Question

What does `(1+2/x-15/x^2)/(1+4/x-5/x^2)` simplify to?

Answer 1

`=(x-3)/(x-1)`

Explanation:

`(1+2/x-15/x^2)/(1+4/x-5/x^2`
`=((x^2+2x-15)/x^2)/((x^2+4x-5)/x^2)`
`=((x^2+2x-15)/cancelx^2)/((x^2+4x-5)/cancelx^2`
`=(x^2+2x-15)/(x^2+4x-5)`
`=(x^2+5x-3x-15)/(x^2+5x-x-5)`
`=(x(x+5)-3(x+5))/(x(x+5)-1(x+5))`
`=((x+5)(x-3))/((x+5)(x-1))`
`=(x-3)/(x-1)`

Answer 2

=`(x-3)/(x-1)`

Explanation:

`(1+2/x-15/x^2)/(1+4/x-5/x^2)`

It is easier to tackle if it is written like this instead:

`color(red)((1/1+2/x-15/x^2)) div color(blue)((1/1+4/x-5/x^2))`

Adding and subtracting fractions, we need the LCD

=`color(red)(((x^2 +2x-15)/x^2)) div color(blue)(((x^2+4x-5)/x^2))`

Factorise the quadratic trinomials

=`color(red)(((x+5)(x-3))/x^2)div color(blue)(((x+5)(x-1))/x^2)`

Divide becomes `xx " by the reciprocal and simplify"`

= `(cancel(x+5)(x-3))/cancelx^2xx cancelx^2/(cancel(x+5)(x-1))`

=`(x-3)/(x-1)`