Let $ax^2+a^2x+2=0$ be a quadratic equation, $a \in R$ and $S$ be the set of values of $a$ for which roots of this equation are imaginary or equal, then answer the following questions: If $\alpha$ and $\beta$ are respectively sum and product of roots of the given equation, then $-\alpha \beta$ is

For ax^2 + a^2x + 2 = 0, roots are imaginary/equal if discriminant D <= 0. D = (a^2)^2 - 4(a)(2) = a^4 - 8a <= 0. a(a^3 - 8) <= 0. a(a-2)(a^2+2a+4) <= 0. Since a^2+2a+4 is always positive, a(a-2) <= 0, so 0 < a <= 2. Sum of roots alpha = -a^2/a = -a. Product of roots beta = 2/a. -alpha*beta = -(-a)*(2/a) = 2. 2 is a prime number.