$x^{2} - (m - 3) x + m = 0 (m \in R)$ be a quadratic equation. Find the value of $m$ for which both the roots are greater than $2$

(9, 10)

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(- 5, 4)

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(6, \infty)

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None of the above.

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Solution

The correct option is A $(9, 10)$
First there exists 2 roots $\Rightarrow (m - 3)^{2} > 4 m$

$\Rightarrow (m - 1) (m - 9) > 0$

$∴ m \in (- \infty, 1) \cup (9, \infty)$

And both roots are greater than 2 $\Rightarrow$ f(2)>0

$\Rightarrow 10 - m > 0$

$\Rightarrow m < 10$

Combining both we get $m \in (9, 10)$

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