Question

evaluate \\((2-5i)(p+q)(i)\\) when \\(p=2\\) and \\(q=5i\\).

\\(\bigcirc\\) \\(29i\\)
\\(\bigcirc\\) \\(29i-20\\)
\\(\bigcirc\\) \\(-21i\\)
\\(\bigcirc\\) \\(29\\)

Explanation:

⚡ Using what you learned: Introduction to Complex Numbers

Step 1: Substitute the given values

Substitute \( p = 2 \) and \( q = 5i \) into the expression \( (2 - 5i)(p + q)(i) \):

$$ (2 - 5i)(2 + 5i)(i) $$

Step 2: Multiply the complex conjugates

Use the difference of squares formula \( (a - bi)(a + bi) = a^2 + b^2 \):

$$ (2 - 5i)(2 + 5i) = 2^2 - (5i)^2 = 4 - 25i^2 $$

Since \( i^2 = -1 \):

$$ 4 - 25(-1) = 4 + 25 = 29 $$

Step 3: Multiply by the remaining term

Multiply the result by the remaining factor \( i \):

$$ 29 \cdot i = 29i $$

Answer:

29i