Question

multiply.

\\(3 + 5i)^2\\

\\(3 + 5i)^2 = \square\\
(type your answer in the form \\(a + bi\\).)

Explanation:

🆕 New Concept Discovered: Operations with Complex Numbers
Working with the imaginary unit \( i \) where \( i^2 = -1 \)

Step 1: Expand the expression

To square the binomial, we can write it as a product of two identical factors:

$$ (3 + 5i)^2 = (3 + 5i)(3 + 5i) $$

Using the FOIL method (First, Outer, Inner, Last) to multiply the terms:

• First: \( 3 \times 3 = 9 \)
• Outer: \( 3 \times 5i = 15i \)
• Inner: \( 5i \times 3 = 15i \)
• Last: \( 5i \times 5i = 25i^2 \)

Combine these parts:

$$ (3 + 5i)^2 = 9 + 15i + 15i + 25i^2 $$

Step 2: Simplify the terms

Combine the imaginary terms in the middle:

$$ 15i + 15i = 30i $$

This gives:

$$ 9 + 30i + 25i^2 $$

Step 3: Substitute \( i^2 = -1 \)

Since \( i \) is the imaginary unit, its square is defined as \( i^2 = -1 \). Substitute this value into the expression:

$$ 9 + 30i + 25(-1) $$

$$ 9 + 30i - 25 $$

Step 4: Combine real parts

Combine the real numbers to write the final answer in the standard complex form \( a + bi \):

$$ (9 - 25) + 30i $$

$$ -16 + 30i $$

Answer:

-16 + 30i