If the equation $|x^2+bx+c|=k$ has four real roots, then
- $b^2-4c > 0$ and $0 < k < \dfrac {4c-b^2}{4}$
- $b^2-4c < 0$ and $0 < k < \dfrac {4c-b^2}{4}$
- $b^2-4c > 0$ and $k > \dfrac {4c-b^2}{4}$
-
none of these
The equation |x^2+bx+c|=k represents the intersection of the parabola y=x^2+bx+c and the horizontal lines y=k and y=-k. For four real roots, the vertex of the parabola must be below the x-axis (b^2-4c > 0) and the value of k must be between 0 and the absolute value of the y-coordinate of the vertex, which is (4c-b^2)/4.
The absolute value equation |x^2 + bx + c| = k breaks down into two separate quadratic equations: x^2 + bx + c = k and x^2 + bx + c = -k. For the original equation to have exactly four distinct real roots, both of these resulting quadratic equations must have two distinct real roots, requiring their individual discriminants to be strictly greater than zero. The first discriminant is b^2 - 4(c - k) > 0, which simplifies to k > c - b^2/4. The second discriminant is b^2 - 4(c + k) > 0, which simplifies to k < -c + b^2/4, or k < (b^2 - 4c)/4. Because k must be an absolute value, k > 0 is also required. The correct set of conditions is b^2 - 4c > 0 along with (b^2 - 4c)/4 > k > (4c - b^2)/4, a combination that is fundamentally flawed in the provided text choices because the maximum possible value of k must be strictly less than the upper bound provided in options A and B, and none of the options correctly state all constraints without contradiction. The result is none of these.