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}{ 1-2i }\, $ is $ \color{blue}{ 1+2i } $.

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

$$\begin{aligned} \frac{ -3+5i }{ 1-2i } &= \frac{ -3+5i }{ 1-2i } \cdot \frac{ \color{blue}{ 1+2i } }{ \color{blue}{ 1+2i } } = \\[1 em] &= \frac{ \left( -3+5i \right) \cdot \left( 1+2i \right) }{ \left( 1-2i \right) \cdot \left( 1+2i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -3+5i \right) \cdot \left( 1+2i \right) &= -3 \cdot 1 -3 \cdot \left(2 \,i \right) + \left( 5 \,i \right) \cdot \left(1 \right) + \left( 5 \,i \right) \cdot \left(2 \,i \right) = \\[1 em] &= -3 -6 \, i + 5 \, i + 10 \color{blue}{(-1)} = \\[1 em] &= -13-i\end{aligned} $$ $$ \begin{aligned} \left( 1-2i \right) \cdot \left( 1+2i \right) &= 1 \cdot 1 + 1 \cdot \left(2 \,i \right) + \left( -2 \,i \right) \cdot \left(1 \right) + \left( -2 \,i \right) \cdot \left(2 \,i \right) = \\[1 em] &= 1 + 2 \, i -2 \, i -4 \color{blue}{(-1)} = \\[1 em] &= 5\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -3+5i }{ 1-2i } = \frac{ -13-i }{ 5 } = \frac{ -13 }{ 5 } + \frac{ -1 }{ 5 } i= -\frac{ 13 }{ 5 }-\frac{ 1 }{ 5 }i $$

Complex Division Calculator