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