Then I try writing z in polar so we have z = |z|e^i^x , where x is the argument of z. But x = arctan b/a . So I somewhat got stuck in a loop.

Q2: We can't use the auxiliary equation because a is also complex. So it doesn't work. I tried expanding both z and a only to get stuck eventually in an overly complicated equation. Any suggestions on just how to start it?

[HallsofIvy]

I actually have two questions, both are in the complex plane.

1. The problem statement, all variables and given/known data

Q1: Express 962 as a sum of two squares (Hint: 962 = (13)(74)
Q2. Given z,a belong to C (complex). Find a such that the roots of the equation z^2 + az + 1 = 0 have equal absolute values (or modulus).



2. Relevant equations
Well this is about the complex plane so for the first one I think most probably its Euler's Identity. As for the second it could be anything.


3. The attempt at a solution

Q1: It is a sum of squares so the best I can think of is that 962 is a modulus of a vector i.e. z belongs to C, |z| = 962. Because z = a + ib so 962^2 = a^2 + b^2 . Then I try writing z in polar so we have z = |z|e^i^x , where x is the argument of z. But x = arctan b/a . So I somewhat got stuck in a loop.
The problem asks you to find a and b such that a^2+ b^2= 962, NOT 962^2! Although it doesn't have anything to do with this hint, one way to do it is by "brute strength": Is 962- 1 a square? 962- 4? 962- 9?...

Q2: We can't use the auxiliary equation because a is also complex. I have no idea what you mean by this. What "auxiliary equation? And the real numbers are a subset of the complex numbers- saying that "a is a complex number" doesn't mean it can't be real. There is an obvious real number solution for a.

So it doesn't work. I tried expanding both z and a only to get stuck eventually in an overly complicated equation. Any suggestions on just how to start it?

[r.a.c.]

The auxiliary equation is x = \frac{-b \frac{+}{-} \sqrt{4ac - b^2}}{2a} where ax^2+bx+c=0. That's what I know it as anyways. Anyways, even the teacher said you can't do it that way. But if you can find a way, then I'd be more than happy.

You're right, I'm not saying a can't be real, but we also cannot make the assumption that it is purely real or imaginary. Why is there an obvious real number solution?

As for the sum of squares: I tried by brute strength, which did work, but its time consuming and has nothing to do with the hint or what we're studying. It turned out to be 962 = 11^2+29^2.