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}{ 9+3i }\, $ is $ \color{blue}{ 9-3i } $.

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

$$\begin{aligned} \frac{ 20+12i }{ 9+3i } &= \frac{ 20+12i }{ 9+3i } \cdot \frac{ \color{blue}{ 9-3i } }{ \color{blue}{ 9-3i } } = \\[1 em] &= \frac{ \left( 20+12i \right) \cdot \left( 9-3i \right) }{ \left( 9+3i \right) \cdot \left( 9-3i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 20+12i \right) \cdot \left( 9-3i \right) &= 20 \cdot 9 + 20 \cdot \left(-3 \,i \right) + \left( 12 \,i \right) \cdot \left(9 \right) + \left( 12 \,i \right) \cdot \left(-3 \,i \right) = \\[1 em] &= 180 -60 \, i + 108 \, i -36 \color{blue}{(-1)} = \\[1 em] &= 216+48i\end{aligned} $$ $$ \begin{aligned} \left( 9+3i \right) \cdot \left( 9-3i \right) &= 9 \cdot 9 + 9 \cdot \left(-3 \,i \right) + \left( 3 \,i \right) \cdot \left(9 \right) + \left( 3 \,i \right) \cdot \left(-3 \,i \right) = \\[1 em] &= 81 -27 \, i + 27 \, i -9 \color{blue}{(-1)} = \\[1 em] &= 90\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 20+12i }{ 9+3i } = \frac{ 216+48i }{ 90 } = \frac{ 216 }{ 90 } + \frac{ 48 }{ 90 } i= \frac{ 12 }{ 5 }+\frac{ 8 }{ 15 }i $$

Complex number calculator