Не решая уравнение 2x^2+3x-13=0, найдите значение выражения 1/x_1^2+1/x_2^2.

\[2x^{2} + 3x - 13 = 0\]

\[x^{2} + \frac{3}{2}x - \frac{13}{2} = 0\]

\[x_{1} + x_{2} = - \frac{3}{2};\ \ x_{1} \cdot x_{2} = - \frac{13}{2};\]

\[\frac{1}{x_{1}^{2}} + \frac{1}{x_{2}^{2}} = \frac{x_{2}^{2} + x_{1}^{2}}{x_{1}^{2}x_{2}^{2}} =\]

\[= \frac{61}{4}\ :\left( - \frac{13}{2} \right)^{2} = \frac{61}{4} \cdot \frac{4}{169} = \frac{61}{169}.\]

\[\left( x_{1} + x_{2} \right)^{2} = x_{1}^{2} + 2x_{1}x_{2} + x_{2}^{2} = \left( - \frac{3}{2} \right)^{2}\]

\[x_{1}^{2} + x_{2}^{2} = \frac{9}{4} - 2 \cdot \left( - \frac{13}{2} \right) =\]

\[= \frac{9}{4} + 13 = \frac{61}{4}.\]

\[Ответ:\frac{61}{169}.\]