If the expression `(1+2i)/(4 + 2i)` is rewritten as a complex number in the form of `a + bi,` what is the value of a ? (Note `i=sqrt-1)`

Answer

To solve the expression \((1 + 2i)/(4 + 2i)\) and rewrite it in the form \(a + bi\), we will follow these steps:

### Step 1: Multiply by the Conjugate
To eliminate the imaginary part from the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(4 + 2i\) is \(4 - 2i\).

\[ \frac{1 + 2i}{4 + 2i} \cdot \frac{4 - 2i}{4 - 2i} \]

### Step 2: Expand the Numerator
Now, we will expand the numerator:

\[ (1 + 2i)(4 - 2i) = 1 \cdot 4 + 1 \cdot (-2i) + 2i \cdot 4 + 2i \cdot (-2i) \]

Calculating each term:

- \(1 \cdot 4 = 4\)
- \(1 \cdot (-2i) = -2i\)
- \(2i \cdot 4 = 8i\)
- \(2i \cdot (-2i) = -4i^2 = 4\) (since \(i^2 = -1\))

Adding these together:

\[ 4 - 2i + 8i + 4 = 8 + 6i \]

### Step 3: Expand the Denominator
Now, we will expand the denominator:

\[ (4 + 2i)(4 - 2i) = 4^2 - (2i)^2 = 16 - 4i^2 \]

Since \(i^2 = -1\):

\[ 16 - 4(-1) = 16 + 4 = 20 \]

### Step 4: Combine the Results
Now we can write the expression as:

\[ \frac{8 + 6i}{20} \]

### Step 5: Simplify
We can simplify this by dividing both the real and imaginary parts by 20:

\[ \frac{8}{20} + \frac{6i}{20} = \frac{2}{5} + \frac{3}{10}i \]

### Step 6: Identify \(a\)
In the form \(a + bi\), we see that:

\[ a = \frac{2}{5} \]

Thus, the value of \(a\) is \(\frac{2}{5}\).

---