$\text{ The greatest value of a non-negative real number }\lambda \text{ for which both the equations }2x^2+(\lambda -1)x+8=0\text{ and }{x}^2-8x+\lambda +4=0$

$\text{ have real roots is }$

$\text{ For real roots, }(\lambda -1{)}^2-64\ge 0\text{ and }64-4(\lambda +4)\ge 0$

$\begin{array}{l}\Rightarrow (\mathrm{\lambda }-1{)}^2\ge 64\text{ and }48-4\mathrm{\lambda }\ge 0\\ \Rightarrow \mathrm{\lambda }-1\ge 8\text{ or }\mathrm{\lambda }-1\le -8\text{ and }12\ge \mathrm{\lambda }\\ \Rightarrow \mathrm{\lambda }\ge 9\text{ or }\mathrm{\lambda }\le -7\text{ and }\mathrm{\lambda }\le 12\\ \text{ Hence greatest value of }\mathrm{\lambda }=12\text{. }\end{array}$