Question : Suppose the quadratic polynomial $p(x)=ax^2 + bx + c~$ has positive coefficients $a, b, c$ in arithmetic progression in that order.If $p(x)=0$ has integer roots $\alpha$ and $\beta$, then what is the value of $\alpha + \beta + \alpha \beta$ ?
My attempt: I am not aware of any inequality of some sort if it exists, but I could not get far by trying substitutions over these unknown variables.
If $b=a+y$ and $c=a+2y$, then matching coefficients of $ax^2+bx+c=a(x-\alpha)(x-\beta)$ yields $a+y=-a(\alpha+\beta)$ and $a+2y=a\alpha \beta$. Subtracting yields $y=a(\alpha\beta+\alpha+\beta)$, and substituting back into the $a+y=\cdots$ equation and dividing by $a$ yields $1+\alpha\beta+\alpha+\beta=-\alpha-\beta,$ which is equivalent to $(2+\alpha)(2+\beta)=3$. As $\alpha$ and $\beta$ are integers there are a small number of solutions that can be found manually, and only one such that $\alpha$ and $\beta$ are negative, as is required by positivity of the coefficients: $\{\alpha,\beta\}=\{-5,-3\}$. Hence $\alpha\beta+\alpha+\beta=7$.