A) \[9{{x}^{2}}+3x+2=0\]
B) \[9{{x}^{2}}-\text{ }3x+2\text{ }=0\]
C) \[9{{x}^{2}}+\text{ }3x-2\text{ }=\text{ }0\]
D) \[9{{x}^{2}}-\text{ }3x+\text{ }2\text{ }=0\]
Correct Answer: C
Solution :
Given, \[\alpha \]and \[\beta \]are the roots of the equation \[{{x}^{2}}+5x+4=0\]. \[\therefore \] \[\alpha +\beta =-\text{5 and }\alpha \beta =4\] Now, \[\frac{\alpha +2}{3}+\frac{\beta +2}{3}=\frac{\alpha +\beta +4}{3}\] \[=\frac{-5+4}{3}=-\frac{1}{3}\] and \[\left( \frac{\alpha +2}{3} \right)\left( \frac{\beta +2}{3} \right)=\frac{\alpha \beta +2(\alpha +\beta )+4}{9}\] \[=\frac{4+2(-5)+4}{9}=-\frac{2}{9}\] \[\therefore \]Required equation is \[{{x}^{2}}-\](sum of roots) \[x+\]Product of roots = 0 \[\Rightarrow \] \[{{x}^{2}}+\frac{1}{3}x-\frac{2}{9}=0\] \[\Rightarrow \] \[9{{x}^{2}}+3x-2=0\]
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