We are given a quadratic equation $y = ax^2 + bx + c$, and the slopes at the points $(3,2)$ and $(2,3)$ are given as 34 and 12, respectively.

Step 1: Derivative of the parabola (Slope equation)

The slope of the parabola at any point is given by the derivative of $y$ with respect to $x$:

$\frac{dy}{dx} = 2ax + b$

Step 2: Using the slope conditions

We are given that at the point $(3, 2)$, the slope is 34, and at $(2, 3)$, the slope is 12. So, we can set up two equations based on these conditions:

1. For $x = 3$, the slope is 34: $2a(3) + b = 34$ Simplifying: $6a + b = 34 \quad \text{(Equation 1)}$

2. For $x = 2$, the slope is 12: $2a(2) + b = 12$ Simplifying: $4a + b = 12 \quad \text{(Equation 2)}$

Step 3: Solving the system of equations

Now we have the system of equations: $6a + b = 34 \quad \text{(Equation 1)}$ $4a + b = 12 \quad \text{(Equation 2)}$

We subtract Equation 2 from Equation 1 to eliminate $b$:

$(6a + b) - (4a + b) = 34 - 12$ $2a = 22$ $a = 11$

Thus, the value of $a$ is $\boxed{11}$.

Would you like further clarification or more details on any step?

---

Here are 5 related questions for further practice:

1. Find the value of $b$ after finding $a = 11$.
2. Determine the equation of the parabola by finding $c$ as well.
3. What is the general formula for the slope of a quadratic function at any point $x$?
4. How would the slopes change if the quadratic had no linear term, i.e., $b = 0$?
5. If the parabola passes through the origin, what would be the value of $c$?

Tip: When given the slope of a curve at specific points, always consider taking the derivative to create equations involving the given information.