In general a quadratic curve may have one, two or no roots, as shown.
For the curve
the number of roots depends entirely on the discriminant
If
there are no roots.
If
there is one root.
If
there are two roots.
We are typically asked: Find the set of values of
for which the curve
has
no roots:
always, for every value of
so for every value of k the curve has no roots.
Example: Find the set of values offor which the curve
has two roots.
We solve for
So
or 
Example: Find the set of values offor which the curvehas one root.
We solve for
So
or 