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

HINT
Note that $$(a-c)^2+(b-d)^2\ge0$$so that $$a^2+c^2+b^2+d^2\ge2ac+2bd$$

This deals with the easy inequality required. Other answers give a sharper result, which is possible because it is impossible for $a=c$ and $b=d$ with the given conditions.
