A square garden is surrounded by a path of uniform width. If the path and the garden...

Area of square garden = x
$$area\ =\ l^2$$
$$\sqrt{l^{^2}}=\sqrt{x}$$
$$l=\sqrt{x}=>\ length\ of\ each\ side\ =\sqrt{x}$$

Garden is surrounded by a path of uniform width, so the shape of the garden and the path will also form a square when combined.

Let the width of path = n
The length of the combined path and garden = length of garden + width of the path on right side + width of the path on the left side
$$combined\ length=\sqrt{x}+n+n$$
$$=\sqrt{x}+2n$$
$$area\ of\ path\ =x$$
$$area\ of\ garden=x$$
$$combined\ area\ =\left(combined\ length\right)^2=x+x\left[\sin ce\ combined\ shape\ is\ a\ square\right]$$
$$\left(combined\ length\right)^2=2x$$
$$where\ combined\ length\ =\ \sqrt{x}+2n$$
$$\sqrt{\left(x+2n\right)^2}=\sqrt{2x}$$
$$\sqrt{x+2n}=\sqrt{2x}$$
$$\frac{2n}{2}=\frac{\sqrt{2x}-\sqrt{x}}{2}$$
$$n=\frac{\sqrt{2x}-\sqrt{x}}{2}$$
$$\frac{\sqrt{2x}}{2}-\frac{\sqrt{x}}{2}$$
$$Answer\ =\ E$$