An algebra problem by Răzvan Birişan

Algebra Level pending

Let $$P(x) =x^2-\left(\dfrac{a^2}{2}+5a+1\right)x+a+3$$ be a quadratic polynomial such that $$P(x_1)=P(x_2)=0$$ and $$x_1+x_2=13$$.

Is there a real value for $$a$$ for which $$x_1$$ and $$x_2$$ are natural numbers?

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