what is the quotient of the complex number $4 - 3i$ divided by its conjugate?\
$\frac{24}{25} - \frac{7}{25}i$ $\frac{7}{25} + \frac{24}{25}i$ $\frac{7}{25} - \frac{24}{25}i$ $\frac{24}{25} + \frac{7}{25}i$

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Accepted Answer

# Explanation:
## Step1: Find the conjugate of $4 - 3i$
The conjugate of a complex number $a - bi$ is $a + bi$. So, the conjugate of $4 - 3i$ is $4 + 3i$.

## Step2: Set up the division
We need to find $\frac{4 - 3i}{4 + 3i}$. To simplify this, we multiply the numerator and denominator by the conjugate of the denominator, which is $4 - 3i$.

## Step3: Multiply the numerator and denominator
$$
\begin{align*}
\frac{(4 - 3i)(4 - 3i)}{(4 + 3i)(4 - 3i)}&=\frac{16 - 12i - 12i + 9i^2}{16 - (3i)^2}\
&=\frac{16 - 24i + 9(-1)}{16 - 9(-1)} \quad (\text{since } i^2 = -1)\
&=\frac{16 - 24i - 9}{16 + 9}\
&=\frac{7 - 24i}{25}\
&=\frac{7}{25} - \frac{24}{25}i
\end{align*}
$$

# Answer: $\frac{7}{25} - \frac{24}{25}i$ (corresponding to the third option)

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Question

Explanation:

Step1: Find the conjugate of \(4 - 3i\)

Step2: Set up the division

Step3: Multiply the numerator and denominator

Answer: