The question says-if roots of $x^2 -2mx +m^2 -1$ lie in $(-2,4)$, Prove that $m$ lies in $(-1,3)$. This was pretty easy, but then I tried to work backwards. What if the question read-"If $m$ lies between $(-1,3)$, prove that the roots of $x^2 -2mx +m^2 -1$ lie in $(-2,4)$". I tried a lot, but I really don't know how to proceed. Can anyone help me out?(This is not a homework problem, though!)
Thanks in advance!!
EDIT- The 'forward' method.
Suppose the roots are the $\alpha$ and $\beta$. It is given that they lie in $(-2,4)$. Let the given expression be $f(x)$. It is easy to see that $f(-2)$ and $f(4)$ are both greater than $0$.( you can draw the X axis, draw an upward parabola(since a>0 here) intersecting it at $(\alpha,0)$ and $(\beta,0)$. As they lie between -2 and 4, mark (-2,0) and (4,0) on either side of the parabola's intersections. Draw perpendiculars from these points to the parabola. You will see that both $f(-2)$ and $f(4)$ are positive.) Now, expand out $f(-2)$ and $f(4)$, and put them both $>0$. Solve the inequalities.