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

Dec 24, 2016

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

or

$\frac{{a}^{2} + a + 15}{{a}^{2} + 3 a - 18}$

#### 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 $\left(a - 3\right) \left(a + 6\right)$.

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

$\left(\textcolor{red}{\frac{\left(a + 6\right)}{\left(a + 6\right)}} \times \frac{a}{\left(a - 3\right)}\right) - \left(\textcolor{b l u e}{\frac{\left(a - 3\right)}{\left(a - 3\right)}} \times \frac{5}{\left(a + 6\right)}\right)$

((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:

$\frac{{a}^{2} + a + 15}{{a}^{2} + 6 a - 3 a - 18}$

$\frac{{a}^{2} + a + 15}{{a}^{2} + \left(6 - 3\right) a - 18}$

$\frac{{a}^{2} + a + 15}{{a}^{2} + 3 a - 18}$