# Question #8380d

Feb 27, 2018

Did you mean to put the negative sign after the divide sign?
If so, $\frac{4 - 3 i + 2 + 6 i}{-} \left(\left(3 - i\right) \left(3 - i\right)\right) = - \frac{3}{10} - \frac{3}{5} i$.

#### Explanation:

First, simplify the numerator by combining the integer terms and combining the $i$ terms: $4 - 3 i + 2 + 6 i = 3 i + 6$.

Secondly, the denominator: $- \left(\left(3 - i\right) \left(3 - i\right)\right) = - \left(9 - 3 i - 3 i + {i}^{2}\right)$. Since $i = \sqrt{- 1}$, ${i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1$.
Plugging that into the denominator and combining like terms, we get $- \left(9 - 6 i + \left(- 1\right)\right) = - \left(8 - 6 i\right) = - 8 + 6 i = 6 i - 8$

Our fraction is now $\frac{3 i + 6}{6 i - 8}$

This may be okay for your teacher, but the standard form for complex numbers is $a \pm b i$, so let's get our answer in that format by multiplying the numerator and denominator by the conjugate of the denominator: $\frac{\left(3 i + 6\right) \left(6 i + 8\right)}{\left(6 i - 8\right) \left(6 i + 8\right)}$
We get $\frac{18 {i}^{2} + 24 i + 36 i + 48}{- 36 - 64} = \frac{30 + 60 i}{-} 100$ and put it in the form of $a \pm b i$ then reduce it:
$\frac{30}{-} 100 - \frac{60 i}{-} 100 = - \frac{3}{10} - \frac{3}{5} i$

Hope that helps! Have fun; imaginary numbers are cool!