Compact procedure for second-degree inequalities

An operational summary to avoid mistakes: four steps and a mnemonic trick to remember when the solution is “internal” and when “external” to the zeros.

In brief — Compact procedure for $ax^2+bx+c \gtrless 0$

1. Compute $\Delta$ and look at the sign of $a$.
2. Draw the little diagram: parabola up or down, with two zeros, one zero, or none.
3. Colour the part of the graph that satisfies the required inequality.
4. Read off the intervals on the $x$-axis.

Mnemonic trick: “internal if you have opposite signs between $a$ and the inequality, external if you have matching signs” (holds only in the case $\Delta>0$). Example: $a>0$ with $<0$ → opposite signs → “internal interval” between the zeros.

Links

Topics: Inequalities
Concepts: Second-degree inequality · Parabola
Skills: Solving inequalities · Using formulae