While reading the proof of Cauchy-schwarz inequality, I didn't get one step. The step is as below,
by positivity axiom, for any real number $t$
This imply $$0≤at^2 + bt+c$$ where $a=⟨u,u⟩$, $b=2⟨u,v⟩$ and $c=⟨v,v⟩$
After this they had written, this inequality implies that the quadratic polynomial has either no real roots or repeated real roots!
I didn't get this! How the quadratic polynomial $at^2 + bt+c$ has either no real roots or repeated real root?