Another quiz

I first saw this quiz in the 50s, and despite being in engineering for 30 years, I have never found an "elegant" solution to this problem.

Assume a 4` X 4` box is resting against the floor and wall. A 12` ladder is resting against the wall and floor and touches the box at
it`s corner. How high is the ladder up on the wall? (2 answers, since the ladder can have two heights).

Math majors usually resort to graphical analysis, to get the answer.
This is not what I consider to be "elegant".

Well yes, I survived, but many people over 35 are, in fact, dead.