Which answer could be the result of two complex conjugates being multiplied together?(1 point)

Question

Responses

16i
16 i

25+4i
25 plus 4 i

20
20

9−9i

Answer 1

When two complex conjugates are multiplied together, the result is always a non-negative real number. This is because if you have a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \). The product of a complex number and its conjugate is:

\[ z \cdot \overline{z} = (a + bi)(a - bi) = a^2 + b^2 \]

Since \( a^2 \) and \( b^2 \) are both non-negative, their sum is also non-negative.

Now, let's evaluate the given options:

1. 16i - This is purely imaginary, not a non-negative real number.
2. 25 + 4i - This is a complex number with a non-zero imaginary part, not a non-negative real number.
3. 20 - This is a non-negative real number.
4. 9 - 9i - This has an imaginary part and is not a non-negative real number.

Since the only option that matches the criteria of being a non-negative real number is 20, that is the correct answer.