Question

Find the condition on the complex constants $\alpha ,\beta$ if $z^2+\alpha z+\beta =0$ has a real root.

Answer 1

$z^2+\alpha z+\beta =0$

Let the root is $z_1$ and $z_2$

So, $z_1+z_2=-\alpha ,z_1{z}_2=\beta$

Let $z_1$ is a real root $⟹z_1^2+\alpha z_1+\beta =0$

Now taking conjugate on both sides

$z_1^2+\overline{\alpha }{z}_1+\overline{\beta }=\overline{0}$

From the above two equations we get

$z_1=\left(\frac{-(\beta -\overline{\beta })}{\alpha -\overline{\alpha }}\right)$