O Level Additional Maths Notes: Using Discriminants to Find the Number of Roots of a Quadratic Curve
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 two roots.
Example: Find the set of values offor
which the curve
has
two roots.
We solve for
So
solutions exist for all values of
Example: Find the set of values offor
which the curve
has
one root.
We solve for
So
or