SOLUTION: Hi, I'm struggling with this question on a past paper: A curve has equation y = 2x^2 - x - 1 and a line has equation y = k(2x -3) , where k is a constant. The x-coordinate

Question 927857: Hi, I'm struggling with this question on a past paper:
A curve has equation y = 2x^2 - x - 1 and a line has equation y = k(2x -3) , where
k is a constant.
The x-coordinate of any point of intersection of the curve and the line satisfies the equation
2x^2 - (2k +1)x + 3k - 1 = 0
The curve and the line intersect at two distinct points.
Show that 4k^2 - 20k + 9 > 0.
If someone would be able to explain the steps needed that would be great, as I'm drawing a complete blank on this.
Many thanks,
R
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
We are told that "The curve and the line intersect at two distinct points." so that means has two distinct roots/solutions for x.

That only happens when the discriminant is greater than zero, ie .

In general, the equation has the discriminant

For , we see that , and

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

So because and , we know
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