If the equation `y=3x^(2)+18x-13` is written in the form `y=a(x-h)^(2)+k`, what are the values of h and k?

Options

A. `h=3, k=14`
B. `h=-3, k=-40`
C. `h=3, k=-13`
D. `h=-3, k=-22`

Answer

To rewrite the equation \( y = 3x^2 + 18x - 13 \) in the form \( y = a(x - h)^2 + k \), we will complete the square. Here’s how to do it step by step:

### Step 1: Factor out the coefficient of \( x^2 \)
The first step is to factor out the coefficient of \( x^2 \) from the first two terms. In this case, the coefficient is 3.

\[ y = 3(x^2 + 6x) - 13 \]

### Step 2: Complete the square
Next, we need to complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of \( x \) (which is 6), square it, and add and subtract that value inside the parentheses.

Half of 6 is 3, and squaring it gives us \( 3^2 = 9 \).

\[ y = 3(x^2 + 6x + 9 - 9) - 13 \]
\[ y = 3((x + 3)^2 - 9) - 13 \]

### Step 3: Distribute and simplify
Now, distribute the 3 and simplify the equation.

\[ y = 3(x + 3)^2 - 27 - 13 \]
\[ y = 3(x + 3)^2 - 40 \]

### Step 4: Identify \( h \) and \( k \)
Now, we can compare this equation with the standard form \( y = a(x - h)^2 + k \).

From \( y = 3(x + 3)^2 - 40 \):
- \( a = 3 \)
- \( h = -3 \) (since \( x + 3 = x - (-3) \))
- \( k = -40 \)

Thus, the values of \( h \) and \( k \) are:
\[ h = -3, \quad k = -40 \]

### Final Answer
The values of \( h \) and \( k \) are:
- \( h = -3 \)
- \( k = -40 \)