Question

Simplify. (In the problems where you combine fractions with unlike denominators leave the denominators in factored form.)

Answer 1

`(a)/(a - 4)`

Explanation:

You are subtracting fractions, so as usual with fraction subtraction, you need a common denominator. To get a common denominator, you need to factor each denominator.

`(3a)/(a + 4) - (2a^2 - 16a)/(a^2 - 16) =`

Factor the right denominator. It is a difference of two squares.

`= (3a)/(a + 4) - (2a^2 - 16a)/((a - 4)(a + 4))`

The left denominator is (a + 4). The right denominator is (a - 4)(a + 4). The LCD is (a - 4)(a + 4), so multiply the left fraction by (a - 4)/(a - 4).

`= (3a)/(a + 4) * (a - 4)/(a - 4) - (2a^2 - 16a)/((a - 4)(a + 4))`

`= (3a^2 - 12a)/((a - 4)(a + 4)) - (2a^2 - 16a)/((a - 4)(a + 4))`

Be careful when you do the subtraction of the numerators, 3a² - 12a minus 2a² - 16a. 3a² - 12a - (2a² - 16a). The minus of the subtraction changes each sign inside the parentheses, so you end up with 3a² - 12a - 2a² + 16a. Be especially careful with the +16a.

`= (3a^2 - 12a - (2a^2 - 16a))/((a - 4)(a + 4))`

`= (3a^2 - 12a - 2a^2 + 16a)/((a - 4)(a + 4))`

`= (a^2 + 4a)/((a - 4)(a + 4))`

`= (a(a + 4))/((a - 4)(a + 4))`

`= (a)/(a - 4)`