Question

cc.2.1.hs.f.6
simplify \\(\frac{4 - 2i}{7 + 3i}\\).
remember to use the conjugate.
\\(\bigcirc\\) \\(\frac{11}{29} - \frac{13}{29}i\\)
\\(\bigcirc\\) \\(\frac{11}{29} - \frac{14}{29}i\\)
\\(\bigcirc\\) \\(\frac{13}{29} - \frac{17}{29}i\\)
\\(\bigcirc\\) \\(\frac{17}{20} - \frac{1}{20}i\\)

Explanation:

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

The conjugate of \(7 + 3i\) is \(7 - 3i\). So we have:
\[
\frac{4 - 2i}{7 + 3i} \times \frac{7 - 3i}{7 - 3i}
\]

Step2: Expand the numerator using the distributive property (FOIL method).

\[

$$\begin{align*} (4 - 2i)(7 - 3i)&=4\times7+4\times(-3i)-2i\times7+(-2i)\times(-3i)\\ &=28 - 12i - 14i + 6i^2 \end{align*}$$

\]
Since \(i^2=- 1\), we substitute:
\[
28-26i + 6\times(-1)=28-26i - 6=22 - 26i
\]

Step3: Expand the denominator using the formula \((a + b)(a - b)=a^2 - b^2\).

\[
(7 + 3i)(7 - 3i)=7^2-(3i)^2=49 - 9i^2
\]
Substitute \(i^2 = - 1\):
\[
49-9\times(-1)=49 + 9=58
\]

Step4: Now we have the fraction \(\frac{22 - 26i}{58}\). Simplify by dividing numerator and denominator by 2.

\[
\frac{22\div2-26i\div2}{58\div2}=\frac{11 - 13i}{29}=\frac{11}{29}-\frac{13}{29}i
\]

Answer:

A. \(\frac{11}{29}-\frac{13}{29}i\)