WOW you dont have to do any of that numerical solve, this is EASY as!
ok so after you simplify that 3rd line in the solutions you posted, you get to this:
-260p^2 x (p-1)^(37) x (35p+3)
Now, before letting it equal 0.47, notice that it's essentially like addition of ordinates
So, to make it easier put this in:
-260p^2 x (p-1)^(37) x (35p+3) | p=x (this way it comes up as an expression that you can graph)
then you have:
-260x^2 x (x-1)^(37) x (35x+3)
So, just drag that or copy that line into Graphs&Tab... (to graph it)
-- if you want you can set restrictions (or just pick the answer that lies between 0 and 1)
once you've graphed it, go to: Analysis>G-Solve>x-Cal
-- make y= 0.47
------------>MAKE SURE YOU ZOOM IN FIRST OR ELSE CALC WONT GIVE ANSWER<---------
then you get x= 0.052056...
therefore x= 0.0521 (4dp)
ALTOGETHER TAKES ABOUT 3 SECONDS AS OPPOSED TO 10 HOURS!!!!!!