Mathematics - Quadratic Equation Question with Solution | TestHub

Let $\alpha ,\beta$ be the roots of the equation. $\therefore \alpha +\beta =a-2$ and $\alpha \beta =-(a+1)$ Now, ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$ $=(a-2{)}^2+2(a+1)=(a-1{)}^2+5$ $\therefore {\alpha }^2+{\beta }^2$ will be minimum if $(a-1{)}^2=0$, i.e. $a=1$.