The number of solutions of the equations x + y + z =12 ; x ^2+ y ^2+ z ^2=50 ; x ^3+ y ^3+ z ^3=216 is

Step-by-step solution

we have x+y+z=12 ...(i) x ^2+ y ^2+ z ^2=50...(ii) and x ^3+ y ^3+ z ^3=216 ...(iii) aligned & ( x + y + z )^2- ( x ^2+ y ^2+ z ^2 )=2( xy + yz + zx ) & xy + yz + zx =47 aligned now x^3+y^3+z^3-3 x y z=(x+y+z) (x^2+y^2+z^2-x y-y z-z x ) xyz =60 now from equn (iii) y ^3+ z ^3=216- x 3 (12- x ) (50- x ^2 )- [47-12 x + x ^2 ] =216- x ^3 on solving we get x^3-12 x^2+47 x-60=0...(iv) by hit at trial aligned & x=3 is a root of equation IV & (x-3) (x^2-9 x+20 )=0 & (x-3)(x-4)(x-5)=0 & x=3,4,5 aligned now no of solution ar