По теореме Виета:

[tex]{x}^{2} + bx + c = 0\\ x_{1} + x_{2} = - b\\ x_{1} x_{2} = c[/tex]

[tex] {x}^{2} + 5x - 7 = 0 \\ x_{1} + x_{2} = - 5 \\ x_{1} x_{2} = - 7 \\ \\ {x_{1}}^{2} + {x}_{2}^{2} = {x_{1}}^{2} + 2x_{1}x_{2} + x {_{2}}^{2} - 2x_{1}x_{2} = \\ = (x_{1} + x_{2}) {}^{2} - 2x_{1}x_{2} = ( - 5) {}^{2} - 2 \times ( - 7) = \\ = 25 + 14 = 39 \\ \\ \frac{1}{x_{1}} + \frac{1}{x_{2}} = \frac{x_{2} + x_{1}}{x_{1}x_{2}} = \frac{ - 5}{ - 7} = \frac{5}{7} [/tex]