Question

question 3

\\((-3 + 7i)(-8 - 2i)\\)

\\(\bigcirc -38 - 64i\\)
\\(\bigcirc 10 - 50i\\)
\\(\bigcirc 24 - 64i\\)
\\(\bigcirc 38 - 50i\\)

Explanation:

🆕 New Concept Discovered: Multiplying Complex Numbers
Using the FOIL method and the fact that \( i^2 = -1 \)

Step 1: Expand using FOIL

To multiply two binomials, we distribute each term in the first parentheses by each term in the second parentheses:

$$ (-3 + 7i)(-8 - 2i) = (-3)(-8) + (-3)(-2i) + (7i)(-8) + (7i)(-2i) $$

Multiply the individual terms:

• First: \( (-3)(-8) = 24 \)
• Outer: \( (-3)(-2i) = 6i \)
• Inner: \( (7i)(-8) = -56i \)
• Last: \( (7i)(-2i) = -14i^2 \)

Combine these results:

$$ 24 + 6i - 56i - 14i^2 $$

Step 2: Simplify the imaginary terms

Combine the like terms containing \( i \):

$$ 6i - 56i = -50i $$

Now the expression is:

$$ 24 - 50i - 14i^2 $$

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:

$$ 24 - 50i - 14(-1) $$

$$ 24 - 50i + 14 $$

Step 4: Combine the real numbers

Add the real parts together to get the final standard form \( a + bi \):

$$ (24 + 14) - 50i $$

$$ 38 - 50i $$

Answer:

\( 38 - 50i \) (the fourth option)