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):

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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: \(-\frac{\boldsymbol{1}}{\boldsymbol{5}}-\frac{\boldsymbol{7}}{\boldsymbol{5}}i\)