The problem asks us to find the constant $a$ in the equation of the curve $y = ax^2 + 2x$, given that the gradient of the tangent at $x = 3$ is 0.25.

Step-by-step solution:

1. Find the gradient function (derivative of $y$):

   The gradient of the curve is given by the derivative of $y = ax^2 + 2x$ with respect to $x$: $\frac{dy}{dx} = 2ax + 2$

2. Use the given information:

   We know that the gradient of the tangent at $x = 3$ is 0.25, so we substitute $x = 3$ and $\frac{dy}{dx} = 0.25$ into the derivative equation: $2a(3) + 2 = 0.25$ Simplifying this: $6a + 2 = 0.25$

3. Solve for $a$:

   Now, isolate $a$: $6a = 0.25 - 2$ $6a = -1.75$ $a = \frac{-1.75}{6}$ $a = -\frac{7}{24}$

Thus, the value of $a$ is $-\frac{7}{24}$.

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

Here are 5 follow-up questions to expand this topic:

1. How do you find the equation of a tangent line to a curve at a given point?
2. What is the geometric interpretation of the derivative at a point on a curve?
3. If $a$ were a different value, how would the shape of the parabola change?
4. How would you determine the coordinates of the point where the tangent line touches the curve?
5. What is the significance of the second derivative for this curve?

Tip: Always double-check units and signs, especially when dealing with gradients and slopes, to avoid small mistakes!