We are asked to find the value of $a + b + c$ for a parabola with a vertex at $(9, -14)$, written in the form $y = ax^2 + bx + c$, and which intersects the x-axis at two points.

Step 1: Vertex Form of a Parabola

The vertex form of a parabola is: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. Given that the vertex is $(9, -14)$, we can write the equation as: $y = a(x - 9)^2 - 14$

Step 2: Convert to Standard Form

To match the given standard form $y = ax^2 + bx + c$, expand the vertex form: $y = a(x^2 - 18x + 81) - 14$ $y = ax^2 - 18ax + 81a - 14$ So, in standard form: $y = ax^2 + (-18a)x + (81a - 14)$

Thus, comparing with $y = ax^2 + bx + c$, we have:

• $b = -18a$
• $c = 81a - 14$

Step 3: Find $a + b + c$

Now, calculate $a + b + c$: $a + b + c = a + (-18a) + (81a - 14) = a - 18a + 81a - 14 = 64a - 14$

Step 4: Use the fact that the parabola intersects the x-axis at two points

Since the parabola intersects the x-axis at two points, the discriminant of the quadratic equation must be positive. This implies $b^2 - 4ac > 0$. However, this information is generally to confirm that there are real roots, not for direct calculation. The question asks us to find possible values for $a + b + c$.

Step 5: Check Options

We are given four options for $a + b + c$:

• A) -23
• B) -19
• C) -14
• D) -12

Substitute these into $64a - 14 = \text{option value}$:

$64a - 14 = -23 \quad \Rightarrow \quad 64a = -9 \quad \Rightarrow \quad a = -\frac{9}{64}$ $64a - 14 = -19 \quad \Rightarrow \quad 64a = -5 \quad \Rightarrow \quad a = -\frac{5}{64}$ $64a - 14 = -14 \quad \Rightarrow \quad 64a = 0 \quad \Rightarrow \quad a = 0$ $64a - 14 = -12 \quad \Rightarrow \quad 64a = 2 \quad \Rightarrow \quad a = \frac{2}{64} = \frac{1}{32}$

For real-world situations, $a = 0$ would not result in a parabola, so the valid solution is $a + b + c = -23$.

Thus, the answer is A) -23.

Would you like any further details or clarification?

Here are some related questions to deepen understanding:

1. How does the discriminant determine the number of x-axis intersections for a parabola?
2. What is the significance of the coefficient $a$ in the equation of a parabola?
3. How would the equation change if the parabola did not intersect the x-axis?
4. Can we deduce the exact x-intercepts from this form of the equation?
5. How does the vertex form help in determining the vertex directly from an equation?

Tip: Always check if a parabola intersects the x-axis by analyzing the discriminant $b^2 - 4ac$. If positive, it has two real solutions!