The vertex of the parabola `x^2+8x+12y+4=0` is (i) `(-4,1)` (ii)`(4,-1)` (iii)`(-4,-1)`

Options

A. (-4, 1)
B. (4, -1)
C. (-4, -1)
D. (4, 1)

Explanation

To find the vertex of the parabola given by the equation \( x^2 + 8x + 12y + 4 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ x^2 + 8x + 12y + 4 = 0 \]
We can rearrange it to isolate the \( y \) term:
\[ 12y = -x^2 - 8x - 4 \]
Now, we will simplify this further.

### Step 2: Complete the square for the \( x \) terms
To complete the square for the expression \( x^2 + 8x \), we take the coefficient of \( x \) (which is 8), divide it by 2 (giving us 4), and square it (resulting in 16). We will add and subtract 16 in the equation:
\[ 12y = - (x^2 + 8x + 16 - 16) - 4 \]
This simplifies to:
\[ 12y = - (x + 4)^2 + 16 - 4 \]
\[ 12y = - (x + 4)^2 + 12 \]

### Step 3: Rearrange to standard form
Now, we can rearrange this to express \( y \):
\[ 12y = - (x + 4)^2 + 12 \]
\[ 12y = - (x + 4)^2 + 12 \]
\[ y = -\frac{1}{12}(x + 4)^2 + 1 \]

### Step 4: Identify the vertex
The equation is now in the standard form of a parabola:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola. From our equation, we can see:
- \( h = -4 \)
- \( k = 1 \)

Thus, the vertex of the parabola is:
\[ (-4, 1) \]

### Conclusion
The vertex of the parabola \( x^2 + 8x + 12y + 4 = 0 \) is \( (-4, 1) \), which corresponds to option (i).

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