I can confirm that the solution is indeed about 4.54197, 2.73602 for p, area_max.
The area goes to zero outside of p=(1/2)(1+/- 5 sqrt(5)) because all three roots are real there, so there is no triangle. The improved plot is
here and
program is here.
The form of the three roots as functions of p each contained a square root within a cube root. I had worried that the answer might depend of which roots were selected. (There are 2 x 3 = 6 possibilities.) It turned out that choosing different roots simply permuted the order of the three roots that the equation yielded: the 6 possible listings of the three roots. I probably should have expected that.
And yes, as noted by Jer and myself, the complex roots come as a conjugate pair to make the base of an isosceles triangle tipped on its side.
Edited on October 3, 2021, 11:34 pm
soln
