# How do you divide (-6-i)/(1+5i)?

Jan 8, 2017

I multiply by 1 in form of the complex conjugate of the denominator divided by itself, use the F.O.I.L. method to multiply the numerator, and then simplify.

#### Explanation:

Given: $\frac{- 6 - i}{1 + 5 i}$

Multiply by 1 in the form of $\frac{1 - 5 i}{1 - 5 i}$:

$\frac{- 6 - i}{1 + 5 i} \frac{1 - 5 i}{1 - 5 i}$

The denominator becomes the difference of two squares:

$\frac{\left(- 6 - i\right) \left(1 - 5 i\right)}{{1}^{2} - {\left(5 i\right)}^{2}}$

$\frac{\left(- 6 - i\right) \left(1 - 5 i\right)}{1 - 25 {i}^{2}}$

Use the F.O.I.L method to multiply the numerator:

$\frac{- 6 + 30 i - i + 5 {i}^{2}}{1 - 25 {i}^{2}}$

Substitute -1 for ${i}^{2}$

$\frac{- 6 + 30 i - i - 5}{1 + 25}$

Combine like terms:

$\frac{- 11 + 29 i}{26}$

Divide each term:

$- \frac{11}{26} + \frac{29}{26} i$