<p>For the quadratic \( x^2 + (n+1)x + (n+2) = 0 \) (where <em>n</em> is an integer), the discriminant must be a perfect square and positive. Given \( D = (n+1)^2 - 4n(n+2) = -3n^2 - 6n + 1 = 4 - 3(n+1)^2 \), find the possible values of <em>n</em>.</p>
Step-by-Step Solution
Key Concept: Rewrite the discriminant as D = 4 - 3(n+1)² and recognize that for D to be a positive perfect square, we need 3(n+1)² < 4, which severely restricts possible integer values of n. Then check which values make D a perfect square.
<p><strong>Step 1:</strong> Express discriminant as D = 4 - 3(n+1)²</p><p><strong>Step 2:</strong> For D > 0: 4 - 3(n+1)² > 0 ⟹ 3(n+1)² < 4 ⟹ (n+1)² < 4/3 ≈ 1.33</p><p><strong>Step 3:</strong> For integer n, we need (n+1)² ≤ 1, so (n+1)² ∈ {0, 1}</p><p><strong>Step 4:</strong> Check possibilities:</p><p>• If (n+1)² = 0: n = -1, then D = 4 - 0 = 4 = 2² ✓ (perfect square)</p><p>• If (n+1)² = 1: n = 0 or n = -2</p><p> - For n = 0: D = 4 - 3(1) = 1 = 1² ✓ (perfect square)</p><p> - For n = -2: D = 4 - 3(1) = 1 = 1² ✓ (perfect square)</p><p><strong>Step 5:</strong> Verify no other values work. For |(n+1)| ≥ 2: (n+1)² ≥ 4, so D ≤ 4 - 12 = -8 < 0 ✗</p><p>∴ Answer: <strong>n ∈ {-2, -1, 0}</strong></p>
Correct Answer: A