Real numbers a and b are chosen so that each of two quadratic trinomials x

^{2}+ax+b and x

^{2}+bx+a has two distinct real roots and the product of these trinomials has exactly three distinct real roots.

Determine all possible values of the sum of these three roots.

(In reply to

re: Solution by Steve Herman)

Looking over stuff again, you are right. I just assumed the case with the roots by my earlier workings just trying to find examples. The algebra is almost identical regardless of which of the four ways to choose a pair of roots, and comes to the same conclusion a+b+1=0.

(a,b)=(-1/3,-2/3) is an example where the larger roots of each quadratic are equal.