Key Concepts and Formulas

• Discriminant of a Quadratic: For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is given by $D = b^2 - 4ac$. The quadratic has real roots if $D \ge 0$ and distinct real roots if $D > 0$.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$, the sum of the roots is $r_1 + r_2 = -\frac{b}{a}$ and the product of the roots is $r_1r_2 = \frac{c}{a}$.
• Sum of Squares Identity: $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1r_2$
• Vertex of a Parabola: The vertex of a parabola $f(x) = Ax^2 + Bx + C$ is at $x = -\frac{B}{2A}$.

Step-by-Step Solution

Step 1: Determine the condition for distinct real roots. For the given quadratic equation $3x^2 + (\alpha - 6)x + (\alpha + 3) = 0$ to have distinct real roots, its discriminant $D$ must be greater than 0. $D = (\alpha - 6)^2 - 4(3)(\alpha + 3) > 0$ $D = \alpha^2 - 12\alpha + 36 - 12\alpha - 36 > 0$ $D = \alpha^2 - 24\alpha > 0$ $\alpha(\alpha - 24) > 0$ This inequality holds if both factors are positive or both are negative. Case 1: $\alpha > 0$ and $\alpha - 24 > 0 \implies \alpha > 24$ Case 2: $\alpha < 0$ and $\alpha - 24 < 0 \implies \alpha < 0$ Therefore, the condition for distinct real roots is $\alpha \in (-\infty, 0) \cup (24, \infty)$.

Step 2: Express the sum of squares of the roots in terms of $\alpha$. Let the roots be $r_1$ and $r_2$. Using Vieta's formulas, we have: $r_1 + r_2 = -\frac{\alpha - 6}{3} = \frac{6 - \alpha}{3}$ $r_1r_2 = \frac{\alpha + 3}{3}$ The sum of squares of the roots is: $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1r_2$ $r_1^2 + r_2^2 = \left(\frac{6 - \alpha}{3}\right)^2 - 2\left(\frac{\alpha + 3}{3}\right)$ $r_1^2 + r_2^2 = \frac{36 - 12\alpha + \alpha^2}{9} - \frac{2\alpha + 6}{3}$ $r_1^2 + r_2^2 = \frac{\alpha^2 - 12\alpha + 36}{9} - \frac{6\alpha + 18}{9}$ $r_1^2 + r_2^2 = \frac{\alpha^2 - 12\alpha + 36 - 6\alpha - 18}{9}$ $r_1^2 + r_2^2 = \frac{\alpha^2 - 18\alpha + 18}{9}$ Let $g(\alpha) = \frac{\alpha^2 - 18\alpha + 18}{9}$.

Step 3: Find the value of $\alpha$ that minimizes the sum of squares. To minimize $g(\alpha)$, we find the vertex of the parabola $\alpha^2 - 18\alpha + 18$. The $\alpha$-coordinate of the vertex is: $\alpha_{\text{vertex}} = -\frac{-18}{2(1)} = 9$ Thus, the global minimum of $g(\alpha)$ occurs at $\alpha = 9$.

Step 4: Check if the minimizing value of $\alpha$ satisfies the condition for distinct real roots. The condition for distinct real roots is $\alpha \in (-\infty, 0) \cup (24, \infty)$. Since $9 \notin (-\infty, 0) \cup (24, \infty)$, the minimum value of $g(\alpha)$ is not attained within the required domain.

Step 5: Analyze the behavior of $g(\alpha)$ within the valid domain. Since the vertex is at $\alpha = 9$, the function is decreasing for $\alpha < 9$ and increasing for $\alpha > 9$. Case 1: $\alpha \in (-\infty, 0)$. As $\alpha$ approaches 0 from the left, $g(\alpha)$ approaches $\frac{0^2 - 18(0) + 18}{9} = 2$. However, $\alpha$ cannot be 0. Case 2: $\alpha \in (24, \infty)$. As $\alpha$ approaches 24 from the right, $g(\alpha)$ approaches $\frac{24^2 - 18(24) + 18}{9} = \frac{576 - 432 + 18}{9} = \frac{162}{9} = 18$. However, $\alpha$ cannot be 24. Since the minimum value is never attained for any $\alpha$ in the valid domain, there's no integral value of $\alpha$ that minimizes the sum of the squares of the roots.

Step 6: Determine the set S. Since there is no integral value of $\alpha$ for which the sum of squares of the roots is minimized, the set $S$ is empty.

Common Mistakes & Tips

• Be careful with the inequality sign when solving for the domain of $\alpha$. Using $D \ge 0$ instead of $D > 0$ would lead to a different result.
• Always check if the value that minimizes the sum of squares lies within the valid domain for $\alpha$. If it doesn't, the minimum is not attained at that point.
• Understand the difference between infimum and minimum. The infimum is the greatest lower bound, while the minimum is the actual value attained by the function.

Summary The sum of squares of the roots is expressed as a quadratic function of $\alpha$. The minimum of this quadratic function occurs at $\alpha = 9$, but this value does not satisfy the condition for the quadratic to have distinct real roots. Since no value of $\alpha$ exists that minimizes the sum of squares while satisfying the distinct real roots condition, the set $S$ is an empty set.

Final Answer The final answer is \boxed{A}, which corresponds to option (A) is an empty set.