<p>Consider the equation \(x^2 + 2x - n = 0\), where \(n \in \mathbb{N}\) and \(n \in [5, 100]\). The total number of different values of \(n\) so that the given equation has integral roots is</p>
Step-by-Step Solution
Key Concept: For a quadratic with integral roots, the discriminant must be a perfect square. Here, Δ = 4 + 4n = 4(1 + n), so we need 1 + n to be a perfect square.
<p><strong>Step 1:</strong> For integral roots, discriminant Δ must be a perfect square.</p><p>Δ = 4 + 4n = 4(1 + n)</p><p><strong>Step 2:</strong> For Δ to be a perfect square, 4(1 + n) = k² for some integer k.</p><p>This means 1 + n must be a perfect square. Let 1 + n = m² where m ∈ ℕ.</p><p><strong>Step 3:</strong> Since n ∈ [5, 100], we have 1 + n ∈ [6, 101].</p><p>We need perfect squares in the range [6, 101]:</p><p>m² ∈ {9, 16, 25, 36, 49, 64, 81, 100}</p><p><strong>Step 4:</strong> Corresponding values of n:</p><p>n ∈ {8, 15, 24, 35, 48, 63, 80, 99}</p><p>Counting these values: 8 different values of n.</p><p>∴ Answer: A</p>
Correct Answer: A