Quadratic Equations
Common Roots
Grade 11
Question:
<p>If the equations \(x^2 + 2x + 3 = 0\) and \(ax^2 + bx + c = 0\), \(a, b, c \in \mathbb{R}\), have a common root, then \(a : b : c\) is</p>
<p>\(3 : 2 : 1\)</p>
<p>\(1 : 3 : 2\)</p>
<p>\(3 : 1 : 2\)</p>
<p>\(1 : 2 : 3\)</p>
Step-by-Step Solution
Key Concept: If two quadratic equations share a common root α, then α satisfies both equations simultaneously. Use the condition that the common root makes both equations zero to establish a relationship between coefficients.
<p><strong>Step 1:</strong> Let α be the common root of both equations.</p><p>Then: α² + 2α + 3 = 0 ... (1)</p><p>And: aα² + bα + c = 0 ... (2)</p><p><strong>Step 2:</strong> From equation (1): α² = -2α - 3</p><p>Substitute into equation (2):</p><p>a(-2α - 3) + bα + c = 0</p><p>-2aα - 3a + bα + c = 0</p><p>(-2a + b)α + (-3a + c) = 0</p><p><strong>Step 3:</strong> Since this must hold for the common root α (where α is complex, not a rational number), both the coefficient of α and the constant term must separately equal zero:</p><p>-2a + b = 0 ⟹ b = 2a</p><p>-3a + c = 0 ⟹ c = 3a</p><p><strong>Step 4:</strong> Therefore:</p><p>a : b : c = a : 2a : 3a = 1 : 2 : 3</p><p>∴ Answer: D</p>
Correct Answer: D