Express the following in the form a+bi
`(5-6i)^(2)`

Answer

To express \((5 - 6i)^2\) in the form \(a + bi\), we will follow these steps:

### Step 1: Apply the formula for the square of a binomial
The square of a binomial \((a - b)^2\) can be expanded using the formula:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
In our case, we have \(a = 5\) and \(b = 6i\).

### Step 2: Substitute the values into the formula
Substituting \(a\) and \(b\) into the formula gives:
\[ (5 - 6i)^2 = 5^2 - 2 \cdot 5 \cdot 6i + (6i)^2 \]

### Step 3: Calculate each term
Now we calculate each term:
- \(5^2 = 25\)
- \(-2 \cdot 5 \cdot 6i = -60i\)
- \((6i)^2 = 36i^2\)

### Step 4: Substitute the calculated terms back into the expression
Now substituting these values back, we get:
\[ (5 - 6i)^2 = 25 - 60i + 36i^2 \]

### Step 5: Replace \(i^2\) with \(-1\)
Since \(i^2 = -1\), we replace \(36i^2\) with \(36 \cdot (-1)\):
\[ (5 - 6i)^2 = 25 - 60i - 36 \]

### Step 6: Combine like terms
Now combine the real parts:
\[ 25 - 36 = -11 \]
Thus, we have:
\[ (5 - 6i)^2 = -11 - 60i \]

### Final Result
The expression in the form \(a + bi\) is:
\[ -11 - 60i \]

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