How to find range this expression I am new to quadratic equation and expression, Hote to find the range of this expression. The expression is: $ y = \frac{x^2 -2x +9}{x^2 - 2x -9} $ Where $ x $ is real. What I've done: Cross multiplied and got, $ x^2(y-1) + x(2-2y) - (9y - 9 ) = 0 $ Since x was real, $ D ≥ 0 $ $ y^2 - 2y + 1 ≥ 0 $

$$y=\frac{(x^2-2x-9)+9+9}{x^2-2x-9}=1+\frac{18}{x^2-2x-9}$$ Now we know that $$x^2-2x-9=(x^2-2x+1) -1 -9 = (x-1)^2-10 \ge -10$$

Thus

$$\frac{1}{x^2-2x-9} \in (-\infty,-0.1] \cup(0, +\infty)$$

Thus

$$y \in (-\infty, -0.8] \cup (1, +\infty)$$