Simultaneous Equation Problem

[Codex]

I have a problem regarding simultaneous equation solving. The question gives the equation of a straight line 'y + kx = -2' and also the equation of a hyperbola (I don't have the exact equation written down but it was something like 'y(k-8)=x'). The straight line represents a footpath going between two circular pools. The purpose is to find 'appropriate values of k' so that the path (straight line) does not intercept the pools (lines of the hyperbola). I use the following quadratic formula:

I remember something about using the discriminant (b^2-4ac) and making it equal less than 0 to show there are no real roots/answers. I just can't remember what else to do. Could someone please show me full steps to solve a problem like this?

What I have figured out how to do so far:

Spoiler:

THESE ARE ONLY ROUGH EXAMPLES, I DON'T HAVE THE EXACT HYPERBOLA EQUATION WRITTEN DOWN SO IT IS JUST TO SHOW THE METHOD.
1. Substitute the straight line equation into the hyperbola equation.
2. Form a quadratic equation. I think it was something like kx^2 + x(k+2) + 20.
3. Put this into the discriminant (b^2-4ac): (k+2)^2 - 4 x k x 20.
4. ?????

The second part of the question was to also show the possible 'range of values' for k.

Thank-you to anyone who can assist. I really appreciate it.

[buddy]

they show how to do it here. but kinda hard to say specific steps w/o specific Q. sry

[Codex]

Thanks for the link. However this problem is slightly more complex than just straightforward substitution simultaneous equation solving. It is asking to find possible values for 'k', not just the values of 'x' and 'y'.

[buddy]

It is asking to find possible values for 'k', not just the values of 'x' and 'y'.

Yeh, but its still kinda hard to say what specifically to do with a specific "it" to answer. Can you write back with "it"?