math
posted by Melissa on .
determine the value of k such that g(x)=3x+k intersects the quadratic function f(x)=2x^25x+3 at exactly one point
determine the value(s) of k such that the linear function g(x)=4x+k does not intersect the parabola f(x)=3x^2x+4

2x^2  5x + 3 = 3x + k
2x^2  8x + (3k) = 0
We want both roots of this to be the same. That is, it must be a perfect square.
2(x2)^2 = 2x^2  8x + 8
So, we want 3k=8 or k=5
SO, 3x5 intersects the parabola in exactly one point.

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 + k4 = 0
5^2  4(3)(k4) = 25  12k + 48 = 73  12k < 0
7312k < 0
k > 73/12