The above formula represents the sum of the distances from point (x, 0) to points (4, 5) and (2, 3).

The symmetry point of the point (2,3) about the X axis is (2,3).

Find the abscissa x= 1 1/4 of the intersection of the connecting line of points (4, 5) and (2, -3) and the X axis.

Then when x= 1 1/4, the algebraic expression √(x? ﹣8x﹢4 1)+√(x？ (4x- 13) has the minimum value.

√(x？ ﹣8x﹢4 1)+√(x？ The minimum value -4x- 13) is equal to the distance between point (4,5) and point (2,3).

That is √ [(4-2)? ﹢(5+3)? ]=2√ 17.