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We are given that the parabola has its vertex at \( (9, -14) \) and intersects the x-axis at two points. The equation of the parabola is written in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

### Vertex Form of a Parabola
The vertex form of a parabola is:
\[
y = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex. Since the vertex is at \( (9, -14) \), we can substitute this into the vertex form:
\[
y = a(x - 9)^2 - 14
\]

### Step 1: Convert Vertex Form to Standard Form
We now expand \( y = a(x - 9)^2 - 14 \):
\[
y = a(x^2 - 18x + 81) - 14
\]
\[
y = ax^2 - 18ax + 81a - 14
\]
Thus, in standard form, the equation becomes:
\[
y = ax^2 - 18ax + (81a - 14)
\]
Here, comparing it with \( y = ax^2 + bx + c \), we have:
- \( b = -18a \)
- \( c = 81a - 14 \)

### Step 2: Use the Fact That the Parabola Intersects the X-Axis
Since the parabola intersects the x-axis at two points, it has two real roots. The discriminant \( \Delta \) of the quadratic equation must be positive:
\[
\Delta = b^2 - 4ac > 0
\]
Substituting \( b = -18a \) and \( c = 81a - 14 \) into the discriminant:
\[
\Delta = (-18a)^2 - 4a(81a - 14)
\]
\[
\Delta = 324a^2 - 4a(81a - 14)
\]
\[
\Delta = 324a^2 - 324a^2 + 56a
\]
\[
\Delta = 56a
\]
For the parabola to intersect the x-axis at two points, \( \Delta > 0 \), so \( 56a > 0 \), meaning \( a > 0 \).

### Step 3: Calculate \( a + b + c \)
We are asked to find \( a + b + c \). Using \( b = -18a \) and \( c = 81a - 14 \), we calculate:
\[
a + b + c = a + (-18a) + (81a - 14)
\]
\[
a + b + c = a - 18a + 81a - 14
\]
\[
a + b + c = 64a - 14
\]
We know that \( a > 0 \). Now, we can check the possible values of \( a + b + c \) from the answer choices.

### Step 4: Test the Answer Choices
We need to find which value matches \( 64a - 14 \). Testing the possible values:
- For \( a + b + c = -12 \):
  \[
  64a - 14 = -12 \implies 64a = 2 \implies a = \frac{1}{32}
  \]
This is a positive value of \( a \), so the correct answer is:
\[
\boxed{-12}
\]

Would you like further details or have any questions? 

Here are 5 related questions to explore:
1. How is the discriminant used to determine the nature of roots in a quadratic equation?
2. What is the relationship between the vertex form and standard form of a parabola?
3. How would the equation change if the parabola did not intersect the x-axis?
4. What is the geometric significance of the vertex of a parabola?
5. Can you derive the general solution for the sum \( a + b + c \) for different vertex points?

**Tip**: When converting between vertex form and standard form, always expand carefully and compare coefficients to ensure accuracy.

In the xy-plane, a parabola has vertex (9, -14) and intersects the x-axis at two points. If the equation of the parabola is written in the form y = ax^2 + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c? A) -23, B) -19, C) -14, D) -12

Solve a quadratic equation where a parabola with a vertex at (9, -14) intersects the x-axis at two points. Learn how to use the vertex form and the discriminant to find the value of a + b + c.

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