Question

multiply and simplify the product.
\\((3 - 5i)(-2 + 4i)\\)
select the product.

\\(14 + 2i\\)
\\(14 + 22i\\)
\\(15 + 22i\\)
\\(26 + 2i\\)

Explanation:

🆕 New Concept Discovered: Operations with Complex Numbers
Multiplying binomials containing the imaginary unit \( i \)

Step 1: Distribute the terms

Multiply the two binomials using the FOIL method (First, Outer, Inner, Last):

$$ (3 - 5i)(-2 + 4i) = (3)(-2) + (3)(4i) + (-5i)(-2) + (-5i)(4i) $$

$$ = -6 + 12i + 10i - 20i^2 $$

Step 2: Simplify using \( i^2 = -1 \)

Since the imaginary unit \( i \) is defined as \( \sqrt{-1} \), its square is \( -1 \):

$$ -20i^2 = -20(-1) = 20 $$

Substitute this back into the expression:

$$ -6 + 12i + 10i + 20 $$

Step 3: Combine like terms

Group the real parts together and the imaginary parts together:

$$ (-6 + 20) + (12i + 10i) $$

$$ = 14 + 22i $$

Answer:

\( 14 + 22i \)