Express the following in the form a + ib
`(2+ 5i)/(3-2i) + (2-5i)/(3 + 2i)`

Answer

To solve the expression \((2 + 5i)/(3 - 2i) + (2 - 5i)/(3 + 2i)\) and express it in 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:
\[ \frac{(2 + 5i)(3 + 2i) + (2 - 5i)(3 - 2i)}{(3 - 2i)(3 + 2i)} \]

### Step 3: Simplify the numerator
Now we will expand the numerator:
1. Expand \((2 + 5i)(3 + 2i)\):
\[ = 2 \cdot 3 + 2 \cdot 2i + 5i \cdot 3 + 5i \cdot 2i = 6 + 4i + 15i + 10i^2 = 6 + 19i - 10 \quad (\text{since } i^2 = -1) = -4 + 19i \]

2. Expand \((2 - 5i)(3 - 2i)\):
\[ = 2 \cdot 3 - 2 \cdot 2i - 5i \cdot 3 + 5i \cdot 2i = 6 - 4i - 15i - 10i^2 = 6 - 19i + 10 = 16 - 19i \]

Now, combine the results:
\[ (-4 + 19i) + (16 - 19i) = -4 + 16 + 19i - 19i = 12 \]

### Step 4: Simplify the denominator
Now we simplify the denominator:
\[ (3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4(-1) = 9 + 4 = 13 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:
\[ \frac{12}{13} \]

### Step 6: Express in the form \(a + ib\)
Since there is no imaginary part, we can express this as:
\[ \frac{12}{13} + 0i \]
Thus, \(a = \frac{12}{13}\) and \(b = 0\).

### Final Answer
The expression in the form \(a + ib\) is:
\[ \frac{12}{13} + 0i \]