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.