Cho phương trình [tex]3x^{2}-5x+m=0[/tex] . Xác định m để phương trình có 2 nghiệm thoả mãn [tex]x_{1}^{2}-x_{2}^{2}=\frac{5}{9}[/tex]
$ \Delta = (-5)^2 - 4 . 3 . m = 25 - 12m \\ \Delta > 0 \Leftrightarrow 25 - 12m > 0 \Leftrightarrow m < \frac{25}{12} \\ x_1 + x_2 = \frac53; x_1x_2 = \frac{m}3 \\ x_1^2 - x_2^2 = \frac59 \\\Leftrightarrow (x_1 + x_2)(x_1 - x_2) = \frac59 \\\Leftrightarrow \frac53(x_1 - x_2) = \frac59 \\\Leftrightarrow x_1 - x_2 = \frac13 \\\Leftrightarrow (x_1 - x_2)^2 = \frac19 \\\Leftrightarrow (x_1 + x_2)^2 - 4x_1x_2 = \frac19 \\\Leftrightarrow \frac{25}9 - \frac{4m}3 = \frac19 \\\Leftrightarrow \frac{25 - 12m}9 = \frac19 \\\Leftrightarrow 25 - 12m = 1 \\\Leftrightarrow 12m = 24 \\\Leftrightarrow m = 2 (tm) $