To simplify the expression [tex]\(\frac{1 + 2i}{3 + i}\)[/tex], follow these steps:
### Step 1: Multiply by the Conjugate
To eliminate the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(3 + i\)[/tex] is [tex]\(3 - i\)[/tex].
[tex]\[
\frac{1 + 2i}{3 + i} \times \frac{3 - i}{3 - i} = \frac{(1 + 2i)(3 - i)}{(3 + i)(3 - i)}
\][/tex]
### Step 2: Simplify the Denominator
First, simplify the denominator using the difference of squares:
[tex]\[
(3 + i)(3 - i) = 3^2 - (i)^2 = 9 - (-1) = 9 + 1 = 10
\][/tex]
### Step 3: Expand and Simplify the Numerator
Expand the numerator:
[tex]\[
(1 + 2i)(3 - i) = 1 \cdot 3 + 1 \cdot (-i) + 2i \cdot 3 + 2i \cdot (-i)
\][/tex]
Calculate each term:
[tex]\[
= 3 - i + 6i - 2i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 3 - i + 6i - 2(-1) = 3 - i + 6i + 2 = 3 + 2 + 5i = 5 + 5i
\][/tex]
### Step 4: Combine Results
Now, we have:
[tex]\[
\frac{(1 + 2i)(3 - i)}{10} = \frac{5 + 5i}{10}
\][/tex]
Simplify by dividing both the real and imaginary parts by the denominator:
[tex]\[
= \frac{5}{10} + \frac{5i}{10} = 0.5 + 0.5i
\][/tex]
### Final Answer
Therefore, the simplified form of [tex]\(\frac{1 + 2i}{3 + i}\)[/tex] is:
[tex]\[
\boxed{0.5 + 0.5i}
\][/tex]
So, in the given expression:
[tex]\[
\frac{1}{[?]}+\square i
\][/tex]
We have:
[tex]\[
\frac{1}{2} + \frac{1}{2}i \quad \text{where}\, \frac{1}{2}=0.5 \, \text{and}\, \frac{1}{2}i=0.5i
\][/tex]
