Determine the values of k if the graph of y = 2x^2 - 2x + 3k intersects the x-axis at two distinct points.

Use the discriminant to see if it has two distinct solutions.
b^{2}-4ac
If the discriminant is greater than 0, then there are two distinct solutions.

So, you need to solve
b^{2}-4a(3k)>0 for k