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

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

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

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

$$ \begin{aligned} 5 \cdot \left( 3-i \right) &= 5 \cdot 3 + 5 \cdot \left(-1 \,i \right) = \\[1 em] &= 15 -5 \, i = \\[1 em] &= 15-5i\end{aligned} $$ $$ \begin{aligned} \left( 3+i \right) \cdot \left( 3-i \right) &= 3 \cdot 3 + 3 \cdot \left(-1 \,i \right) + \left( 1 \,i \right) \cdot \left(3 \right) + \left( 1 \,i \right) \cdot \left(-1 \,i \right) = \\[1 em] &= 9 -3 \, i + 3 \, i -1 \color{blue}{(-1)} = \\[1 em] &= 10\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 5 }{ 3+i } = \frac{ 15-5i }{ 10 } = \frac{ 15 }{ 10 } + \frac{ -5 }{ 10 } i= \frac{ 3 }{ 2 }-\frac{ 1 }{ 2 }i $$

Complex Division Calculator