# How do you evaluate (3a -9i +2ai +6)/(a^2+9) + (3-9i+3i+9)/(9+a^2)?

Aug 14, 2017

$\frac{3 a + 2 a i - 15 i + 18}{9 + {a}^{2}}$

#### Explanation:

The first thing we notice with the two expression here is that the denominators are the same since ${a}^{2} + 9 = 9 + {a}^{2}$.

This means that we can "combine" the two fractions to add the numerators:

$\frac{3 a - 9 i + 2 a i + 6 + 3 - 9 i + 3 i + 9}{9 + {a}^{2}}$

Now, we can combine like terms on the numerator. The like terms have been put in the same color, so you just add/subtract them:

$\frac{3 a - \textcolor{g r e e n}{9 i} + 2 a i + \textcolor{red}{6} + \textcolor{red}{3} - \textcolor{g r e e n}{9 i} + \textcolor{g r e e n}{3 i} + \textcolor{red}{9}}{9 + {a}^{2}}$

Rearranging to bring like terms closer together and to make it easier to simplify:

$\frac{3 a + 2 a i - \textcolor{g r e e n}{9 i} - \textcolor{g r e e n}{9 i} + \textcolor{g r e e n}{3 i} + \textcolor{red}{6} + \textcolor{red}{9} + \textcolor{red}{3}}{9 + {a}^{2}}$

$\frac{3 a + 2 a i - 15 i + 18}{9 + {a}^{2}}$