Quadratic Equations

Step 1: Understanding the given inequality
We need to find the values of such that the given inequality: holds for all .

Step 2: Setting up the inequality constraints
Rewriting the given fraction: Multiply both sides by (which is always positive for all real since the discriminant is negative): Expanding both inequalities:

Step 3: Solving for
Rearrange both inequalities: 1. . 2. . For these quadratic inequalities to hold for all real , the discriminants must be negative: 1. . 2. . Solving these quadratic inequalities gives: 1. . 2. . The intersection of these intervals is .

Step 4: Conclusion
Thus, the correct set of values for is:

Given the quadratic equation: Step 1: Identify the Conditions for the Roots Let the roots be and . Since both roots must exceed 3, we need: Step 2: Apply Vieta's Formulas By Vieta's relations: Step 3: Derive the Conditions For both roots to exceed 3: 1. Condition 1: Sum of roots condition From , this implies: 2. Condition 2: Product of roots condition From , Rearranging: Using the quadratic formula to solve this inequality: The two solutions are: Step 4: Identify the Correct Inequality Since the inequality holds true for: Given that from the sum condition, the valid solution is: Step 5: Final Answer