Question

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

Answer 1

Answer:

$\dfrac{a}{a - 4}$

Step-by-step 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.

$\dfrac{3a}{a + 4} - \dfrac{2a^2 - 16a}{a^2 - 16} =$

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

$= \dfrac{3a}{a + 4} - \dfrac{2a^2 - 16a}{(a - 4)(a + 4)}$

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).

$= \dfrac{3a}{a + 4} \times \dfrac{a - 4}{a - 4} - \dfrac{2a^2 - 16a}{(a - 4)(a + 4)}$

$= \dfrac{3a^2 - 12a}{(a - 4)(a + 4)} - \dfrac{2a^2 - 16a}{(a - 4)(a + 4)}$

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.

$= \dfrac{3a^2 - 12a - (2a^2 - 16a)}{(a - 4)(a + 4)}$

$= \dfrac{3a^2 - 12a - 2a^2 + 16a}{(a - 4)(a + 4)}$

$= \dfrac{a^2 + 4a}{(a - 4)(a + 4)}$

$= \dfrac{a(a + 4)}{(a - 4)(a + 4)}$

$= \dfrac{a}{a - 4}$