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-i }\, $ is $ \color{blue}{ 8+i } $.

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

$$\begin{aligned} \frac{ 12-6i }{ 8-i } &= \frac{ 12-6i }{ 8-i } \cdot \frac{ \color{blue}{ 8+i } }{ \color{blue}{ 8+i } } = \\[1 em] &= \frac{ \left( 12-6i \right) \cdot \left( 8+i \right) }{ \left( 8-i \right) \cdot \left( 8+i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 12-6i \right) \cdot \left( 8+i \right) &= 12 \cdot 8 + 12 \cdot \left(1 \,i \right) + \left( -6 \,i \right) \cdot \left(8 \right) + \left( -6 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= 96 + 12 \, i -48 \, i -6 \color{blue}{(-1)} = \\[1 em] &= 102-36i\end{aligned} $$ $$ \begin{aligned} \left( 8-i \right) \cdot \left( 8+i \right) &= 8 \cdot 8 + 8 \cdot \left(1 \,i \right) + \left( -1 \,i \right) \cdot \left(8 \right) + \left( -1 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= 64 + 8 \, i -8 \, i -1 \color{blue}{(-1)} = \\[1 em] &= 65\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 12-6i }{ 8-i } = \frac{ 102-36i }{ 65 } = \frac{ 102 }{ 65 } + \frac{ -36 }{ 65 } i= \frac{ 102 }{ 65 }-\frac{ 36 }{ 65 }i $$

Complex number calculator