Let , be the roots of the quadratic equation 12x^2-20x+3 =0, Z.

Question

Let $\alpha ,\beta $ be the roots of the quadratic equation $12{x}^{2}-20x+3\lambda =0,\lambda \in Z$. If $\frac{1}{2}\leqslant |\beta -\alpha |\leqslant \frac{3}{2}$, then the sum of all possible values of $\lambda $ is :

Accepted Answer

$\begin{gathered} \frac{1}{2}\le |\alpha -\beta |\le \frac{3}{2} \ \frac{1}{4}\le |\alpha -\beta {|}^{2}\le \frac{9}{4} \ \frac{1}{4}\le (\alpha +\beta {)}^{2}-4\alpha \beta \le \frac{9}{4} \ \frac{1}{4}\le \frac{25}{9}-4	imes \frac{\lambda }{4}\le \frac{9}{4} \ -\frac{91}{36}\le -\lambda \le \frac{-19}{36} \ \frac{19}{36}\le \lambda \le \frac{91}{36} \ \lambda =1,2 \ =3 \end{gathered}$

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