both roots lie in the interval $(1, 2)$

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Solution

Condition I: $D \geq 0$
$\Rightarrow$ $m ϵ (- \infty, 1] \cup [9, \infty)$
Condition II: $f (1) > 0$
$\Rightarrow$ $1 - (m - 3) + m > 0$
$\Rightarrow$ $4 > 0$ which is true $\forall m ϵ R$
Condition III: $f (2) > 0$
$\Rightarrow$ $m < 10$
Condition IV: $1 < - \frac{b}{2 a} < 2$

$\Rightarrow$ $1 < \frac{m - 3}{2} < 2$
$\Rightarrow$ $5 < m < 7$
intersection gives $m = ϕ$

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