I saw this video about the proof of Cauchy-Schwarz inequality. The proof has be done by constructing an artificial function
$$ p(t)=\left\| t \vec{v}- \vec {u}\right\|^2 $$
and then this function was evaluated in
$$p \left( \frac{b}{2a} \right) $$
Why is the function constructed like this. The point where the function is evaluated is when the determinant is zero. Why?