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

This is similar ot the other problem.

If the equation crosses the x-axis then y=0.

So you need to find the solutions for

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

using the general equation we get

x=-2+/-(4-4.2.3k)^.5/4

for there to be real values

4-24k>0 (we can't square root a number less than zero, and it can't equal zero as the question says there are two distinct points. Zero would give us two coincident points)

so 4>24k or k<4/24 k<0.1667

Check my working!

To determine the values of 'k' such that the graph of the equation y = 2x^2 - 2x + 3k intersects the x-axis at two distinct points, you need to first consider when the discriminant (b^2 - 4ac) is greater than zero.

In the given equation, the coefficients are:
a = 2
b = -2
c = 3k

The discriminant can be calculated as follows:
D = b^2 - 4ac
= (-2)^2 - 4(2)(3k)
= 4 - 24k

For the graph to intersect the x-axis at two distinct points, the discriminant D must be greater than zero. So we have:
4 - 24k > 0

Solving this inequality, we can isolate 'k':
-24k > -4
k < 1/6

Therefore, the values of 'k' that satisfy the condition for the graph to intersect the x-axis at two distinct points are any values less than 1/6.