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If that equation has an integer root, called $ x_{0}$, then $ x_{0}\not =-3$ and $ p=-\frac{x_{0}^{2}}{x_{0}+3}\in\mathbb Q$. We write $ p=\frac{m}{n}$, here $ \gcd (m,n)=1$.
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orl wrote:
For which real numbers $ p$ does the equation $ x^{2}+px+3p=0$ have integer solutions ?
$ \{p\in \mathbb{R}|\ x^{2}+px+3p=0\ have\ integer\ solution\}=\{-\frac{a^{2}}{a+3}|\ a\in\mathbb{Z}-\{-3\}\}$