Q.Solve $-x^2+3x-2\ge0$.

To solve $ax^2+bx+c<0$ or $>0$: (1) solve $ax^2+bx+c=0$ for its real roots (the critical points); (2) if there are none, the expression keeps one constant sign over all of $R$; (3)-(7) otherwise, the critical points split the number line into intervals, and testing one representative value from each interval (substituting into the original expression) tells you the sign on that whole interval -- the sign can only change AT a critical point, never inside an interval.

Sign shortcut for a factored quadratic. Once factored as $a(x-p)(x-q)$ with $p<q$ and $a>0$: the expression is negative strictly between the roots, $(p,q)$, and positive outside them, $(-\infty,p)\cup(q,\infty)$ (flip this rule if $a<0$). …