Question

Consider the equation $2+|x^2+4x+2|=m,mϵR$ so that the given equation has three solutions is:

• A. $\left\{4\right\}$
• B. $\left\{2\right\}$
• C. $\left\{1\right\}$
• D. $\left\{0\right\}$

Answer 1

The correct option is A $\left\{4\right\}$

Given that $2+|x^2+4x+2|=m,m\in R$
Let $f\left(x\right)=|x^2+4x+2|$ & $g\left(x\right)=m-2$
From the figure it is clear that
$f\left(x\right)$ & $g\left(x\right)$ have exactly three solution when $max\left(f\left(x\right)\right)=g\left(x\right)$
maximum of $f\left(x\right)$ is at $-\frac{b}{2a}=-2$
Therefore, $max(|x^2+4x+2|)=|4-8+2|=2=m-2$
$\Rightarrow m=4$