Germany | Can you solve this ? | A Nice Math Olympiad Problem

[@wes9627]

Substitute 1/x=y-6 and rearrange to (y+1)^4-(y-1)^4-240=2*4(y^3+y)-240=0. y^3+y-30=0 yields y=3. Divide by (y-3): y^2+3y+10 has complex roots, so ignore. Thus, 1/x=3-6=-3 so x=-1/3. Check: (7-3)^4-(5-3)^4=256-16=240

[@NadiehFan]

It is more efficient to substitute 6 + 1/x = m because then you end up with the much simpler depressed cubic equation m³ + m = 30 where it is immediately obvious that m = 3 is the sole real solution since m³ + m is strictly increasing on the real number line.