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Question

If the equation $$|x^2+bx+c|=k$$ has four real roots, then


A
b24c>0 and 0<k<4cb24
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B
b24c<0 and 0<k<4cb24
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C
b24c>0 and k>4cb24
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D
none of these
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Solution

The correct option is B none of these
$$|x^2+bx+c|=k$$
let $$f\left( x \right) =|x^{ 2 }+bx+c|$$ & $$g\left( x \right) =k$$
from the figure:
Since, $$f\left( x \right)=0$$ have real roots
Therefore, $$D={ b }^{ 2 }-4c>0$$     ...(1)
maximum value of $$f\left( x \right)$$ occur at $$x=-\dfrac { b }{ a } $$.

Therefore, $$max\left( f\left( x \right)  \right) =\left| \dfrac { 4c-{ b }^{ 2 } }{ 4 }  \right| =\dfrac { { b }^{ 2 }-4c }{ 4 } $$
from the figure it is clear that, $$f\left( x \right)$$  & $$g\left( x \right)$$ will have four point of intersection when $$0<g\left( x \right) <max\left( f\left( x \right)  \right) $$
$$\Rightarrow 0<k<\dfrac { { b }^{ 2 }-4c }{ 4 } $$
Ans: D

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