Determine all possible triple(s) (p, q, r) of rational numbers that satisfy the following system of simultaneous equations:

p²(q+r) = 5, q²(r + p) = 8, and r²(p + q) = 9

(In reply to Solution by Praneeth)

Nice work, Praneeth!

Except, I am not sure why you focused on just the integral root of

x³+11x²-180=0

Aren't the other two roots real?  
And won't they lead two two other solutions?

/****************************************/

Oh, on second thought, I do understand.  
In order to have a rational solution, the other roots
to Praneeth's cubic equation
would need to be rational, which they are not.

So, the other two(?) solutions to the original 
equations are real, but not rational.  For
instance, from Excel iteration, one of them is:

p  approximately 1.21012153484856
q  approximately 1.63618079682203
r  approximately 1.77820055081319

Edited on May 19, 2008, 10:32 pm

Edited on May 19, 2008, 10:32 pm

Posted by Steve Herman on 2008-05-19 18:37:56