Let f(x)=x^2+a x+b , where a, b R . If f(x)=0 has all its roots imaginary, then the roots of f(x)+f^ (x)+f^ (x)=0 are

Step-by-step solution

Given, f(x)=x^2+a x+b has imaginary roots. Discriminant, D < 0 a^2-4 b < 0 Now, aligned f^ (x) & =2 x+a f^ (x) & =2 aligned array ll & x^2+a x+b+2 x+a+2=0 & x^2+(a+2) x+b+a+2=0 array aligned & x= -(a+2) (a+2)^2-4(a+b+2) 2 & = -(a+2) a^2-4 b-4 2 & Since, a^2-4 b < 0 & a^2-4 b-4 < 0 & aligned Hence, Eq. (i) has imaginary roots.