<s>[INST] $a+b+c = 13$; if $b/a=c/b$, find the maximum and minimum values of $a$ and the corresponding $b$ and $c$ Question :

The sum of $3$ integers $a,b$ and $c$ is $13$. If $\dfrac{b}{a}=\dfrac{c}{b}$, find the maximum and minimum values of $a$ and the corresponding $b$ and $c$.

To tackle this problem I let $x=\dfrac{b}{a}=\dfrac{c}{b}$ because I wanted to create a quadratic equation in order to use the discriminant theorem. 
From the equation above I can deduce that $b=ax$ and $c=ax^2$.
Because $a+b+c=13$.
Therefore; $$a+ax+ax^2=13$$
$$\implies 1+x+x^2-\frac{13}{a} = 0 $$  (where $a \ne 0$, $b \ne 0$, $c \ne 0$)
I can only work up to here. I do not know how to use the discriminant theorem to work out the maximum and minimum of $a$, $b$ and $c$.
 [/INST] ∵ $x=b/a$ is a solution of (1)
∴ (1) has at least one solution.
$△=1-4(1-(13/9))≥0$
$52/a≥3...(2)$
From (2)  $a>0$ and $52/3≥a$
$∴0<a≤52/3$
$1<a≤17$
Minimum of $a=1$ and maximum of $a =17$. 
Sub these values of $a$ into (1) to find the corresponding maximum and minimum of b and c.
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