The vertex of the parabola `x^(2)+12x-9y=0` is

Options

A. (6,-4)
B. (-6,4)
C. (6,4)
D. (-6,-4)

Answer

To find the vertex of the parabola given by the equation \( x^2 + 12x - 9y = 0 \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate the \( y \) term:
\[ x^2 + 12x = 9y \]

### Step 2: Complete the square for the \( x \) terms
To complete the square for the expression \( x^2 + 12x \), we take half of the coefficient of \( x \) (which is 12), square it, and add and subtract it:
\[ x^2 + 12x + 36 - 36 = 9y \]
This simplifies to:
\[ (x + 6)^2 - 36 = 9y \]

### Step 3: Rearrange to standard form
Now, we can rearrange the equation to express \( y \):
\[ (x + 6)^2 = 9y + 36 \]
\[ (x + 6)^2 = 9(y + 4) \]

### Step 4: Identify the vertex
Now, we can compare this with the standard form of a parabola:
\[ (x - h)^2 = 4a(y - k) \]
From our equation \( (x + 6)^2 = 9(y + 4) \), we can identify:
- \( h = -6 \)
- \( k = -4 \)
- \( 4a = 9 \) (which gives \( a = \frac{9}{4} \))

Thus, the vertex of the parabola is:
\[ (-6, -4) \]

### Conclusion
The vertex of the parabola \( x^2 + 12x - 9y = 0 \) is \( (-6, -4) \).

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