Proving an inequality with given equality of the form $a^2+b^2=c$ $$\begin{Bmatrix}
a^2+b^2=2\\ 
c^2+d^2=4
\end{Bmatrix} \to ac+bd\le3 $$ prove that inequality

Another way is by Cauchy Schwarz inequality:
$$8=(a^2+b^2)(c^2+d^2)\ge(ac+bd)^2$$
gives a tighter bound. 
