Finding range of $m$ in $x^2+mx+6$. Find the range of values of $m$ in the quadratic equation $x^2+mx+6=0$ such that both the roots of the equation $\alpha,\beta<1$.
My attempt - it is given that
$\alpha<1$ and $\beta<1$
$\rightarrow \alpha+\beta<2$
But $\alpha+\beta=-m$
Thus $m>-2$.
But this solution doesn't involve the coefficient term i.e. $6$.
Any solution to the above question is appreciated.

You made a logical fallacy as old as time.
What you have proven is:

If $\alpha, \beta<1$, then $m>-2$

What you HAVE NOT PROVEN is:

If $m>-2$, then $\alpha, \beta < 1$.

You can easily see that the second statement is false, since if $m=0>-2$, the equation has no real solutions.
