= Solution
For a lens between an object and a screen, $u+v=D$ and the <thin lens> relation requires $uv=fD$. Eliminating $v$ gives the quadratic equation
$$
u^2-Du+fD=0,\qquad u_\pm=\frac{D\pm\sqrt{D^2-4fD}}2.
$$
For $D>4f$, both roots lie between $0$ and $D$, and they exchange the object and image distances. Both therefore focus the image on the same screen, with reciprocal absolute <magnifications>. The distance between lens positions is
$$
\boxed{l=|u_+-u_-|=\sqrt{D^2-4fD}=\sqrt{D(D-4f)}}.
$$
The existence qualification is essential: for $D=4f$ the two roots coincide and there is only one distinct position; for $D<4f$ there is none. For example, $D=3f$ makes the discriminant negative, providing a concrete counterexample to an unconditional assertion of two positions. This is the basis of the <Bessel lens displacement method>, with $f=(D^2-l^2)/(4D)$ when two distinct positions can be measured.
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