If x + y + z = 6 and xy + yz + zx = 10, then the value of x³ + y³ + z³ - 3xyz is?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Given}} \cr & x + y + z = 6 \cr & xy + yz + zx = 10 \cr & {\text{To find }}{x^3} + {y^3} + {z^3} - 3xyz{\text{ = ?}} \cr & \Rightarrow {\text{ Using formula,}} \cr} $$
  $$ \Rightarrow {\left( {x + y + z} \right)^2} = $$   $${x^2} \,+$$ $$\, {y^2} \,+ $$ $$\, {z^2} \,+ $$ $$\, 2\left( {xy + yz + zx} \right)$$
$$\eqalign{ & \Rightarrow {6^2} = {x^2} + {y^2} + {z^2} + 2 \times 10 \cr & \Rightarrow 36 = {x^2} + {y^2} + {z^2} + 20 \cr & \Rightarrow {x^2} + {y^2} + {z^2} = 16 \cr} $$
  $$ \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = $$     $$\left( {x + y + z} \right)$$  $$\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)$$
$$\eqalign{ & \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 6\left[ {16 - 10} \right] \cr & \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 6 \times 6 \cr & \Rightarrow {x^2} + {y^2} + {z^2} - 3xyz = 36 \cr} $$