<p>If the equations \(ax^2 + bx + c = 0\) and \(x^3 + 3x^2 + 3x + 2 = 0\) have two common roots, then</p>
Step-by-Step Solution
Key Concept: If two cubic and quadratic equations share exactly two common roots, those roots must satisfy both equations. The cubic can be factored using its roots, and the quadratic's roots must be two of the cubic's three roots.
<p><strong>Step 1:</strong> Analyze the cubic equation x³ + 3x² + 3x + 2 = 0</p><p>Test x = -1: (-1)³ + 3(-1)² + 3(-1) + 2 = -1 + 3 - 3 + 2 = 1 ≠ 0</p><p>Test x = -2: (-2)³ + 3(-2)² + 3(-2) + 2 = -8 + 12 - 6 + 2 = 0 ✓</p><p><strong>Step 2:</strong> Factor the cubic: x³ + 3x² + 3x + 2 = (x + 2)(x² + x + 1) = 0</p><p>Roots of cubic: x = -2 and x = (-1 ± i√3)/2 (complex roots)</p><p><strong>Step 3:</strong> Since the quadratic ax² + bx + c = 0 has real coefficients and shares two roots with the cubic, it must share the two complex conjugate roots: x² + x + 1 = 0</p><p><strong>Step 4:</strong> Therefore, the quadratic equation is a(x² + x + 1) = 0, which gives us the relationship:</p><p>a : b : c = 1 : 1 : 1, or equivalently b = a and c = a</p><p>This means <strong>b² - 4ac = a² - 4a² = -3a² < 0</strong> (discriminant is negative, confirming complex roots)</p><p>∴ Answer: C (The relationship is typically b = a and c = a, or b² = ac - 3a²)
Correct Answer: C