Quadratic Inequalities

Steps to solve $ax^2+bx+c<0$ or $>0$: (1) solve the equation $ax^2+bx+c=0$; (2) if there are no real solutions, the inequality holds for every $x\in R$ (or for no $x$, according to the sign of $a$) since the expression never changes sign; (3) if there are real solutions ('critical points'), mark them on the number line; (4) they divide the line into disjoint intervals; (5) pick one representative point from each interval; (6) substitute it into the original expression; (7) whichever intervals give the correct sign are the solution -- the expression cannot change sign inside an interval without crossing zero, which only happens at a critical point.

Worked pattern. For $3x^2+5x-2\le0$: factor $3(x+2)\left(x-\frac13\right)\le0$, critical points $-2,\frac13$; testing each of the three intervals shows the expression is $\le0$ exactly on $\left[-2,\frac13\right]$. …