TITLE: Both solutions to a quadratic make sense -- looking for applications
QUESTION [7 upvotes]: I'm looking for reasonably real, non-abstract applications modeled by quadratic equations where both solutions make sense. I'd like them to be accessible to high school algebra students.
One I come up with is firing a bullet through a sphere -- the two solutions correspond to the entry and exit points of the bullet, that is the intersections of the bullet's line and the sphere's surface. To keep it simple, we could do this instead with a line and circle in the xy plane, but the idea is the same. 
Applications outside of physics would be particularly welcome.

REPLY [4 votes]: Knowing two sides $a$ and $c$ of a triangle and an angle $\alpha$ not between them—i.e. the SSA condition—does not determine the length of the third side $b$. See Figure 3 on this page. (Unfortunately, the convention at the length of the side opposite the vertex $A$ is denoted $a$ is going to make the notation a little confusing below.)
Set the origin at the vertex $A$ with the $x$-axis along $AC$. Then the vertex $B$ is at $(c \cos \alpha, c \sin \alpha)$ and $C$ is at $(b,0)$, and we have $$(c \cos \alpha - b)^2 + (c \sin \alpha - 0)^2 = a^{2},$$ which is a quadratic in $b$ both of whose solutions evidently make sense.