Question 927857
We are told that "The curve and the line intersect at two distinct points." so that means {{{2x^2 -(2k +1)x + 3k-1 = 0}}} has two distinct roots/solutions for x.



That only happens when the discriminant {{{D}}} is greater than zero, ie {{{D>0}}}.



In general, the equation {{{ax^2+bx+c = 0}}} has the discriminant {{{D = b^2 - 4ac}}}



For {{{2x^2 -(2k +1)x + 3k-1 = 0}}}, we see that {{{a = 2}}}, {{{b = -(2k+1)}}} and {{{c = 3k-1}}}



Plug in those values of a,b,c and simplify



{{{D = b^2 - 4ac}}}



{{{D = (-(2k+1))^2 - 4*2(3k-1)}}}



{{{D = (2k+1)^2 - 24k + 8}}}



{{{D = 4k^2+4k+1 - 24k + 8}}}



{{{D = 4k^2-20k+9}}}



So because {{{D>0}}} and {{{D = 4k^2-20k+9}}}, we know {{{4k^2-20k+9>0}}}