E12. Ladder against a wall (Part II)

Here is another ladder against a wall problem, from Coxeter's classic Introduction to Geometry. It is somewhat atypical of the book in that the interesting part seems to be the algebra, rather than the geometry. To make this more of a challenge, try to do it without using a CAS.

A 24-foot long ladder rests against the horizontal ground and a vertical wall in such a way that it touches a cube. The cube is 7 feet on a side and is placed flat on the ground, touching the wall. Find the height of the top of the ladder.

This is a two-dimensional problem which could be stated a little less colorfully in terms of a square and a line segment. It's easy to see that there must be (at least) two answers because of the symmetry of the problem; if a line segment of length 24 passes through (7,7) with endpoints on the positive coordinate axes, the reflection of that line segment in the line y = x will satisfy the same conditions. I found several ways of setting up the problem, all of which result in a 4th degree equation. However, the solutions are quadratic irrationals. In one approach, the biquadratic factors into two quadratics with integer coefficients. In another, a somewhat obvious substitution does the trick.