If are vectors in , then
Remember that the roots of are given by the quadratic formula
In particular, the term must either be negative, yielding two complex roots, or 0, yielding a single real root. Thus
Substituting the values of into the last of these inequalities, it can be seen that
which is equal to the original statement.
- ↑ Intuitively, the graph of is either 'floating above' the horizontal axis, if it has two complex roots, or tangent if it has one real root. Since is non-negative for every , it can't have two real roots because the graph of the function would have to 'pass under' the horizontal axis.