Question

How do you subtract `6/(x+2)-(2x+3)/x`?

Answer 1

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

Explanation:

Before subtracting we require the fractions to have a `color(blue)"common denominator"`

This is achieved as follows.

`(6/(x+2)xxx/x)-((2x+3)/(x)xx((x+2))/((x+2)))`

That is, multiply the numerator/denominator of the first fraction by the denominator of the second fraction and multiply the numerator/denominator of the second fraction by the denominator of the first fraction.

`=(6x)/(x(x+2))-((2x+3)(x+2))/(x(x+2))`

We now have a common denominator and can subtract the terms on the numerator.

`=(6x-(2x^2+7x+6))/(x(x+2))`

`=(6x-2x^2-7x-6)/(x(x+2))=(-2x^2-x-6)/(x(x+2))`

Take out a common factor of - 1 on the numerator.

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