Question

For x $\in$ R, the range of the given expression $\frac{x^2+34x-71}{x^2+2x-7}$ is

Answer 1

R.E.F image

let $y=\frac{x^2+34x-71}{x^2+2x-7}\Rightarrow x^2(y-1)+x(2y-34)-7y+71=0$

As $xϵR,\Rightarrow D⩾0$

$\Rightarrow (2y-34{)}^2+4(y-1\left)\right(7y-71)⩾0$

$\Rightarrow (y-17{)}^2+(7y^2-78y+71)⩾0$

$\Rightarrow 8y^2-112y+306⩾0$

$\Rightarrow 4y^2-56y+153⩾0$

$D=(56{)}^2-16(153)=688>0\Rightarrow Roots\Rightarrow y=\frac{56\pm \sqrt{688}}{8}$

$\Rightarrow$ Range $=yϵ(-\infty ,\alpha ]\cup [\beta ,\infty )$