How do you simplify (1 + 2i) / (3-4i) ?

Mar 24, 2018

$- \frac{1}{5} + \frac{2}{5} i$

Explanation:

$\text{multiply the numerator/denominator by the "color(blue)"conjugate}$
$\text{of the denominator}$

$\text{this ensures that the denominator is real}$

$\text{the conjugate of "3-4i" is } 3 \textcolor{red}{+} 4 i$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{w h i t e}{x} {i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1$

$\Rightarrow \frac{1 + 2 i}{3 - 4 i} \times \frac{3 + 4 i}{3 + 4 i}$

$= \frac{\left(1 + 2 i\right) \left(3 + 4 i\right)}{\left(3 - 4 i\right) \left(3 + 4 i\right)} \leftarrow \textcolor{b l u e}{\text{expand using FOIL}}$

$= \frac{3 + 10 i + 8 {i}^{2}}{9 - 16 {i}^{2}}$

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

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

$= - \frac{1}{5} + \frac{2}{5} i \leftarrow \textcolor{red}{\text{in standard form}}$