Question

question 7 of 10
divide the following complex numbers:
\\(\frac{(5 - i)}{(4 + 3i)}\\)
\\(\bigcirc\\) a. \\(\frac{23}{25} - \frac{19}{25}i\\)
\\(\bigcirc\\) b. \\(\frac{23}{7} - \frac{19}{7}i\\)
\\(\bigcirc\\) c. \\(\frac{17}{7} - \frac{19}{7}i\\)
\\(\bigcirc\\) d. \\(\frac{17}{25} - \frac{19}{25}i\\)

Explanation:

Step1: Multiply numerator and denominator by conjugate of denominator

The conjugate of \(4 + 3i\) is \(4 - 3i\). So we have:
\[
\frac{(5 - i)(4 - 3i)}{(4 + 3i)(4 - 3i)}
\]

Step2: Expand numerator and denominator

First, expand the numerator \((5 - i)(4 - 3i)\):
\[

$$\begin{align*} (5 - i)(4 - 3i)&=5\times4-5\times3i - i\times4+i\times3i\\ &=20 - 15i - 4i+3i^{2}\\ &=20 - 19i+3\times(- 1)\\ &=20 - 19i - 3\\ &=17 - 19i \end{align*}$$

\]
Then, expand the denominator \((4 + 3i)(4 - 3i)\) using the formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 4\) and \(b = 3i\):
\[

$$\begin{align*} (4 + 3i)(4 - 3i)&=4^{2}-(3i)^{2}\\ &=16-9i^{2}\\ &=16 - 9\times(-1)\\ &=16 + 9\\ &=25 \end{align*}$$

\]

Step3: Write the result as a complex number

Now we have \(\frac{17 - 19i}{25}\), which can be written as \(\frac{17}{25}-\frac{19}{25}i\)

Answer:

D. \(\frac{17}{25}-\frac{19}{25}i\)