How do you combine a/(a-3)-5/(a+6)?

1 Answer
Dec 24, 2016

(a^2 + a + 15)/((a - 3)(a + 6)

or

(a^2 + a + 15)/(a^2 + 3a - 18)

Explanation:

To subtract these two fractions, as with any fractions, they need to be over common denominators. In this case the common denominator will be (a - 3)(a + 6).

Therefore, first we need to multiply each fraction by the correct form of 1 to put each fraction over the common denominator:

(color(red)(((a + 6))/((a + 6))) xx a/((a - 3))) - (color(blue)(((a - 3))/((a - 3))) xx 5/((a + 6)))

((a + 6)a)/((a + 6)(a - 3)) - ((a - 3)5)/((a - 3)(a + 6)

We can now expand the terms within parenthesis in each of the numerators:

(a^2 + 6a)/((a + 6)(a - 3)) - (5a - 15)/((a - 3)(a + 6)

We can now combine the fractions by subtracting the numerators and keeping the common denominator:

((a^2 + 6a) - (5a - 15))/((a - 3)(a + 6)

Now we can expand the terms within the parenthesis ensuring we keep the signs of the terms correct:

(a^2 + 6a - 5a + 15)/((a - 3)(a + 6)

And then we can combine like terms in the numerator:

(a^2 + (6 - 5)a + 15)/((a - 3)(a + 6)

(a^2 + a + 15)/((a - 3)(a + 6)

Or, is we want to expand the term in the denominator by cross multiplying we get:

(a^2 + a + 15)/(a^2 + 6a - 3a - 18)

(a^2 + a + 15)/(a^2 + (6 - 3)a - 18)

(a^2 + a + 15)/(a^2 + 3a - 18)