How would you determine the values of k if the graph of y=2x^2-2x+3k intersects the x-axis at two distinct points?


I was wondering too just aside from the question how do you check if an equation is congruent to another, by hand? I had to find the equation of a parabola that opens downward from (1,-3) and is congruent to y=2(x-4)^2 and I came up with the equation y= -2(x-3)^2+1 and though the parabolas are in the same position accept for mine being positive one up into the right upper quadrant I was wondering how I could check?

Thank you.

Intersecting the x-axis means y=0.

2x^2-2x+3k=0.

By the quadratic formula,
x=( 2 plusorminus squareroot(4 - 4*2*3*k) )/4.

Intersecting at two distinct points means the term under the squareroot sign must be positive:

4 - 24k > 0.

k < 1/6.