Solving Quadratic Inequalities

Strategy: factor, find roots, test intervals

Quadratic inequalities are solved by: (1) factor the quadratic, (2) find the roots (boundary points), (3) test a point in each interval, (4) shade the intervals that satisfy the inequality.

Step-by-step: Solve $x^2 - 5x + 6 < 0$

Step 1 — Factor: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Step 2 — Find the roots (zeros): $x = 2$ and $x = 3$. These divide the number line into three intervals: $(-\infty, 2)$, $(2, 3)$, $(3, \infty)$.

Step 3 — Test a point from each interval:

• Interval $(-\infty, 2)$: try $x = 0$. $(0-2)(0-3) = (-2)(-3) = +6 > 0$ ✗
• Interval $(2, 3)$: try $x = 2.5$. $(2.5-2)(2.5-3) = (0.5)(-0.5) = -0.25 < 0$ ✓
• Interval $(3, \infty)$: try $x = 4$. $(4-2)(4-3) = (2)(1) = +2 > 0$ ✗

Step 4 — Solution: Only the middle interval satisfies $< 0$.

$2 < x < 3 \quad \text{or in interval notation: } (2, 3)$

Why this works graphically

The parabola $y = x^2 - 5x + 6$ opens upward (positive $x^2$ coefficient). It crosses the x-axis at $x = 2$ and $x = 3$. The parabola is below the x-axis (negative) only between the roots — exactly the interval $(2, 3)$.

Quick rule for parabolas

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

Try it in the visualization

Adjust $b$ and $c$. The parabola shows the positive and negative regions, with the solution interval highlighted on both the graph and the number line.