iran-konkur 2022 Q106

iran-konkur · Other · konkur-riazi_1401_specialized Discriminant and conditions for roots Root relationships and Vieta's formulas
106-- $\alpha$ and $\beta$ are roots of the equation $x^2 + 6x + a = 0$. If $0 < \alpha < \beta < 0$ and $12\sqrt{2} + 85 = 12\sqrt{2} + 85$, and $3\alpha^2 + 2\beta^2 = 12\sqrt{2} + 85$, what is the value of $a$?
(1) $1$ (2) $\dfrac{13}{4}$ (3) $\dfrac{21}{5}$ (4) $2$

\textbf{106--} $\alpha$ and $\beta$ are roots of the equation $x^2 + 6x + a = 0$. If $0 < \alpha < \beta < 0$ and $12\sqrt{2} + 85 = 12\sqrt{2} + 85$, and $3\alpha^2 + 2\beta^2 = 12\sqrt{2} + 85$, what is the value of $a$?

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\hspace{1cm} (1) $1$ \hspace{2cm} (2) $\dfrac{13}{4}$ \hspace{2cm} (3) $\dfrac{21}{5}$ \hspace{2cm} (4) $2$

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