Question

If the equation ${\text{Z}}^{\text{2}}\text{-}\left(\text{3 + i}\right)\text{z + m + 2i = 0,m}\in \text{R}$ has a real root, then the value of m is

Answer 1

$z^2-(3+i)z+m+2i=0$

has real roots then

$D\ge 0\Rightarrow (3+i{)}^2-4(m+2i)\ge 0$

$\Rightarrow 9-1+6i-4m-8i\ge 0$

$\Rightarrow 8-2i-4m\ge 0$

$\Rightarrow 4-i\ge 2m$

$\Rightarrow m\le \frac{4-i}{2}$