Quadratic Inequalities

Solving quadratic inequalities: factor, find roots, test intervals

Step 1 — Factor: $x^2 - 4x - 5 = (x - 5)(x + 1)$.

Step 2 — Find roots: $x = 5$ and $x = -1$. These divide the number line into three intervals.

Step 3 — Test each interval:

• $x = -2$: $(-7)(-1) = +7 > 0$ ✓
• $x = 0$: $(-5)(1) = -5 < 0$ ✗
• $x = 6$: $(1)(7) = +7 > 0$ ✓

Step 4 — Solution: $x < -1$ or $x > 5$, i.e., $(-\infty, -1) \cup (5, \infty)$.

Quick rule for parabolas

For $ax^2 + bx + c > 0$ with $a > 0$: the parabola is positive outside the roots. For $< 0$: between the roots. If $a < 0$, everything flips.

Try it in the visualization

The parabola shows positive (above x-axis) and negative (below) regions. The solution is highlighted. Adjust $b$ and $c$ to change the roots and see the solution region shift. Since the parabola opens up ($a > 0$), the function is positive outside the roots: $x < -1$ or $x > 5$.