[tex] x^{2} -2x-3 = 0 \\ \\ x^{2} -3x+x-3= 0\\ \\ x(x-3)+1(x-3)=0 \\ \\(x+1)(x-3)=0 =\ \textgreater \ x_1=-1; \ x_2=3. \\ \\ \\
x_1+x_2= -1 + 3 = 2 \\ \\
x_1*x_2 = -1*3= -3 \\ \\
\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_2}{x_1*x_2} + \frac {x_1}{x_1*x_2} = \frac{x_1+x_2}{x_1*x_2} = \frac{2}{-3} = \frac{-2}{3} \\ \\
x_1^{2} + x_2^{2} = (-1)^{2} + 3^{2} = 1 + 9 = 10 [/tex]
