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प्रश्न
The set of all values of m for which both the roots of the equation \[x^2 - (m + 1)x + m + 4 = 0\] are real and negative, is
विकल्प
\[( - \infty , - 3] \cup [5, \infty )\]
[−3, 5]
(−4, −3]
(−3, −1]
उत्तर
\[m \in ( - 4, - 3]\] The roots of the quadratic equation \[x^2 - (m + 1)x + m + 4 = 0\] will be real, if its discriminant is greater than or equal to zero.
\[\therefore \left( m + 1 \right)^2 - 4\left( m + 4 \right) \geq 0\]
\[ \Rightarrow \left( m - 5 \right)\left( m + 3 \right) \geq 0\]
\[ \Rightarrow m \leq - 3 \text { or } m \geq 5 . . . (1)\]
\[\Rightarrow m + 1 < 0\]
\[ \Rightarrow m < - 1 . . . (2)\]
and product of zeros >0
\[\Rightarrow m + 4 > 0\]
\[ \Rightarrow m > - 4 . . . (3)\]
From (1), (2) and (3), we get,
\[m \in ( - 4, - 3]\]
Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.
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