what is number of quadratic equation that are unchanged by squaring their roots is

There are four such cases x²  =0 root 0

(x-1)²=0  root 1

x(x+1)=0  roots  0 and 1

x²+x+1=0 roots ω and ω² 

let x² +bx +c=0 ............1

not let another equation whoose roots are square of this equation

so   X=x²   or   x=√X

 put value of x

  X +b√X  +c =0

 or X +c =-b√X

square

   X² +c² +2cX =b²X

   X² +(2c-b²)X + c² =0..................2

root of equation 2 is the square of root of equation 1

both equation will be same if their coefficient are in proportion

1/1  =b/(2c-b²)  =c/c ²

b= 2c-b^{2     ................3}            

c=c^{2  ....................3}

 from equation 3  c=0 or 1

 for c =0  b= 0 and -1

  for c=1  b= 1 and -2 

 so four combination are possible