1. For what values of a will the straight line y=ax+1 have ONE intersection with the parabola y=-x^2-x-8?

Answer 1

the line intersects where

-x^2-x-8 = ax+1
x^2 + (a+1)x + 9 = 0

In order for there to be a single intersection, the discriminant must be zero. That is,

(a+1)^2 - 36 = 0

See what you can do with that.

http://www.wolframalpha.com/input/?i=plot+-x%5E2-x-8,+y%3D5x%2B1,+y%3D-7x%2B1,+-5%3C%3Dx%3C%3D5