A quadratic equation with roots $m$ and $n$ where $mn=4$ satisfies the equation $\frac {m}{m-1} + \frac {n}{n-1}=\frac {a^2 -7}{a^2 -4}$ . We have to find the number of integral values of $a$ for $m$, $n \in (1,4)$.
By comparing the given I found the quadratic equation as $x^2 -(a^2 +1)x +4 =0$.
But now after that I am stuck
Can anybody help me out how to proceed?
Source : https://i.stack.imgur.com/vOpDl.gif
Hint:
The condition for the roots to lie within $(a,b)$ [i.e. $(1,4)$] should be used.
$f(a)>0$, $f(b)>0$, $x$ coordinate of vertex between $a$ and $b$ , $D \geq 0$ (discriminant)