Case Delta negative — parabola always positive

One of the cases that confuse students: “but if the parabola never touches the $x$-axis, what is the solution?”. With $\Delta<0$ the parabola has no zeros and lies entirely on one side, according to the sign of $a$.

Example — Case $\Delta < 0$, parabola always positive

Solve $x^2+2x+5 > 0$.

$\Delta = 4-20 = -16 < 0$: the parabola does not intersect the $x$-axis. Since $a=1>0$, it lies always above.

The parabola lies entirely above the $x$-axis: it is positive for every value of $x$.

Solution: $\boxed{\forall x\in\mathbb{R}}$ (the set of all real numbers). If the request had been $x^2+2x+5 \le 0$, the answer would have been $\boxed{\emptyset}$ (no solution).

Links

Topics: Inequalities
Concepts: Discriminant · Second-degree inequality · Parabola
Skills: Reasoning by cases · Solving inequalities