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.

As you see solving for $x$, one can start from the system $$\begin {cases} y=mx \\ \\ y=-x^2-6 \end {cases}$$ studying the intersections of a generic line passing through the origin ($m$ is just its slope) with a (fixed) parabola.
After checking the lines of tangency with $m=\pm \sqrt {24}$ and the line passing through the point $(1,-7)$, it is easy to deduce that both the roots are less than $1$ for every (and only) $m \ge \sqrt {24}$ .
