<p>If <strong>x</strong><sup>2</sup> + 3<strong>x</strong> + 5 = 0 and <strong>a</strong><strong>x</strong><sup>2</sup> + <strong>b</strong><strong>x</strong> + <strong>c</strong> = 0 have a common root and <strong>a</strong>, <strong>b</strong>, <strong>c</strong> ∈ ℕ, the minimum value of <strong>a</strong> + <strong>b</strong> + <strong>c</strong> is</p>
Step-by-Step Solution
Key Concept: When two quadratic equations share a non-real root, they must share both complex conjugate roots, meaning one is a scalar multiple of the other.
<p><strong>Step 1:</strong> The roots of x<sup>2</sup> + 3x + 5 = 0 are non-real (discriminant = 9 − 20 = −11 < 0)</p><p><strong>Step 2:</strong> Since the roots are non-real complex conjugates, if they share one root with ax<sup>2</sup> + bx + c = 0, they must share both roots</p><p><strong>Step 3:</strong> Therefore, ax<sup>2</sup> + bx + c must be a multiple of x<sup>2</sup> + 3x + 5</p><p><strong>Step 4:</strong> So a/1 = b/3 = c/5 = k for some constant k</p><p><strong>Step 5:</strong> Thus a = k, b = 3k, c = 5k where k ∈ ℕ</p><p><strong>Step 6:</strong> Minimum occurs when k = 1: a = 1, b = 3, c = 5</p><p><strong>Step 7:</strong> Therefore a + b + c = 1 + 3 + 5 = 9</p><p>∴ Answer is (b)</p>
Correct Answer: b