(2) If $z \in \mathbf{C}$, for the quadratic equation in $x$, $x^{2}-z x+4+3 \mathrm{i}=0$ to have real roots, find the complex number $z$ that minimizes the modulus of $z$.
(Solution: $z= \pm\left(\frac{9 \sqrt{5}}{5}+\frac{3 \sqrt{5}}{5} \mathrm{i}\right)$ )