Question Number 10067 by konen last updated on 22/Jan/17

((a×b×c)/(a+b))=−(1/2) ((a×b×c)/(b+c))=(1/2) ((a×b×c)/(a+c))=1 ⇒a×b×c=?

Answered by mrW1 last updated on 22/Jan/17

let u=a×b×c ⇒a+b=−2u ...(i) ⇒b+c=2u ...(ii) ⇒a+c=u ...(iii) (ii)−(iii): b−a=u ...(iv) (i)+(iv): 2b=−u b=−(u/2) c=2u−b=2u+(u/2)=((5u)/2) a=−2u−b=−2u+(u/2)=−((3u)/2) u=a×b×c=(((−3u)/2))×(((−u)/2))×(((5u)/2))=((15)/8)u^3 (((15)/8)u^2 −1)u=0 u≠0 ⇒((15)/8)u^2 −1=0 u^2 =(8/(15)) ⇒a×b×c=u=±(√(8/(15)))=±2(√(2/(15)))