what is number of quadratic equation that are unchanged by squaring their roots is
There are four such cases x^{2 }=0 root 0
(x-1)^{2}=0 root 1
x(x+1)=0 roots 0 and 1
x^{2}+x+1=0 roots ω and ω^{2}
let x^{2} +bx +c=0 ............1
not let another equation whoose roots are square of this equation
so X=x^{2 }or x=√X
put value of x
X +b√X +c =0
or X +c =-b√X
square
X^{2} +c^{2 }+2cX =b^{2}X
X^{2} +(2c-b^{2})X + c^{2 }=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-b^{2}) =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