Question

How do you combine `(3x+1)/(x-2) - (4x+1)/(x-3)`?

Answer 1

`-(x^2+x+1)/((x-2)(x-3))`

Explanation:

We have
`(3x+1)/(x-2)-(4x+1)/(x-3)=((3x+1)(x-3)-(4x+1)(x-2))/((x-2)(x-3))`
Expanding the numerator:
`(3x+1)(x-3)-(4x+1)(x-2)=3x^2+x-9x-3-(4x^2+x-8x-2)=`
`3x^2-8x-3-4x^2+7x+2=-x^2-x-1`

Answer 2

`-(x^2+x+1)/((x-3)(x-2)`

Explanation:

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

`color(green)( color(white)("dd")[(3x+1)/(x-2)color(red)(xx1)]color(white)("ddd")-color(white)("dd")[(4x+1)/(x-3)color(red)(xx1)]`

`color(green)([(3x+1)/(x-2)color(red)(xx(x-3)/(x-3))]color(white)("d")-[(4x+1)/(x-3)color(red)(xx(x-2)/(x-2))])`

`color(green)([(3x^2-9x+x-3)/((x-3)(x-2))]-[(4x^2-8x+x-2)/((x-3)(x-2))])`

`color(green)([(3x^2-8x-3)/((x-3)(x-2))]-[(4x^2-7x-2)/((x-3)(x-2))])`

`color(white)("ddddddddd")color(green)( (-x^2-x-1)/((x-3)(x-2) )`

`color(white)("ddddddddd")color(green)( (-(x^2+x+1))/((x-3)(x-2) )`

`color(white)("dddddddd")color(green)( -(x^2+x+1)/((x-3)(x-2) )`