A level maths questions on inequalities and simultaneous equations

I have mock exams next week and have been trying to revise inequalities and sim eqs but I am finding any questions using a constant, k and the discriminant very difficult. If I post some of said questions I would appreciate some help :)

Inequalities:

'By considering the discriminant, or otherwise, find the range of values of k that give each of these equations two distinct real roots.' [I got as far as thinking that the discriminant must be greater than 0?]

a) x²+3x+k=0

b) 3x²+kx+2

c) k(x²+1)=x-k

'Find the range of values for k that give these equations no real roots'

a) x²+6x+k=0

b) 2x²+kx+1=0

c) (k+1)x²+4kx+9=0

'Show that x²+2kx+9 is greater than or equal to 0 for all real values of x, if k² is less than or equal to 9.

Simultaneous equations:

'Find the possible values of k if y=2x+k meets y=x²-2x-7'

a) in two distinct real points b) in just one point

'Find the range of values of k for which kx+y=3 meets x²+y²=5 in two distinct points'

'Find the range of values for qhich y=kx-2 is tangent to the curve y=x²-8x+7'

Thanks for reading through this, sorry it is so long!