Determine the number of zeros in the first quadrant

Question

This is a homework question:

$$f(z) = z^2 - z + 1$$

sorry for the poor code!

Answer 1

The zeros can be computed by quadratic formula, yield $z=\frac{1}{2}\pm \sqrt{3}i$. Thus the number of zeros on the first quadrant is one, that is, $z=\frac{1}{2}+\sqrt{3}i$.

Answer 2

The sum of the zeroes for this quadratic polynomial is 1. Since the polynomial has real coefficients, the zeroes must be complex conjugates $ \ a \pm bi \ $ . So we have the sum $ \ 2a = 1 \ \Rightarrow \ a = \frac{1}{2} \ $ , which tells us that the zeroes lie in the first and fourth quadrants; the fact that they are conjugates places one in each of those quadrants.

(Just to eliminate the possibility that the zeroes might be on the real axis, the product of the roots, $ \ a^2 + b^2 \ $ , is also 1 , so $ \ b \neq \ 0 \ $ . )