Question

use the complex conjugate to divide these complex numbers.\\(\frac{2 - 6i}{4 + 2i}\\)\\(-\frac{?}{\square} - \frac{\square}{\square}i\\)\\(\text{remember to simplify.}\\)

Explanation:

Step1: Multiply numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of \(4 + 2i\) is \(4 - 2i\). So we multiply the fraction \(\frac{2 - 6i}{4 + 2i}\) by \(\frac{4 - 2i}{4 - 2i}\):
\[
\frac{(2 - 6i)(4 - 2i)}{(4 + 2i)(4 - 2i)}
\]

Step2: Expand the numerator and the denominator.

First, expand the numerator using the distributive property (FOIL method):
\[

$$\begin{align*} (2 - 6i)(4 - 2i)&=2\times4 - 2\times2i - 6i\times4 + 6i\times2i\\ &=8 - 4i - 24i + 12i^2 \end{align*}$$

\]
Since \(i^2=-1\), we substitute that in:
\[
8 - 28i + 12\times(-1)=8 - 28i - 12=-4 - 28i
\]
Now, expand the denominator using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), where \(a = 4\) and \(b = 2i\):
\[
4^2-(2i)^2 = 16 - 4i^2
\]
Substitute \(i^2 = -1\):
\[
16 - 4\times(-1)=16 + 4 = 20
\]
So now our fraction is \(\frac{-4 - 28i}{20}\).

Step3: Simplify the fraction.

We can split the fraction into two parts:
\[
\frac{-4}{20}-\frac{28i}{20}
\]
Simplify each fraction:
\[
-\frac{1}{5}-\frac{7}{5}i
\]

Answer:

The first numerator is \(1\), the first denominator is \(5\), the second numerator is \(7\), and the second denominator is \(5\). So filling in the boxes: \(-\frac{\boldsymbol{1}}{\boldsymbol{5}}-\frac{\boldsymbol{7}}{\boldsymbol{5}}i\)