For any real value of $x$, if $\frac{11 x^2+12 x+6}{x^2+4 x+2}
otin(a, b]$ then the value x for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

Question

For any real value of $x$, if $\frac{11 x^2+12 x+6}{x^2+4 x+2} 
otin(a, b]$ then the value x for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

Aicharya · Accepted Answer

$\frac{11 x^2+12 x+6}{x^2+4 x+2}=y$ $\begin{aligned} & \Rightarrow(11-y) x^2+(12-4 y) x+6-2 y=0 \ & 	ext { For real value of } x, D \geq 0 \ & \Rightarrow(12-4 y)^2-4(11-y)(6-2 y) \geq 0 \ & \Rightarrow y^2+2 y-15 \geq 0 \ & \Rightarrow(y+5)(y-3) \geq 0 \ & \Rightarrow y \leq-5 	ext { or } y \geq 3 \ & \frac{11 x^2+12 x+6}{x^2+4 x+2} 
otin(-5,3] \Rightarrow b=3, a=-5 \ & \frac{11 x^2+12 x+6}{x^2+4 x+2}=11 \ & \Rightarrow 12 x+6=44 x+22 \Rightarrow-16=32 x \ & \Rightarrow x=\frac{-1}{2}\end{aligned}$

For any real value of $x$, if $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b]$ then the value x for which…

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