# How do you divide (5-6i)/(-5+10i)?

Sep 4, 2016

$- \frac{17}{25} - \frac{4}{25} i$

#### Explanation:

We have: $\frac{5 - 6 i}{- 5 + 10 i}$

Let's multiply both the numerator and the denominator by the complex conjugate of the denominator:

$= \frac{5 - 6 i}{- 5 + 10 i} \cdot \frac{- 5 - 10 i}{- 5 - 10 i}$

$= \frac{\left(5\right) \left(- 5\right) + \left(5\right) \left(- 10 i\right) + \left(- 6 i\right) \left(- 5\right) + \left(- 6 i\right) \left(- 10 i\right)}{{\left(- 5\right)}^{2} - {\left(10 i\right)}^{2}}$

$= \frac{- 25 - 50 i + 30 i + 60 {i}^{2}}{25 - 100 {i}^{2}}$

Let's apply the fact that ${i}^{2} = - 1$:

$= \frac{- 25 + \left(60 \cdot - 1\right) - 20 i}{25 - \left(100 \cdot - 1\right)}$

$= \frac{- 85 - 20 i}{125}$

$= \frac{- 17 - 4 i}{25}$

$= - \frac{17}{25} - \frac{4}{25} i$