Multiple choice

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

  1. $(9,10)$
  2. $(-5,4)$
  3. $(6,\infty)$

  4. None of the above.
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A Correct answer
Explanation

For roots > 2: 1) D >= 0: (m-3)^2 - 4m >= 0 => m^2 - 6m + 9 - 4m >= 0 => m^2 - 10m + 9 >= 0 => (m-9)(m-1) >= 0. m <= 1 or m >= 9. 2) f(2) > 0: 4 - 2(m-3) + m > 0 => 4 - 2m + 6 + m > 0 => 10 - m > 0 => m < 10. 3) Vertex > 2: (m-3)/2 > 2 => m-3 > 4 => m > 7. Combining: m >= 9 and m < 10. So (9, 10).

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