<s>[INST] What are the roots of this equation? I have a quadratic equation $ ax^2 +bx+c =0 $, where $ a,b,c $ all are positives and are in Arithmetic Progression. 
Also, the roots $\alpha$ and $\beta$ are integers.
I need to find out $ \alpha + \beta + \alpha\beta $.
I have tried taking $ a = a' - d , b = a' , c = a' + d $ because I have supposed $ a' $ is the first term and $ d $ is common difference, I used sum of roots and product of roots rule, but nothing helped..
 [/INST] We see that the equation $$ax^2+(a+d)x+a+2d=0$$
has integer roots, which says $1+\frac{d}{a}\in\mathbb Z$.
Let $\frac{d}{a}=k$.
Hence, we have $$x^2+(1+k)x+1+2k=0$$
has integer roots, which gives
$$(1+k)^2-4(1+2k)=n^2,$$ where $n$ is non-negative integer number,
which gives
$$k^2-6k-3=n^2$$ or
$$(k-3)^2=n^2+12$$ or
$$(k-3-n)(k-3+n)=12$$ and the rest is smooth:
Since $n$ is non-negative integer, we obtain two cases only.


*

*$n-k+3=6$ and $n+k-3=-2$, which gives $k=-1$;

*$n-k+3=-2$ and $n+k-3=6$, which gives $k=7$.
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