Question

determine the value of k such that g(x)=3x+k intersects the quadratic function f(x)+2x^2-5x+3

Answer 1

we want 2x^2 - 5x + 3 = 3x + k

2x^2 - 8x + 3-k = 0

Now, this equation has solutions as long as the discriminant is not negative

64 - 8(3-k) >= 0
40 + 3k >= 0

So, you can pick any value for k >= -40/3.

Think of the graphs. You have a parabola that open up, and a line with slope=3. If the line is too low, it will never touch the parabola. Raise it high enough, and it will always intersect the parabola, which extends upward forever.

Now, if you want a specific k, then you need to specify where the line intersects, or the slope at the intersection, or something.