My books says to prove that the following inequality is true, and to use it to prove Cauchy-Schwarz: $$(a_1x+b_1)^2+(a_2x+b_2)^2+(a_3x+b_3)^2+\dots+(a_nx+b_n)^2 \ge 0$$
This is easy to prove because by the trivial inequality each term on the LHS is $\ge 0$. However, to prove Cauchy-Schwarz using this this book gives the hint:
Write the left side as a quadratic equation in $x$ and note that a quadratic equation is non negative for all $x$ if and only if the discriminant is non positive.
If you multiply this out and set the discriminant $\le 0$, the Cauchy-Schwarz inequality follows very straightforwardly.
My question is: If there were no restrictions on $a$, $b$, or $x$ before I multiplied it out, why are there restrictions on them now?
I know that before I was considering a series of quadratics and now I'm considering one giant one, but still it's the same inequality as before.