Question

For real values of $x,$ the value of the expression $\frac{11x^2-12x-6}{x^2+4x+2}$

• A. lies between $-17$ and $-3$
• B. does not lie between $-17$ and $-3$
• C. lies between $3$ and $17$
• D. does not lie between $3$ and $17$

Answer 1

Answer: (B)

Let $y=\frac{11x^2-12x-6}{x^2+4x+2}$
$y{x}^2+4yx+2y=11x^2-12x-6$
$\left(y-11\right)x^2+\left(4y+12\right)x+\left(2y+6\right)=0$
$x\in R\Rightarrow D\ge 0$
$\Rightarrow {\left(4\left(y+3\right)\right)}^2-4\left(y-11\right)2\left(y+3\right)\ge 0$
$\Rightarrow 8\left(y+3\right)\left\{2\left(y+3\right)-\left(y-11\right)\right\}\ge 0$
$\Rightarrow 8\left(y+3\right)\left(y+17\right)\ge 0$
$\Rightarrow y\le -17$ or $y\ge -3$
$\Rightarrow y$ does not lie between $-17$ and $-3$