How to show if a linear function through a point passed through a curve twice?

Question

Show that any linear function passing through P(0, 3) will meet the curve $$f(x)=2x^2-x-2$$ twice

I tried using the discriminant but the discriminant becomes $$m^2-2m+41$$ and from there you get two values as m however that means that only the gradients for those values satisfy it right? I'm quite confused

Answer 1

The discriminant is:

$D=b^2-4ac$

Note that this value doesn't tell you the roots but rather tells you how many there are.

If $D>0$, then there are 2 roots.

If $D=0$, then there is 1 root.

If $D<0$, then there are no roots.

Answer 2

Note that the discriminant is always positive because its discriminant (when viewed as a function of $m$) is always negative. Therefore the equation $2x^2-x-2=mx+3$ always has two solutions, and the question is solved.