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}{ 12+7i }\, $ is $ \color{blue}{ 12-7i } $.

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

$$\begin{aligned} \frac{ 153+41i }{ 12+7i } &= \frac{ 153+41i }{ 12+7i } \cdot \frac{ \color{blue}{ 12-7i } }{ \color{blue}{ 12-7i } } = \\[1 em] &= \frac{ \left( 153+41i \right) \cdot \left( 12-7i \right) }{ \left( 12+7i \right) \cdot \left( 12-7i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 153+41i \right) \cdot \left( 12-7i \right) &= 153 \cdot 12 + 153 \cdot \left(-7 \,i \right) + \left( 41 \,i \right) \cdot \left(12 \right) + \left( 41 \,i \right) \cdot \left(-7 \,i \right) = \\[1 em] &= 1836 -1071 \, i + 492 \, i -287 \color{blue}{(-1)} = \\[1 em] &= 2123-579i\end{aligned} $$ $$ \begin{aligned} \left( 12+7i \right) \cdot \left( 12-7i \right) &= 12 \cdot 12 + 12 \cdot \left(-7 \,i \right) + \left( 7 \,i \right) \cdot \left(12 \right) + \left( 7 \,i \right) \cdot \left(-7 \,i \right) = \\[1 em] &= 144 -84 \, i + 84 \, i -49 \color{blue}{(-1)} = \\[1 em] &= 193\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 153+41i }{ 12+7i } = \frac{ 2123-579i }{ 193 } = \frac{ 2123 }{ 193 } + \frac{ -579 }{ 193 } i= 11-3i $$

Complex Division Calculator