A lighthouse stands at L(3, 4) and a boat sails along the $x$-axis. At the instant the boat is exactly 5 km from the lighthouse, what are its possible positions?

A point on the $x$-axis has the form $(x$, 0), so $(x-3{)}^2+(0-4{)}^2=5^2$, giving $(x-3{)}^2=25-16=9$ and $x-3=\pm 3$. Hence $x=0$ or $x=6$: the boat is either at the origin or at (6, 0), symmetric about the foot of the perpendicular from L. That foot, (3, 0), is only 4 km from the lighthouse, which is the closest the boat ever gets; taking one square root only would hide the second position.