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(B) Given the quadratic equation $x^{2}+3ax+c=0$.Let $\alpha$ and $\beta$ be the roots of the equation.From the properties of roots,we have the sum of

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$\sqrt{\frac{5+2c}{9}}$

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(B) Given the quadratic equation $x^{2}+3ax+c=0$.Let $\alpha$ and $\beta$ be the roots of the equation.From the properties of roots,we have the sum of roots $\alpha+\beta = -3a$ and the product of roots $\alpha\beta = c$.We are given $\alpha^{2}+\beta^{2}=5$.We know the identity $\alpha^{2}+\beta^{2} = (\alpha+\beta)^{2} - 2\alpha\beta$.Substituting the known values into the identity: $5 = (-3a)^{2} - 2(c)$.$5 = 9a^{2} - 2c$.$9a^{2} = 5 + 2c$.$a^{2} = \frac{5+2c}{9}$.Therefore,$a = \pm \sqrt{\frac{5+2c}{9}}$.Comparing with the given options,the correct value is $\sqrt{\frac{5+2c}{9}}$.

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+3ax+c=0$ and if $\alpha^{2}+\beta^{2}=5$,find the value of $a$.

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