# How do you divide -2-2i div 5-2i?

Feb 1, 2016

$- \frac{6}{29} - \frac{14}{29} i$

#### Explanation:

When dividing complex numbers ,multiply numerator and

denominator by the complex conjugate of the denominator.

If a + bi is a complex number then$\textcolor{red}{\text{ a - bi is conjugate}}$

This ensures the denominator is real.

as $\left(a + b i\right) \left(a - b i\right) = {a}^{2} + {b}^{2} \textcolor{b l a c k}{\text{ which is real}}$

[Note:  i^2 =( sqrt-1 )^2 = -1 ]

the conjugate of 5 - 2i is 5 + 2i .

$\Rightarrow \frac{\left(- 2 - 2 i\right) \left(5 + 2 i\right)}{\left(5 - 2 i\right) \left(5 + 2 i\right)}$

( distribute to obtain) $\frac{- 10 - 14 i - 4 {i}^{2}}{25 - 4 {i}^{2}}$

$= \frac{- 6 - 14 i}{29} = - \frac{6}{29} - \frac{14}{29} i$