Question

question 4

multiply the conjugates:

\\(-7 + i)(-7 - i)\\

Explanation:

🆕 New Concept Discovered: Multiplying Complex Conjugates
Multiplying complex numbers to get a real number

Step 1: Identify the pattern

The expression is a product of complex conjugates in the form \( (a + bi)(a - bi) \).

Here, we have:

$$ a = -7 $$

$$ b = 1 $$

When we multiply conjugates, the middle terms cancel out, following the difference of squares pattern:

$$ (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 i^2 $$

Since \( i^2 = -1 \), this simplifies to:

$$ a^2 + b^2 $$

Step 2: Substitute the values

Using the formula \( a^2 + b^2 \):

$$ (-7)^2 + (1)^2 $$

Step 3: Simplify the expression

Calculate the squares and add them together:

$$ (-7)^2 = 49 $$

$$ (1)^2 = 1 $$

$$ 49 + 1 = 50 $$

Answer:

50