To solve the expression [tex]\((2 + i)(5 - i)\)[/tex], let's do it step-by-step.
Given the complex numbers: [tex]\(2 + i\)[/tex] and [tex]\(5 - i\)[/tex].
We need to multiply these two complex numbers together:
[tex]\[
(2 + i)(5 - i)
\][/tex]
Let's expand the expression using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(2 + i)(5 - i) = 2 \cdot 5 + 2 \cdot (-i) + i \cdot 5 + i \cdot (-i)
\][/tex]
Compute each term individually:
- [tex]\(2 \cdot 5 = 10\)[/tex]
- [tex]\(2 \cdot (-i) = -2i\)[/tex]
- [tex]\(i \cdot 5 = 5i\)[/tex]
- [tex]\(i \cdot (-i) = -i^2\)[/tex]
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
-i^2 = -(-1) = 1
\][/tex]
Now substitute back into our expanded expression:
[tex]\[
10 - 2i + 5i + 1
\][/tex]
Combine the real and imaginary parts separately:
- Real parts: [tex]\(10 + 1 = 11\)[/tex]
- Imaginary parts: [tex]\(-2i + 5i = 3i\)[/tex]
So the result of the expression is:
[tex]\[
11 + 3i
\][/tex]
Therefore, the equivalent expression is:
A. [tex]\(11 + 3i\)[/tex]
