Finding Intersection Points of $y = 2x^2$ and $y = 4x$

To find where the curves $y = 2x^2$ and $y = 4x$ intersect, we set the equations equal to each other.

Solving the System of Equations

1. Set the expressions for $y$ equal:

   $2x^2 = 4x$

2. Rearrange the equation to form a quadratic equation:

   $2x^2 - 4x = 0$

3. Factor the equation:

   $2x(x - 2) = 0$

4. Solve for $x$ by setting each factor to zero:
   • $2x = 0 \implies x = 0$
   • $x - 2 = 0 \implies x = 2$
5. Find the corresponding $y$ values using $y = 4x$:
   • For $x = 0$, $y = 4(0) = 0$. The point is $(0, 0)$.
   • For $x = 2$, $y = 4(2) = 8$. The point is $(2, 8)$.

Conclusion on Intersection Points

The calculations show two distinct solutions for $x$, leading to two distinct intersection points, $(0, 0)$ and $(2, 8)$. Therefore, the curves intersect at exactly two points.