Quadratic Inequations

A quadratic inequation in one variable is any statement of the form $ax^2+bx+c>0$, $\geq 0$, $<0$, or $\leq 0$, with $a,b,c$ real and $a\neq0$. Unlike a quadratic equation, which typically has just two solutions, a quadratic inequation usually has infinitely many solutions — an entire interval (or union of intervals) of real numbers — and solving it means describing that set precisely.

The sign-change rule from earlier gives a completely mechanical algebraic method: factor $ax^2+bx+c$ (or find its roots $\alpha\le\beta$ via the quadratic formula), then read off the sign pattern — same sign as $a$ outside $[\alpha,\beta]$, opposite sign as $a$ strictly between the roots (with the boundary points included or excluded depending on whether the inequality is strict). There is also a graphical method: plot $y=ax^2+bx+c$, and the solution set of, say, $f(x)<0$ is simply the set of $x$-values where the parabola dips below the $x$-axis.

When a problem asks for two (or more) inequations to hold simultaneously, solve each one separately to get two solution intervals, then intersect them — the final answer is the overlap.

Worked example. Solve $x^2 - 5x + 6 < 0$, and also find where $x^2-5x+6 \geq 0$.

Factor: $x^2-5x+6=(x-2)(x-3)$, with roots $2$ and $3$, and $a=1>0$. Between the roots the expression takes the opposite sign to $a$ (i.e. negative), and outside the roots it matches $a$'s sign (positive). So: …