Minimum value of the given function 
Minimum value of $$\sqrt{2x^2+2x+1} +\sqrt{2x^2-10x+13}$$ is $\sqrt{\alpha}$ then $\alpha$ is________ .

Attempt
Wrote the equation as sum of distances from point A(-1,2) and point B(2,5) as
$$\sqrt{(x+1)^2 +(x+2-2)^2} +\sqrt{(x-2)^2 + (x+2-5)^2}$$ 
Hence the point lies on the line y=x+2 it is the minimum sum of distance from the above given two points.
But from here I am not able to get the value of x and hence $\alpha$. Any suggestions?

Your method is also really good. Consider the two points $A(-1,2)$ and $B(2,5)$. We need to find the point $(x,x+2)$ on the line $y=x+2$ for which the sum $AD+BD$ is minimum (see the graph below).
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You reflect the point $A$ over the line $y=x+2$ to find the point $C(0,1)$. The line through $C$ and $B$ is $y=2x+1$. The two lines intersect at $D(1,3)$, which is the solution of the given problem. Hence, when $x=1$, the sum will be minimum $\sqrt{20}$.
