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

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

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

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

$$ \begin{aligned} \left( 3-2i \right) \cdot \left( 4+3i \right) &= 3 \cdot 4 + 3 \cdot \left(3 \,i \right) + \left( -2 \,i \right) \cdot \left(4 \right) + \left( -2 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 12 + 9 \, i -8 \, i -6 \color{blue}{(-1)} = \\[1 em] &= 18+i\end{aligned} $$ $$ \begin{aligned} \left( 4-3i \right) \cdot \left( 4+3i \right) &= 4 \cdot 4 + 4 \cdot \left(3 \,i \right) + \left( -3 \,i \right) \cdot \left(4 \right) + \left( -3 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 16 + 12 \, i -12 \, i -9 \color{blue}{(-1)} = \\[1 em] &= 25\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 3-2i }{ 4-3i } = \frac{ 18+i }{ 25 } = \frac{ 18 }{ 25 } + \frac{ 1 }{ 25 } i= \frac{ 18 }{ 25 }+\frac{ 1 }{ 25 }i $$

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