Question

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

• A. $b^2-4c>0$ and $0<k<\frac{4c-b^2}{4}$
• B. $b^2-4c<0$ and $0<k<\frac{4c-b^2}{4}$
• C. $b^2-4c>0$ and $k>\frac{4c-b^2}{4}$
• D. none of these

Answer 1

Answer: (D)

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 $x=-\frac{b}{a}$.

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