Let $\mathrm{\alpha },\mathrm{\beta }$ be real and z be a complex number. If ${\mathrm{z}}^2+\alpha z+\mathrm{\beta }=0$ has two distinct roots on the line $\mathrm{Re}\left(\mathrm{z}\right)=1$, then it is necessary that

Since $\mathrm{\alpha },\mathrm{\beta }$ are real, Real part of the different roots is 1, sum…and product of the roots are real
$\Rightarrow$The roots may be 1+ ai and $1-\mathrm{ai},\mathrm{a}\in {\mathrm{R}}^{\ast }$
$\Rightarrow \mathrm{\beta }=1+{\mathrm{a}}^2>1\Rightarrow \mathrm{\beta }\in (1,\mathrm{\infty })$