Beroov S.L.

The numbers $a$ and $b$ are such that each of the two quadratic trinomials $x^{2} + a x + b$ and $x^{2} + b x + a$ has two distinct roots, and the product of these trinomials has exactly three distinct roots. Find all possible values of the sum of these three roots.
From the condition, it follows that the quadratic polynomials $x^{2}+a x+b$ and $x^{2}+b x+a$ have a common root $x_{0}$, as well as roots $x_{1}$ and $x_{2}$ different from it, respectively; in particular, $a \neq b$. The common root is also a root of the difference of these quadratics, that is, $(a-b)\left(x_{0}-1\right)=0$. Thus, $x_{0}=1$. Substituting this root into any quadratic, we get $1+a+b=0$. By Vieta's formulas, $x_{0}+x_{1}=-a, x_{0}+x_{2}=-b$, from which $x_{0}+x_{1}+x_{2}=-x_{0}-a-b=-(1+a+b)=0$.

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