Answer :
To solve the problem of dividing the complex numbers [tex]\((6 - 2i)\)[/tex] by [tex]\((4 + i)\)[/tex], you can follow these steps:
1. Identify the Numbers:
- The numerator is [tex]\(6 - 2i\)[/tex].
- The denominator is [tex]\(4 + i\)[/tex].
2. Multiply by the Conjugate:
- Multiply both the numerator and the denominator by the conjugate of the denominator, which is [tex]\(4 - i\)[/tex]. This will help eliminate the imaginary part in the denominator.
3. Perform the Multiplication:
- The conjugate of [tex]\(4 + i\)[/tex] is [tex]\(4 - i\)[/tex].
- Multiply:
[tex]\((6 - 2i) \times (4 - i) = (6 \times 4) + (6 \times -i) + (-2i \times 4) + (-2i \times -i)\)[/tex].
- Simplify:
[tex]\(= 24 - 6i - 8i + 2i^2\)[/tex].
Remember that [tex]\(i^2 = -1\)[/tex], so [tex]\(2i^2 = 2(-1) = -2\)[/tex].
- Further Simplify:
[tex]\(= 24 - 6i - 8i - 2 = 22 - 14i\)[/tex].
4. Simplify the Denominator:
- Calculate: [tex]\((4 + i) \times (4 - i) = 4^2 - i^2\)[/tex].
- [tex]\(= 16 - (-1) = 16 + 1 = 17\)[/tex].
5. Combine the Results:
- Now, place the results over each other:
[tex]\(\frac{(22 - 14i)}{17}\)[/tex].
6. Separate into Real and Imaginary Parts:
- Real part: [tex]\(\frac{22}{17}\)[/tex].
- Imaginary part: [tex]\(-\frac{14}{17}i\)[/tex].
Thus, the final answer is:
[tex]\[
\frac{22}{17} - \frac{14}{17}i
\][/tex]
Comparing this with the given options, it matches with option b.) [tex]\(\frac{22}{17} - \frac{14}{17}i\)[/tex].
1. Identify the Numbers:
- The numerator is [tex]\(6 - 2i\)[/tex].
- The denominator is [tex]\(4 + i\)[/tex].
2. Multiply by the Conjugate:
- Multiply both the numerator and the denominator by the conjugate of the denominator, which is [tex]\(4 - i\)[/tex]. This will help eliminate the imaginary part in the denominator.
3. Perform the Multiplication:
- The conjugate of [tex]\(4 + i\)[/tex] is [tex]\(4 - i\)[/tex].
- Multiply:
[tex]\((6 - 2i) \times (4 - i) = (6 \times 4) + (6 \times -i) + (-2i \times 4) + (-2i \times -i)\)[/tex].
- Simplify:
[tex]\(= 24 - 6i - 8i + 2i^2\)[/tex].
Remember that [tex]\(i^2 = -1\)[/tex], so [tex]\(2i^2 = 2(-1) = -2\)[/tex].
- Further Simplify:
[tex]\(= 24 - 6i - 8i - 2 = 22 - 14i\)[/tex].
4. Simplify the Denominator:
- Calculate: [tex]\((4 + i) \times (4 - i) = 4^2 - i^2\)[/tex].
- [tex]\(= 16 - (-1) = 16 + 1 = 17\)[/tex].
5. Combine the Results:
- Now, place the results over each other:
[tex]\(\frac{(22 - 14i)}{17}\)[/tex].
6. Separate into Real and Imaginary Parts:
- Real part: [tex]\(\frac{22}{17}\)[/tex].
- Imaginary part: [tex]\(-\frac{14}{17}i\)[/tex].
Thus, the final answer is:
[tex]\[
\frac{22}{17} - \frac{14}{17}i
\][/tex]
Comparing this with the given options, it matches with option b.) [tex]\(\frac{22}{17} - \frac{14}{17}i\)[/tex].
