<p>If \(b_1 b_2 = 2(c_1 + c_2)\), then at least one of the equations \(x^2 + b_1 x + c_1 = 0\) and \(x^2 + b_2 x + c_2 = 0\) has</p>
Step-by-Step Solution
Key Concept: If both equations had no real roots, their discriminants would both be negative (b₁² < 4c₁ and b₂² < 4c₂). Adding these inequalities and using the given constraint b₁b₂ = 2(c₁ + c₂) leads to a contradiction, forcing at least one equation to have real roots.
<p><strong>Step 1:</strong> Assume, for contradiction, that both equations have no real roots.</p><p>Then: Δ₁ = b₁² - 4c₁ < 0 ⟹ b₁² < 4c₁</p><p>And: Δ₂ = b₂² - 4c₂ < 0 ⟹ b₂² < 4c₂</p><p><strong>Step 2:</strong> Add the inequalities:</p><p>b₁² + b₂² < 4(c₁ + c₂)</p><p><strong>Step 3:</strong> Apply the given constraint b₁b₂ = 2(c₁ + c₂):</p><p>b₁² + b₂² < 4 · (b₁b₂)/2 = 2b₁b₂</p><p><strong>Step 4:</strong> Rearrange:</p><p>b₁² + b₂² < 2b₁b₂</p><p>b₁² - 2b₁b₂ + b₂² < 0</p><p>(b₁ - b₂)² < 0</p><p><strong>Step 5:</strong> This is impossible since (b₁ - b₂)² ≥ 0 for all real b₁, b₂.</p><p>Therefore, at least one equation <strong>has real roots</strong>.</p><p>∴ Answer: B</p>
Correct Answer: B