Given that `i = sqrt(-1),` which of the following is equivalent to `(6i^(2)-7i)+ (3+6i)` ?

Options

A. `6i^(2) +i+3`
B. `-3-i`
C. `-9-i`
D. `10`

Answer

To solve the expression \( (6i^{2} - 7i) + (3 + 6i) \), we will follow these steps:

### Step 1: Substitute \( i^{2} \)
We know that \( i = \sqrt{-1} \), hence \( i^{2} = -1 \). We will substitute this value into the expression.

\[ 6i^{2} = 6(-1) = -6 \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression by substituting \( 6i^{2} \):

\[ (6i^{2} - 7i) + (3 + 6i) = (-6 - 7i) + (3 + 6i) \]

### Step 3: Combine like terms
Next, we will combine the constant terms and the terms with \( i \):

- Constant terms: \(-6 + 3 = -3\)
- Terms with \( i \): \(-7i + 6i = -i\)

Putting it all together, we have:

\[ -3 - i \]

### Final Answer
Thus, the expression \( (6i^{2} - 7i) + (3 + 6i) \) simplifies to:

\[ -3 - i \]