<s>[INST] Find the value of $\alpha + \beta + \alpha\beta$ from the given data Suppose the quadratic polynomial 
$P(x) = ax^2 + bx +c $
has positive coefficients $ a, b, c $ in an arithmetic progression in that order. If $P(x) = 0$ has integer roots $\alpha $ and $\beta $, then $\alpha + \beta + \alpha\beta $ equals?
1) $3$
2) $5$
3) $7$
4) $14$
It's a single choice correct problem. All genuine answers are welcome :)
The question is from a famous Indian Scholarship test 'KVPY' for high school students. The official website doesn't provide any solutions to it that is why I am asking for help.
My go on the question 
we know that in a quadratic equation $ax^2 + bx +c$ with roots $\alpha, \beta$ the sum of roots is $\alpha +\beta = \frac{(-b)}{a} $ and product is $\alpha\beta = \frac{c}{a} $
So $\alpha + \beta + \alpha\beta  = \frac{(-b+c)}{a}$
and since $a,b,c$ are in arithmetic progression $\frac{(a+c)}{2} = b$
Something that I tried was also this -
$  a, b, c     $   are in AP (arithmetic progression)
$a-b , 0, c-b$ are in  AP (subtracting by $b$)
$1-\frac{b}{a}, 0, \frac{(c-b)}{a}$ are in AP  (dividing by $a$)
Now I can't proceed further. After a bit of plugging and chugging I still cant get an integer as an answer...
 [/INST] Any quadratic (or for that matter polynomial) can be written in terms of its roots:
$$P(x)=a(x-\alpha)(x-\beta)=ax^2-a(\alpha+\beta)x+a\alpha\beta$$
So $\alpha+\beta+\alpha\beta=-\frac{b}{a}+\frac{c}{a}=\frac{-b+c}{a}$
We also know $a,b,c$ form an AP so lets rewritten them as $a,a+d,a+2d$ where $d$ is the common difference.
So $\alpha+\beta+\alpha\beta=\frac{-(a+d)+a+2d}{a}=\frac{d}{a}$
As the roots are integers then $d$ is a multiple of $a$ and hence so is $b$ and $c$. Without loss of generality we can let $a=1$.
Also we need $b^2-4ac=n^2,n\in\mathbb{Z}$
Subbing in $a=1$ and $d$ into this gives:
$$(1+d)^2-4(1+2d)=n^2$$
$$1+2d+d^2-4-8d=n^2$$
$$d^2-6d-3=n^2$$
$$d^2-6d-9+12=n^2$$
$$(d-3)^2+12=n^2$$
$$(d-3)^2-n^2=12$$
$$(d-3+n)(d-3-n)=12$$
Let the factors of $12$ be $\left(f,\frac{12}{f}\right)$.
So $d-3+n=f$ and $d-3-n=\frac{12}{f}$ hence $2(d-3)=f+\frac{12}{f}$
So $d=\frac{f}{2}+\frac{6}{f}+3$
Possible values of $f$ can only be $2$,$6$ as $d$ is a positive integer.
Both leads to $d=1+3+3=7$ and $d=3+1+3=7$.
Hence $\alpha+\beta+\alpha\beta=\frac{-(a+d)+a+2d}{a}=\frac{d}{a}=7$
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