Question

multiply and simplify the product.
\\((8 - 5i)^2\\)
select the product.

• 39
• 89
• 39 - 80i
• 89 - 80i

Explanation:

🆕 New Concept Discovered: Multiplying Binomials and Special Products
Squaring a binomial using the pattern \( (a - b)^2 = a^2 - 2ab + b^2 \).

Step 1: Identify the expression and the pattern

The given expression is:

$$ (8 - 5i)^2 $$

This is a binomial squared in the form \( (a - b)^2 \), which expands to:

$$ a^2 - 2ab + b^2 $$

Here, let \( a = 8 \) and \( b = 5i \).

Step 2: Expand the expression

Apply the values of \( a \) and \( b \) to the pattern:

$$ (8 - 5i)^2 = (8)^2 - 2(8)(5i) + (5i)^2 $$

Simplify each term:

1. First term:

$$ (8)^2 = 64 $$

1. Middle term:

$$ -2(8)(5i) = -80i $$

1. Last term:

$$ (5i)^2 = 25i^2 $$

Combine them:

$$ 64 - 80i + 25i^2 $$

Step 3: Simplify using the definition of \( i^2 \)

Recall that the imaginary unit \( i \) has the property:

$$ i^2 = -1 $$

Substitute \( -1 \) for \( i^2 \):

$$ 64 - 80i + 25(-1) $$

$$ 64 - 80i - 25 $$

Combine the real numbers:

$$ (64 - 25) - 80i $$

$$ 39 - 80i $$

Answer:

\( 39 - 80i \)