Consider the equation $x^2 + 2x - n = 0$, where $n \in \mathbb{N}$ and $n \in [5, 100]$. The number of different values of $n$ so that the given equation has integral roots, is
Step-by-Step Solution
Key Concept: The discriminant condition determines the range of parameter values for which real roots exist
Consider $z^2 - 47z + k = 0$. For real roots, the discriminant must be non-negative: $47^2 - 4k \geq 0$, which gives $k \leq 552$. Therefore, $k$ can take values $1, 2, 3, \ldots, 552$. The product of all real roots across all valid values of $k$ is $1 \times 2 \times 3 \times 4 \times \cdots \times 552 = 552!$.
Correct Answer: 552!