Question

How do you simplify `(x^2 – x – 6 )/(x^2 – 9x + 18 )`?

Answer 1

`(x^2-x-6)/(x^2-9x+18)=(x+2)/(x-6)=1+8/(x-6)`

excluding `x=3`

Explanation:

Factor the numerator and denominator. Cancel the common factor `(x-3)`, noting that `x=3` is an excluded value. Simplify...

`(x^2-x-6)/(x^2-9x+18)`

`=(color(red)(cancel(color(black)((x-3))))(x+2))/(color(red)(cancel(color(black)((x-3))))(x-6))`

`=(x+2)/(x-6)`

`=(x-6+8)/(x-6)`

`=1+8/(x-6)`

excluding `x=3`.

The value `x=3` must be excluded since the numerator and denominator of the original rational expression are both zero, resulting in an undefined value, whereas the simplified expression is well defined when `x=3`.