High School

Multiply and simplify the product: [tex](8-5i)^2[/tex]

Select the product:

A. 39
B. 89
C. 39 - 80i
D. 89 - 80i

Answer :

To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], we need to apply the formula for the square of a binomial:

[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]

For the expression [tex]\((8 - 5i)^2\)[/tex], set [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Plug these values into the formula:

1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]

2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times 5i = -80i
\][/tex]

3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2
\][/tex]

Since [tex]\(i^2 = -1\)[/tex], this becomes:
[tex]\[
25 \times -1 = -25
\][/tex]

Now, combine all these results together:

[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]

Simplify by combining the real terms:

[tex]\[
64 - 25 = 39
\][/tex]

Thus, the simplified expression is:

[tex]\[
39 - 80i
\][/tex]

So, the correct answer is [tex]\(39 - 80i\)[/tex].