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:

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

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

$$ \begin{aligned} \left( 3+4i \right) \cdot \left( 1+2i \right) &= 3 \cdot 1 + 3 \cdot \left(2 \,i \right) + \left( 4 \,i \right) \cdot \left(1 \right) + \left( 4 \,i \right) \cdot \left(2 \,i \right) = \\[1 em] &= 3 + 6 \, i + 4 \, i + 8 \color{blue}{(-1)} = \\[1 em] &= -5+10i\end{aligned} $$ $$ \begin{aligned} \left( 1-2i \right) \cdot \left( 1+2i \right) &= 1 \cdot 1 + 1 \cdot \left(2 \,i \right) + \left( -2 \,i \right) \cdot \left(1 \right) + \left( -2 \,i \right) \cdot \left(2 \,i \right) = \\[1 em] &= 1 + 2 \, i -2 \, i -4 \color{blue}{(-1)} = \\[1 em] &= 5\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 3+4i }{ 1-2i } = \frac{ -5+10i }{ 5 } = \frac{ -5 }{ 5 } + \frac{ 10 }{ 5 } i= -1+2i $$

Complex Division Calculator