1. The problem is to understand the applications of quadratic equations and inequalities. 2. Quadratic equations are equations of the form $ax^2 + bx + c = 0$ where $a \neq 0$. 3. Quadratic inequalities involve expressions like $ax^2 + bx + c > 0$, $< 0$, $\geq 0$, or $\leq 0$. 4. Applications include: - Projectile motion in physics, where the path of an object follows a quadratic equation. - Calculating areas and optimizing dimensions in geometry and design. - Economics for profit maximization and cost minimization problems. - Engineering for stress and strain analysis. - Solving problems involving speed, time, and distance where relationships are quadratic. 5. Quadratic inequalities help determine ranges of values for which certain conditions hold, such as safety limits or feasible regions in optimization. 6. Understanding the sign of the quadratic expression (positive or negative) is key to solving inequalities. 7. The discriminant $\Delta = b^2 - 4ac$ helps determine the nature of roots and solution intervals for inequalities. 8. In summary, quadratic equations and inequalities model many real-world scenarios involving parabolic relationships and constraints.