<p>69. If the equation \(|x^2 + bx + c| = k\) has four real roots, then</p>
<p>(1) \(b^2 - 4c > 0\) and \(0 < k < \dfrac{4c - b^2}{4}\)</p>
<p>(2) \(b^2 - 4c < 0\) and \(0 < k < \dfrac{4c - b^2}{4}\)</p>
<p>(3) \(b^2 - 4c > 0\) and \(k > \dfrac{4c - b^2}{4}\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: The equation |x² + bx + c| = k has four real roots when the parabola y = x² + bx + c crosses the horizontal lines y = k and y = -k, requiring the parabola's vertex to be below the x-axis and k to equal the absolute value of the minimum (which is negative).
<p><strong>Step 1:</strong> Let f(x) = x² + bx + c. The equation |f(x)| = k becomes two cases: f(x) = k and f(x) = -k.</p><p><strong>Step 2:</strong> For exactly 4 real roots, the parabola must intersect both lines y = k and y = -k twice each. This requires the parabola to have its vertex below the x-axis.</p><p><strong>Step 3:</strong> The vertex value is: f_min = c - b²/4. For 4 real roots, we need f_min < 0, meaning c - b²/4 < 0, or <strong>b² - 4c > 0</strong>.</p><p><strong>Step 4:</strong> Additionally, k must satisfy: <strong>0 < k < |f_min|</strong>, where |f_min| = (b² - 4c)/4.</p><p><strong>Step 5:</strong> The critical condition is that the discriminant of the original quadratic is positive AND the vertex lies below the x-axis AND 0 < k < (b² - 4c)/4.</p><p>∴ Answer: b² > 4c and the minimum value of x² + bx + c is negative</p>
Correct Answer: 2