Question

Find it :-
$f\left(x\right)=\frac{x^2+34x-71}{x^2+2x-7}.$

Answer 1

$f\left(x\right)=\frac{x^2+34x-71}{x^2+2x-7}$

The basic method to fiend quadratic -

Suppose,

$y=\frac{x^2+34x-71}{x^2+2x-7}$

$y{x}^2+2xy-7y=x^2+34x-71$ (cross-multiy)

$x^2(1-y)+2x(17-y)+7y-71=0$ $\left(xϵR\right)$

$4(17-y{)}^2-4(1-y\left)\right(7y-71)\ge 0$ $(D\ge 0)$

$4({17}^2+y^2-34y)-4(-7y^2-71+78y)\ge 0$

$y^2-34y+{17}^2+7y^2+71-78yy\ge 0$

$8y^2-112y+360\ge 0$

$y^2-14y+45\ge 0$

$(y-5)(y-9)\ge 0$

$yϵ(-\infty ,d)\cup (g,\infty )$