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}{ 10-20i }\, $ is $ \color{blue}{ 10+20i } $.

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

$$\begin{aligned} \frac{ -200i }{ 10-20i } &= \frac{ -200i }{ 10-20i } \cdot \frac{ \color{blue}{ 10+20i } }{ \color{blue}{ 10+20i } } = \\[1 em] &= \frac{ \left( -200i \right) \cdot \left( 10+20i \right) }{ \left( 10-20i \right) \cdot \left( 10+20i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -200i \right) \cdot \left( 10+20i \right) &= + \left( -200 \,i \right) \cdot \left(10 \right) + \left( -200 \,i \right) \cdot \left(20 \,i \right) = \\[1 em] &= -2000 \, i -4000 \color{blue}{(-1)} = \\[1 em] &= 4000-2000i\end{aligned} $$ $$ \begin{aligned} \left( 10-20i \right) \cdot \left( 10+20i \right) &= 10 \cdot 10 + 10 \cdot \left(20 \,i \right) + \left( -20 \,i \right) \cdot \left(10 \right) + \left( -20 \,i \right) \cdot \left(20 \,i \right) = \\[1 em] &= 100 + 200 \, i -200 \, i -400 \color{blue}{(-1)} = \\[1 em] &= 500\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -200i }{ 10-20i } = \frac{ 4000-2000i }{ 500 } = \frac{ 4000 }{ 500 } + \frac{ -2000 }{ 500 } i= 8-4i $$

Complex Division Calculator