Posted by Melissa on Monday, October 24, 2011 at 4:40pm.
2x^2 - 5x + 3 = 3x + k
2x^2 - 8x + (3-k) = 0
We want both roots of this to be the same. That is, it must be a perfect square.
2(x-2)^2 = 2x^2 - 8x + 8
So, we want 3-k=8 or k=-5
SO, 3x-5 intersects the parabola in exactly one point.
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If the line does not intersect the parabola, then f(x)-g(x) = 0 must have a negative discriminant.
The line intersects the parabola when
4x+k = -3x^2 - x + 4
3x^2 + 5x + k-4 = 0
5^2 - 4(3)(k-4) = 25 - 12k + 48 = 73 - 12k < 0
73-12k < 0
k > 73/12
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