Quadratic Inequality 32D336

1. **State the problem:** Solve the inequality $$x^2 - 10x + 24 \ge 0$$. 2. **Identify the type of problem:** This is a quadratic inequality. 3. **Recall the formula:** To solve quadratic inequalities, first find the roots of the quadratic equation $$x^2 - 10x + 24 = 0$$. 4. **Find the roots:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-10$$, and $$c=24$$. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-10)^2 - 4 \times 1 \times 24 = 100 - 96 = 4$$. Calculate the roots: $$x = \frac{-(-10) \pm \sqrt{4}}{2 \times 1} = \frac{10 \pm 2}{2}$$. So, $$x_1 = \frac{10 - 2}{2} = \frac{8}{2} = 4$$ $$x_2 = \frac{10 + 2}{2} = \frac{12}{2} = 6$$. 5. **Analyze the inequality:** Since $$a=1 > 0$$, the parabola opens upwards. 6. **Determine intervals:** The quadratic expression is $$\ge 0$$ outside the roots and $$< 0$$ between the roots. 7. **Write the solution:** $$x \le 4 \quad \text{or} \quad x \ge 6$$. **Final answer:** $$(-\infty, 4] \cup [6, \infty)$$.