If x² + y² + 2x + 1 = 0, then the value of x³¹ + y³⁵ is?

Solution (By Examveda Team)

$$\eqalign{ & {x^2} + {y^2} + 2x + 1 = 0 \cr & \Rightarrow {x^2} + 2x + 1 + {y^2} = 0 \cr & \Rightarrow {\left( {x + 1} \right)^2} + {y^2} = 0 \cr} $$
Hence both terms are squares and there addition is zero.
So, it can be possible only when both terms are zeros.
$$\eqalign{ & \therefore x + 1 = 0 \cr & \Rightarrow x = - 1 \cr & y = 0 \cr & \therefore {x^{31}} + {y^{35}} \cr & \Rightarrow {\left( { - 1} \right)^{31}} + {\left( 0 \right)^{35}} \cr & \Rightarrow - 1 \cr} $$