Question

for what values of k in y=3x+k with the curve y=-x^2-3x+6 intersect the line at 2 points

Answer 1

let's intersect them:
-x^2 - 3x + 6 = 3x + k
-x^2 - 6x + 6-k = 0
x^2 + 6x + k-6 = 0

to have 2 distinct real solutions , the discriminant must be > 0
b^2 - 4ac > 0
36 - 4(1)(k-6) > 0
36 -4k + 24 > 0 > 0
-4k > -60
k < 15

Notice in my graph, when k = 0, the straight line is a tangent, so for k< 15, the line would drop down and you would have 2 intersection points

http://www.wolframalpha.com/input/?i=plot+y+%3D+-x%5E2+-+3x+%2B+6+%2C+y+%3D+3x+%2B+15