We assume everyone knows that a real polynomial *p* of degree 2 , of the form
*p*(*t*) = *ax*^{2} + *bx* + *c* , with *a* ≠ 0 ,
has a graph taking on one of the following approximate shapes: ∩ , ∪ ,
and that to find the *roots* of such a polynomial, i.e., the values of *x*
at which the graph of *p* crosses the *x*-axis, one calculates the (*real*)
values given by the well-known **quadratic formula**
–b ± √(*b*^{2} – 4*ac*)
————————————————— .
2*a*

Of course the graph of *p* may need not ever actually cross the *x*-axis,
perhaps because it never touches it at all, or perhaps because it touches it at just
one point; these situations come about exactly when the expression inside the radical
sign, that is, the expression *b*^{2} – 4*ac* , is negative
or zero, i.e., *b*^{2} – 4*ac* __<__ 0 , or what amounts
to the same thing,
*b*^{2} __<__ 4*ac* .
This is the case, in particular, when *p*(*t*) __>__ 0 for all real *t*^{ }.

It is the above observation that serves us best as we seek to establish the Cauchy-Schwartz
Inequality, which asserts (of two vectors *u* and *v* , whether both column
vectors in **R**^{n} or both row vectors in **R**_{n} ) that
|*u* • *v*| __<__ ||*u*|| ||*v*|| .
First,^{ }we dispose of the (easy) case in which
*u* = **0** ^{ }—
here both sides are obviously just 0 .

Next, given two vectors *u* ≠ **0** and *v* ,
the formula
*p*(*t*) = ||*tu* + *v*||^{2} pretty obviously defines a
real-valued function *p* for which it’s also pretty obvious that
*p*(*t*) __>__ 0 for all real *t* .

We^{ }now evaluate *p*(*t*) to discover that *p*
is a polynomial of degree 2 (not less, because from *u* ≠ **0** we can
easily infer that the coefficient of *t*^{2} below is nonzero too):
*p*(*t*) = ||*tu* + *v*||^{2} =
(*tu* + *v*) • (*tu* + *v*) =
(*u* • *u*)*t*^{2} +
2(*u* • *v*)*t* + *v* • *v* .
To finish, we use *a* = *u* • *u* ,
*b* = 2(*u* • *v*) , and *c* = *v* • *v*
to translate the *b*^{2} __<__ 4*ac* conclusion
to [2(*u* • *v*)]^{2} __<__
4(*u* • *u*)(*v* • *v*) ;
rewriting, dividing both sides by 4, and then taking square roots, we obtain sequentially
4 (*u* • *v*)^{2} __<__
4 ||*u*||^{2} ||*v*||^{2} ,
(*u* • *v*)^{2} __<__
||*u*||^{2} ||*v*||^{2} , and finally
|(*u* • *v*)| __<__ ||*u*|| ||*v*|| ,
as desired.