Question

The subject is operations on rational expressions.

The instructions are add or subtract the following expressions. Remember to find a common denominator when necessary. Reduce all answers to lowest terms.

Answer 1

`(2(x-1)(2x+9))/((x-3)(x-2))`

Explanation:

`(4x)/((x-3))+(6)/((x+2))`

=
`(4x(x+2))/((x-3)(x+2))+(6(x-3))/((x+2)(x-3))`

Now we have done the denominators of each term of the expression equal.

Further we add the terms,

`(4x(x+2))/((x-3)(x+2))+(6(x-3))/((x+2)(x-3))`

=
`(4x(x+2)+6(x-3))/((x-3)(x+2))`

=
`(4x^(2)+8x+6x-18)/((x-3)(x+2))`

=
`(4x^(2)+14x-18)/((x-3)(x-2))`

Now factorize the numerator of the fraction.

4x² + 14x - 18 = 2(2x² + 7x - 9)

= 2(2x² + 9x - 2x - 9)

= 2[x(2x + 9) - 1(2x + 9)]

= 2(x - 1)(2x + 9)

Therefore,
`(2(x-1)(2x+9))/((x-3)(x-2))` will be the answer.