$O \le {x^2} - x + 1 \le 1$ $\Rightarrow {x^2} - x \le 0$ $\Rightarrow x \in [0,1]$ Also, $0 < {\sin ^{ - 1}}\left( {{{2x - 1} \over 2}} \right) \le {\pi \over 2}$ $\Rightarrow 0 < {{2x - 1} \over 2} \le 1$ $\Rightarrow 0 < 2x - 1 \le 2$ $1 < 2x \le 3$ ${1 \over 2} < x \le {3 \over 2}$ Taking intersection $x \in \left( {{1 \over 2},1} \right]$ $\Rightarrow \alpha = {1 \over 2},\beta = 1$ $\Rightarrow \alpha + \beta = {3 \over 2}$