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}{ 3+4i }\, $ is $ \color{blue}{ 3-4i } $.

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

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

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

$$ \begin{aligned} \left( 1+2i \right) \cdot \left( 3-4i \right) &= 1 \cdot 3 + 1 \cdot \left(-4 \,i \right) + \left( 2 \,i \right) \cdot \left(3 \right) + \left( 2 \,i \right) \cdot \left(-4 \,i \right) = \\[1 em] &= 3 -4 \, i + 6 \, i -8 \color{blue}{(-1)} = \\[1 em] &= 11+2i\end{aligned} $$ $$ \begin{aligned} \left( 3+4i \right) \cdot \left( 3-4i \right) &= 3 \cdot 3 + 3 \cdot \left(-4 \,i \right) + \left( 4 \,i \right) \cdot \left(3 \right) + \left( 4 \,i \right) \cdot \left(-4 \,i \right) = \\[1 em] &= 9 -12 \, i + 12 \, i -16 \color{blue}{(-1)} = \\[1 em] &= 25\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1+2i }{ 3+4i } = \frac{ 11+2i }{ 25 } = \frac{ 11 }{ 25 } + \frac{ 2 }{ 25 } i= \frac{ 11 }{ 25 }+\frac{ 2 }{ 25 }i $$

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