<s>[INST] Minimizing in 3 variables 
Find the least possible value of the fraction $\dfrac{a^2+b^2+c^2}{ab+bc}$, where $a,b,c > 0$.

My try:
$a^2+b^2+c^2 = (a+c)^2 - 2ac + b^2$,
$= (a+c)/b  +b/(a+c)  -2ac/b(a+c)$
AM > GM
$3\sqrt[3]{-2ac/b(a+c)}$
And I cant somehow move on.
 [/INST] Let $\dfrac{a^2+b^2+c^2}{ab+bc}=k>0$ as $a,b,c>0$
$\iff b^2-kb(a+c)+a^2+c^2=0$ which is a Quadratic Equation in $b$
As $b$ is real, the discriminant must be $\ge0$
i.e., $$k^2(a+c)^2-4(a^2+c^2)\ge0\iff k^2\ge\dfrac{4(a^2+c^2)}{(a+c)^2}$$
the equality occurs if $a=\dfrac{k(a+c)}2$
Now $2(a^2+c^2)-(a+c)^2=(a-c)^2\ge0\implies2(a^2+c^2)\ge(a+c)^2$
$\implies k^2\ge2$
the equality occurs if $a=c$
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