Express ` ( 4 - 5/2 i)^(2)` in the form of a + ib.

Answer

To express \( (4 - \frac{5}{2} i)^{2} \) in the form of \( a + ib \), we can follow these steps:

### Step 1: Identify the components
We have \( a = 4 \) and \( b = \frac{5}{2} \).

### Step 2: Use the formula for squaring a binomial
We will use the formula \( (a - b)^2 = a^2 - 2ab + b^2 \).

### Step 3: Substitute the values into the formula
Substituting \( a = 4 \) and \( b = \frac{5}{2} i \):
\[ (4 - \frac{5}{2} i)^{2} = 4^{2} - 2 \cdot 4 \cdot \frac{5}{2} i + \left(\frac{5}{2} i\right)^{2} \]

### Step 4: Calculate each term
1. Calculate \( 4^{2} \):
\[ 4^{2} = 16 \]
2. Calculate \( -2 \cdot 4 \cdot \frac{5}{2} i \):
\[ -2 \cdot 4 \cdot \frac{5}{2} i = -20 i \]
3. Calculate \( \left(\frac{5}{2} i\right)^{2} \):
\[ \left(\frac{5}{2} i\right)^{2} = \left(\frac{5}{2}\right)^{2} i^{2} = \frac{25}{4} (-1) = -\frac{25}{4} \]

### Step 5: Combine the results
Now, we combine all the terms:
\[ (4 - \frac{5}{2} i)^{2} = 16 - 20 i - \frac{25}{4} \]

### Step 6: Convert to a common denominator
To combine \( 16 \) and \( -\frac{25}{4} \), we convert \( 16 \) to a fraction with a denominator of 4:
\[ 16 = \frac{64}{4} \]
Thus,
\[ (4 - \frac{5}{2} i)^{2} = \frac{64}{4} - \frac{25}{4} - 20 i = \frac{39}{4} - 20 i \]

### Step 7: Write in the form \( a + ib \)
Now we can express the result in the form \( a + ib \):
\[ (4 - \frac{5}{2} i)^{2} = \frac{39}{4} - 20 i \]
where \( a = \frac{39}{4} \) and \( b = -20 \).

### Final Answer
\[ (4 - \frac{5}{2} i)^{2} = \frac{39}{4} - 20 i \]

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