Question

проверьте, правильно ли я решил?

Answer 1

$1. x^2 + 2x - 1 = 0\\ D = 4 + 4 = 8\\ x_1 = frac{-2+sqrt{8}}{2} = frac{-2 + 2sqrt{2}}{2} = -1 + sqrt{2}\\ x_2 = frac{-2-sqrt{8}}{2} = frac{-2 - 2sqrt{2}}{2} = -1 - sqrt{2}\\ (x - x_1)(x - x_2) = (x - ( -1 + sqrt{2}))(x - (-1 - sqrt{2})) = \\ = (x + 1 - sqrt{2})(x + 1 + sqrt{2})$

$2. x^2 - 2(sqrt{2} - 1)x + 3 - 2sqrt{2} =\\ = x^2 - 2(sqrt{2} - 1)x + 2 - 2sqrt{2} + 1 =\\ = x^2 - 2(sqrt{2} - 1)x + (sqrt{2} - 1)^2 = \\ = (x - (sqrt{2} - 1))^2 = (x - sqrt{2} + 1)^2 = (x + 1 - sqrt{2})^2$

$3. frac{x^2 + 2x - 1}{x^2 - 2(sqrt{2} - 1)x + 3 - 2sqrt{2}} = frac{(x + 1 - sqrt{2})(x + 1 + sqrt{2})}{(x + 1 - sqrt{2})^2} = \\ = boxed{frac{x + 1 + sqrt{2}}{x + 1 - sqrt{2}}}$