Second-degree inequalities

For second-degree inequalities we use the parabola: the sign of $ax^2+bx+c$ is read directly from the graph, according to whether the parabola lies above or below the $x$-axis. Everything comes down to three pieces of information: the sign of $a$, the sign of the discriminant $\Delta$ and the position of any zeros. Combining them gives the six fundamental cases, which this section presents first in a diagram and then with worked examples, distinguishing the “standard” cases ($\Delta>0$) from the “degenerate” ones ($\Delta=0$ or $\Delta<0$).

• The sign of the parabola
• The six cases of the parabola
• Case a positive, Delta positive
• Case a negative, Delta positive
• Case Delta negative: parabola always positive
• Case Delta zero: tangent parabola
• Case a negative, Delta negative: always negative
• Compact procedure