<p>Let \(a, b\), and \(c\) be real numbers such that \(4a + 2b + c = 0\) and \(ab > 0\). Then the equation \(ax^2 + bx + c = 0\) has</p>
Step-by-Step Solution
Key Concept: Use the constraint 4a + 2b + c = 0 to express c in terms of a and b, then analyze the discriminant and roots using the condition ab > 0 to determine the nature of roots.
<p><strong>Step 1:</strong> From the constraint 4a + 2b + c = 0, express c = −4a − 2b.</p><p><strong>Step 2:</strong> Substitute into the quadratic equation: ax² + bx + (−4a − 2b) = 0.</p><p><strong>Step 3:</strong> Calculate the discriminant:</p><p>Δ = b² − 4a(−4a − 2b) = b² + 16a² + 8ab = (b + 4a)² > 0</p><p>Since Δ = (b + 4a)² and ab > 0 (a and b have the same sign), we have b + 4a ≠ 0, so Δ > 0.</p><p><strong>Step 4:</strong> Since Δ > 0, the equation has two distinct real roots.</p><p><strong>Step 5:</strong> The roots are x = [−b ± (b + 4a)]/(2a).</p><p>• x₁ = [−b + b + 4a]/(2a) = 4a/(2a) = 2</p><p>• x₂ = [−b − b − 4a]/(2a) = (−2b − 4a)/(2a) = (−b − 2a)/a</p><p><strong>Step 6:</strong> One root is always x = 2. The other root equals (−b − 2a)/a. Since ab > 0, the second root is negative (when a > 0, b > 0) or the roots have specific real distinct nature.</p><p>∴ Answer: The equation has two distinct real roots, with one root equal to 2.</p>
Correct Answer: C