Step 1: Determine the conjugate of the denominator. ( to find the conjugate just change the sign of the imaginary part ).

In this example, the conjugate of $ \color{orangered}{ 8+4i }\, $ is $ \color{blue}{ 8-4i } $.

Step 2: Multiply both the numerator and denominator by the conjugate:

$$\begin{aligned} \frac{ 8-6i }{ 8+4i } &= \frac{ 8-6i }{ 8+4i } \cdot \frac{ \color{blue}{ 8-4i } }{ \color{blue}{ 8-4i } } = \\[1 em] &= \frac{ \left( 8-6i \right) \cdot \left( 8-4i \right) }{ \left( 8+4i \right) \cdot \left( 8-4i \right) } \end{aligned} $$

Step 3: Simplify numerator and denominator (use $\color{blue}{i^2 = -1}$)

$$ \begin{aligned} \left( 8-6i \right) \cdot \left( 8-4i \right) &= 8 \cdot 8 + 8 \cdot \left(-4 \,i \right) + \left( -6 \,i \right) \cdot \left(8 \right) + \left( -6 \,i \right) \cdot \left(-4 \,i \right) = \\[1 em] &= 64 -32 \, i -48 \, i + 24 \color{blue}{(-1)} = \\[1 em] &= 40-80i\end{aligned} $$ $$ \begin{aligned} \left( 8+4i \right) \cdot \left( 8-4i \right) &= 8 \cdot 8 + 8 \cdot \left(-4 \,i \right) + \left( 4 \,i \right) \cdot \left(8 \right) + \left( 4 \,i \right) \cdot \left(-4 \,i \right) = \\[1 em] &= 64 -32 \, i + 32 \, i -16 \color{blue}{(-1)} = \\[1 em] &= 80\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 8-6i }{ 8+4i } = \frac{ 40-80i }{ 80 } = \frac{ 40 }{ 80 } + \frac{ -80 }{ 80 } i= \frac{ 1 }{ 2 }-i $$

Complex Division Calculator