(3 - 2i)² ⋅ (i / (i - 2)) =

Steps to Solve

1. Square the complex number ( (3 - 2i) )

To evaluate ( (3 - 2i)^2 ), we use the formula for squaring a binomial:

$$(a - b)^2 = a^2 - 2ab + b^2$$

Here, ( a = 3 ) and ( b = 2i ):

$$ (3 - 2i)^2 = 3^2 - 2 \cdot 3 \cdot 2i + (2i)^2 $$

Calculating each term:

$$ = 9 - 12i + 4(-1) = 9 - 12i - 4 = 5 - 12i $$

2. Evaluate the division ( \frac{i}{i - 2} )

To simplify ( \frac{i}{i - 2} ), we can multiply the numerator and denominator by the conjugate of the denominator:

$$ \frac{i(i + 2)}{(i - 2)(i + 2)} $$

Calculating the denominator:

$$(i - 2)(i + 2) = i^2 - 4 = -1 - 4 = -5$$

Calculating the numerator:

$$ i(i + 2) = i^2 + 2i = -1 + 2i $$

Thus,

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

3. Combine the results

Now, we multiply ( (5 - 12i) ) with ( \left( \frac{1}{5} - \frac{2}{5}i \right) ):

$$ (5 - 12i) \cdot \left( \frac{1}{5} - \frac{2}{5}i \right) $$

Distributing each part:

$$ = 5 \cdot \frac{1}{5} - 5 \cdot \frac{2}{5}i - 12i \cdot \frac{1}{5} + 12i \cdot \frac{2}{5}i $$

Calculating each term:

1. ( 5 \cdot \frac{1}{5} = 1 )
2. ( -5 \cdot \frac{2}{5}i = -2i )
3. ( -12i \cdot \frac{1}{5} = -\frac{12}{5}i )
4. ( 12i \cdot \frac{2}{5}i = \frac{24}{5}(-1) = -\frac{24}{5} )

Now, combine all terms:

$$ 1 - 2i - \frac{12}{5}i - \frac{24}{5} = 1 - \frac{24}{5} + \left(-2 - \frac{12}{5}\right)i $$

Convert ( 1 ) to a fraction:

$$ 1 = \frac{5}{5} \Rightarrow \frac{5 - 24}{5} = -\frac{19}{5} $$

And combining the imaginary parts:

$$ -2 = -\frac{10}{5} \Rightarrow -\frac{10}{5} - \frac{12}{5} = -\frac{22}{5}i $$

Final result:

$$ -\frac{19}{5} - \frac{22}{5}i $$