`y=x^2+16x+28`
The equation above represents the graph of a parabola in the xy-plane. Which of the following represents an equivalent form of the equation that includes the minimum value of y as a constant?

Options

A. `y-28=x(x+16)`
B. `y=x^2+2(8x+14)`
C. `y=x(x+16)+28`
D. `y=(x+8)^2-36`

Answer

To rewrite the equation \( y = x^2 + 16x + 28 \) in a form that highlights the minimum value of \( y \), we will complete the square. Here’s how to do it step-by-step:

### Step 1: Identify the quadratic equation
The given equation is:
\[ y = x^2 + 16x + 28 \]

### Step 2: Rearrange the quadratic terms
We will focus on the quadratic part \( x^2 + 16x \) first.

### Step 3: Complete the square
To complete the square for \( x^2 + 16x \), we take the coefficient of \( x \), which is 16, divide it by 2 to get 8, and then square it to get \( 8^2 = 64 \).

Now we can rewrite \( x^2 + 16x \) as:
\[ x^2 + 16x = (x + 8)^2 - 64 \]
This is because \( (x + 8)^2 = x^2 + 16x + 64 \), and we subtract 64 to keep the equation balanced.

### Step 4: Substitute back into the equation
Now we substitute this back into the original equation:
\[ y = (x + 8)^2 - 64 + 28 \]

### Step 5: Simplify the equation
Now simplify the constant terms:
\[ y = (x + 8)^2 - 36 \]

### Step 6: Identify the minimum value
The expression \( (x + 8)^2 \) reaches its minimum value of 0 when \( x = -8 \). Therefore, the minimum value of \( y \) occurs at:
\[ y = 0 - 36 = -36 \]

### Final Form
Thus, the equivalent form of the equation that includes the minimum value of \( y \) as a constant is:
\[ y = (x + 8)^2 - 36 \]