To solve the given expression [tex]\((5 + 3i) - (5 + 3i)(5 - 5i)\)[/tex], we will break it down step-by-step.
First, let's define the complex numbers involved:
- [tex]\( z_1 = 5 + 3i \)[/tex]
- [tex]\( z_2 = 5 - 5i \)[/tex]
### Step 1: Calculate the first term, which is simply [tex]\( z_1 \)[/tex]:
[tex]\[ z_1 = 5 + 3i \][/tex]
### Step 2: Calculate the product of [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]:
[tex]\[
z_1 \cdot z_2 = (5 + 3i)(5 - 5i)
\][/tex]
To perform this multiplication, we apply the distributive property (FOIL method):
[tex]\[
(5 + 3i)(5 - 5i) = 5 \cdot 5 + 5 \cdot (-5i) + 3i \cdot 5 + 3i \cdot -5i
\][/tex]
Simplifying each term:
[tex]\[
= 25 - 25i + 15i - 15i^2
\][/tex]
Since [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
-15i^2 = -15(-1) = 15
\][/tex]
Now, simplifying further:
[tex]\[
25 - 25i + 15i + 15
\][/tex]
Combine the real and imaginary parts:
[tex]\[
(25 + 15) + (-25i + 15i) = 40 - 10i
\][/tex]
So, the result of [tex]\( z_1 \cdot z_2 \)[/tex] is:
[tex]\[ 40 - 10i \][/tex]
### Step 3: Calculate the value of the expression:
[tex]\[
(5 + 3i) - (40 - 10i)
\][/tex]
Subtract the corresponding real parts and imaginary parts separately:
[tex]\[
= (5 - 40) + (3i - (-10i))
\][/tex]
[tex]\[
= -35 + 13i
\][/tex]
Thus, the value of the expression [tex]\((5 + 3i) - (5 + 3i)(5 - 5i)\)[/tex] is:
[tex]\[ -35 + 13i \][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-35 + 13i}
\][/tex]
Option B matches this result. So, the correct answer is:
B. [tex]\(-35 + 13i\)[/tex]
