challenging questions on quadratic functions

Question

1)m is a real number; P(x)=(m-2)x²-3(m+2)x+6m

for all x∈¶R, decide m's minimum integar(Z) when p(x)>0

2) p(x)=0 range of numbers m can have when the roots are different and real(Δ>0)

3)p(x)=0 roots are real and difference of the roots are 3 ;find m.

Answer 1

The easiest way to investigate this function is to graph it...hopefully on a graphing calculator which allows "sliders".

For P(x) > 0 m > 2; put another way for m>2, p(x) is strictly positive and has no real roots.

When m<2, p(x)<0 and again there are no real roots....that is

The graph suggests that there are no roots when m is other than 2 and when m=2, the graph is linear with root x=1

The question suggests that I may be missing something. If you find that I have misled you, I would appreciate some feedback to tell me what I missed. Thank you.