The number of solutions for z^3+ z =0 , is

Step-by-step solution

Given, z^3+ z =0 Let z=x+i y aligned & (x+i y)^3+x-i y=0 & x^3+(i y)^3+3 x^2 i y+3 x(i y)^2+x-i y=0 & x^3-y^3 i+3 x^2 y i-3 x y^2+x-i y=0 & (x^3-3 x y^2+x )+ (-y^3+3 x^2 y-y ) i=0 aligned On equating real and imaginary parts, we get aligned & x^3-3 x y^2+x=0 & or -y^3+3 x^2 y-y=0 & x (x^2-3 y^2+1 )=0 & or -y (y^2-3 x^2+1 )=0 & x=0 and x^2-3 y^2+1=0 & or y=0 and y^2-3 x^2+1=0 & Now, & x^3-3 y^2+1=y^2-3 x^2+1 & 4 x^2=4 y^2 & x= y & y^2-3 y^2+1=0 & 2 y^2=1 y= 1 2 & x= 1 2 & aligned Hence, their solutions are (0,0) and