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}{ 2i }\, $ is $ \color{blue}{ -2i } $.

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

$$\begin{aligned} \frac{ -15-20i }{ 2i } &= \frac{ -15-20i }{ 2i } \cdot \frac{ \color{blue}{ -2i } }{ \color{blue}{ -2i } } = \\[1 em] &= \frac{ \left( -15-20i \right) \cdot \left( -2i \right) }{ 2i \cdot \left( -2i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -15-20i \right) \cdot \left( -2i \right) &= -15 \cdot \left(-2 \,i \right) + \left( -20 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= + 30 \, i + 40 \color{blue}{(-1)} = \\[1 em] &= -40+30i\end{aligned} $$ $$ \begin{aligned} 2i \cdot \left( -2i \right) &= + \left( 2 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= -4 \color{blue}{(-1)} = \\[1 em] &= 4\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -15-20i }{ 2i } = \frac{ -40+30i }{ 4 } = \frac{ -40 }{ 4 } + \frac{ 30 }{ 4 } i= -10+\frac{ 15 }{ 2 }i $$

Complex Division Calculator