<s>[INST] Minimum value of given expression What is the minimum value of the $$ \frac {x^2 + x + 1 }  {x^2 - x + 1 } \ ?$$
I have solved by equating it to m and then discriminant greater than or equal to zero and got the answer, but can algebraic manipulation is possible 
 [/INST] Here is an 'algebra solution':
$\frac {x^2 + x + 1 }  {x^2 - x + 1 } = \frac{(x+1)^2-(x+1)+1}{(x+1)^2-3(x+1)+3} = \frac{1}{3} + \frac{2}{3} \frac{(x+1)^2}{x^2-x+1} = \frac{1}{3} + \frac{2}{3} \frac{(x+1)^2}{(x-\frac{1}{2})^2+\frac{3}{4}}$.
Since the last term is greater than zero when $x\neq -1$, we see that the minimum is

 $\frac{1}{3}$.

Of course, this is cheating since I know the answer from J.D.'s solution and this suggests the way to expand the expression.
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