Answer :
To solve the problem [tex]\((8 - 5i)^2\)[/tex], follow these steps:
1. Understand the Problem:
We need to multiply the complex number [tex]\((8 - 5i)\)[/tex] by itself.
2. Use the Formula for Squaring a Binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
Here [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate Each Part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25(i^2)\)[/tex], and since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25(-1) = -25\)[/tex].
4. Combine the Parts:
Combine the real and imaginary parts:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]
5. Conclusion:
The simplified product is [tex]\(39 - 80i\)[/tex].
Therefore, the correct choice from the given options is [tex]\(39 - 80i\)[/tex].
1. Understand the Problem:
We need to multiply the complex number [tex]\((8 - 5i)\)[/tex] by itself.
2. Use the Formula for Squaring a Binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
Here [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate Each Part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25(i^2)\)[/tex], and since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25(-1) = -25\)[/tex].
4. Combine the Parts:
Combine the real and imaginary parts:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]
5. Conclusion:
The simplified product is [tex]\(39 - 80i\)[/tex].
Therefore, the correct choice from the given options is [tex]\(39 - 80i\)[/tex].
