# How do you simplify (2 - i) /(5 + i)?

Dec 22, 2015

Multiply the numerator and denominator by the conjugate of the denominator to find

$\frac{2 - i}{5 + i} = \frac{9}{26} - \frac{7}{26} i$

#### Explanation:

The conjugate of a complex number $a + b i$ is $a - b i$. The product of a complex number and its conjugate is a real number. We will use that fact here.

$\frac{2 - i}{5 + i} = \frac{2 - i}{5 + i} \cdot \frac{5 - i}{5 - i}$

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

$= \frac{10 - 2 i - 5 i - 1}{25 - 5 i + 5 i + 1}$

$= \frac{9 - 7 i}{26}$

$= \frac{9}{26} - \frac{7}{26} i$