Find the difference of the complex numbers.

[tex]\[ (5 + 9i) - (-4 - 2i) \][/tex]

A. [tex]\( 1 + 11i \)[/tex]

B. [tex]\( 9 + 7i \)[/tex]

C. [tex]\( 1 + 7i \)[/tex]

D. [tex]\( 9 + 11i \)[/tex]



Answer :

To solve for the difference of the complex numbers [tex]\( (5 + 9i) \)[/tex] and [tex]\( (-4 - 2i) \)[/tex], follow these steps:

1. Understand the problem:
We need to find the difference of two complex numbers. Recall that a complex number has a real part and an imaginary part, and they are generally written in the form [tex]\( a + bi \)[/tex], where [tex]\( a \)[/tex] is the real part, and [tex]\( bi \)[/tex] is the imaginary part.

2. Set up the expression:
We are given the complex numbers [tex]\( (5 + 9i) \)[/tex] and [tex]\( (-4 - 2i) \)[/tex] and asked to find their difference:
[tex]\[ (5 + 9i) - (-4 - 2i) \][/tex]

3. Distribute the negative sign:
When subtracting complex numbers, distribute the negative sign across the second complex number:
[tex]\[ (5 + 9i) - (-4 - 2i) = (5 + 9i) + (4 + 2i) \][/tex]

4. Combine like terms:
Add the real parts together and the imaginary parts together:
[tex]\[ \text{Real part:} \quad 5 + 4 = 9 \][/tex]
[tex]\[ \text{Imaginary part:} \quad 9i + 2i = 11i \][/tex]

5. Write the result:
Combine the results from step 4 to form the difference:
[tex]\[ 9 + 11i \][/tex]

Thus, the difference of the complex numbers [tex]\( (5 + 9i) \)[/tex] and [tex]\( (-4 - 2i) \)[/tex] is [tex]\( 9 + 11i \)[/tex].

The correct answer is:
[tex]\[ \boxed{9 + 11i} \][/tex]
Therefore, the answer aligns with option D.