$$\begin{gathered} \begin{array}{l}Thevalueof\text{'}a\text{'}forwhichthesumofthesquaresoftherootsoftheequation \\ {x}^2-x(a-2)-(a-1)=0assumestheleastvalueis\\ \end{array} \end{gathered}$$

$\begin{array}{l}GivenEquation{x}^2-(a-2)x-(a-1)=0\\ Let\alpha and\beta areriitsthen\\ \alpha +\beta =a-2,\alpha \beta =1-a\\ {\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta \\ ={(a-2)}^2-2(1-a)\\ =a^2+4-4a-2+2a\\ =a^2-2a+2\\ Letf\left(a\right)=a^2-2a+2\\ f^{\text{'}}\left(a\right)=2a-2\\ for\max imumor\min imum{f}^{\text{'}}\left(a\right)=0\\ \Rightarrow 2a-2=0\Rightarrow a=1\end{array}$