If the slope of the parabola is y = ax^2+bx+c where a, b, c ∈ R{0} at the points (3,2) and (2,3) are 34 and 12 respectively, then find the value of a.

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?

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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.

If the slope of the parabola is y = ax^2 + bx + c where a, b, c ∈ R{0} at the points (3,2) and (2,3) are 34 and 12 respectively, then find the value of a.

Learn how to determine the value of 'a' in a quadratic equation by using given slope conditions at specific points. This step-by-step solution involves key concepts from calculus and algebra, including derivatives and differentiation rules. Suitable for students in Grades 10-12.

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