Convert ` [(3+2i)/( 3-2i)+ (3 -2i)/(3+2i)]` in the form of `(a+ib)`.

Explanation

To convert the expression \(\frac{3+2i}{3-2i} + \frac{3-2i}{3+2i}\) into the form \(a + ib\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((3 - 2i)(3 + 2i)\).

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \frac{(3 + 2i)(3 + 2i) + (3 - 2i)(3 - 2i)}{(3 - 2i)(3 + 2i)} \]

### Step 3: Expand the numerators
Now we will expand both numerators:
1. For \((3 + 2i)(3 + 2i)\):
\[ (3 + 2i)^2 = 3^2 + 2 \cdot 3 \cdot 2i + (2i)^2 = 9 + 12i - 4 = 5 + 12i \]

2. For \((3 - 2i)(3 - 2i)\):
\[ (3 - 2i)^2 = 3^2 - 2 \cdot 3 \cdot 2i + (2i)^2 = 9 - 12i - 4 = 5 - 12i \]

### Step 4: Combine the expanded numerators
Now we combine the results:
\[ (5 + 12i) + (5 - 12i) = 5 + 12i + 5 - 12i = 10 \]

### Step 5: Expand the denominator
Now we will expand the denominator:
\[ (3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - (-4) = 9 + 4 = 13 \]

### Step 6: Combine everything
Now we can combine the results:
\[ \frac{10}{13} \]

### Step 7: Write in the form \(a + ib\)
Since there is no imaginary part, we can write:
\[ \frac{10}{13} + 0i \]

Thus, in the form \(a + ib\), we have:
\[ a = \frac{10}{13}, \quad b = 0 \]

### Final Answer:
\[ \frac{10}{13} + 0i \]

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