kill(all)$

van=facebook.com/LearningCalculus$

f(x,a,b):=2*(a^2+b^2)*x^2 + 2*(a+b)*x + 1;

s:solve(f(x,a,b),x);

imag:imagpart(s);

solve(imag[1],a);
solve(imag[2],a);

f(x,a,a);
f(x,b,b);

solve(f(x,a,a),x);

      When a !=  b the roots of f(x,a,b) are imaginary.
      When a  =  b the roots of f(x,a,b) are real and equal
       and x  = -1/(2a)
      