$25x^2-20x+5= (5x)^2-2.5x.2+2^2+1= (5x-2)^2+1 \ge 1$
$\Rightarrow \sqrt{25x^2-20x+5}\ge 1$
$25x^2-20x+10= (5x)^2-2.5x.2+2^2+6= (5x-2)^2+6\ge 6$
$\Rightarrow \sqrt{25x^2-20x+10}\ge \sqrt{6}$
$\Rightarrow E=\sqrt{25x^2-20x+5}+\sqrt{25x^2-20x+10}\ge \sqrt{6}+1$
$minE=\sqrt{6}+1\Leftrightarrow 5x-2=0$
$\Leftrightarrow x=\frac{2}{5}$