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}{ 7-5i }\, $ is $ \color{blue}{ 7+5i } $.

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

$$\begin{aligned} \frac{ 3-2i }{ 7-5i } &= \frac{ 3-2i }{ 7-5i } \cdot \frac{ \color{blue}{ 7+5i } }{ \color{blue}{ 7+5i } } = \\[1 em] &= \frac{ \left( 3-2i \right) \cdot \left( 7+5i \right) }{ \left( 7-5i \right) \cdot \left( 7+5i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 3-2i \right) \cdot \left( 7+5i \right) &= 3 \cdot 7 + 3 \cdot \left(5 \,i \right) + \left( -2 \,i \right) \cdot \left(7 \right) + \left( -2 \,i \right) \cdot \left(5 \,i \right) = \\[1 em] &= 21 + 15 \, i -14 \, i -10 \color{blue}{(-1)} = \\[1 em] &= 31+i\end{aligned} $$ $$ \begin{aligned} \left( 7-5i \right) \cdot \left( 7+5i \right) &= 7 \cdot 7 + 7 \cdot \left(5 \,i \right) + \left( -5 \,i \right) \cdot \left(7 \right) + \left( -5 \,i \right) \cdot \left(5 \,i \right) = \\[1 em] &= 49 + 35 \, i -35 \, i -25 \color{blue}{(-1)} = \\[1 em] &= 74\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 3-2i }{ 7-5i } = \frac{ 31+i }{ 74 } = \frac{ 31 }{ 74 } + \frac{ 1 }{ 74 } i= \frac{ 31 }{ 74 }+\frac{ 1 }{ 74 }i $$

Complex number calculator