Answer :
To solve [tex]\((8 - 5i)^2\)[/tex] and find the product in the form [tex]\(a + bi\)[/tex], follow these steps:
1. Expand the Expression:
Start by using the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. Here, you apply the formula as:
[tex]\((8 - 5i)^2 = (8)^2 - 2 \times 8 \times 5i + (5i)^2\)[/tex].
2. Calculate Each Part:
- Square the Real Part (8):
[tex]\(8^2 = 64\)[/tex].
- Calculate the Imaginary Part Coefficient:
[tex]\(-2 \times 8 \times 5i = -80i\)[/tex].
- Square the Imaginary Part (5i):
[tex]\((5i)^2 = 25(i^2)\)[/tex].
Since [tex]\(i^2 = -1\)[/tex], it follows that [tex]\(25(i^2) = 25 \times -1 = -25\)[/tex].
3. Combine the Results:
Sum all the calculated parts:
[tex]\[
64 + (-25) + (-80i) = 39 - 80i
\][/tex]
4. Select the Product:
The calculated expression is [tex]\(39 - 80i\)[/tex], which matches the option [tex]\(\boxed{39 - 80i}\)[/tex].
1. Expand the Expression:
Start by using the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]. Here, you apply the formula as:
[tex]\((8 - 5i)^2 = (8)^2 - 2 \times 8 \times 5i + (5i)^2\)[/tex].
2. Calculate Each Part:
- Square the Real Part (8):
[tex]\(8^2 = 64\)[/tex].
- Calculate the Imaginary Part Coefficient:
[tex]\(-2 \times 8 \times 5i = -80i\)[/tex].
- Square the Imaginary Part (5i):
[tex]\((5i)^2 = 25(i^2)\)[/tex].
Since [tex]\(i^2 = -1\)[/tex], it follows that [tex]\(25(i^2) = 25 \times -1 = -25\)[/tex].
3. Combine the Results:
Sum all the calculated parts:
[tex]\[
64 + (-25) + (-80i) = 39 - 80i
\][/tex]
4. Select the Product:
The calculated expression is [tex]\(39 - 80i\)[/tex], which matches the option [tex]\(\boxed{39 - 80i}\)[/tex].
