Корни:
$$\renewcommand{\arraystretch}{1.7}\begin{array}x_1 = \left\{\frac{3}{2} + \frac{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + \frac{56}{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}} + \frac{50}{3}}}{2}\right\}\\x_2 = \left\{x\; \middle|\; x \in \left\{- \frac{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}}{2} + \frac{3}{2} - \frac{i \sqrt{- \frac{50}{3} - \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{56}{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}}}}{2}, - \frac{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}}{2} + \frac{3}{2} + \frac{i \sqrt{- \frac{50}{3} - \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{56}{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}}}}{2}\right\} \wedge - x + \sqrt{x + 1} + \sqrt{2 x - 1} = 0 \right\}\end{array}$$
Пошаговое решение
График:
Исследование функции:
$$\renewcommand{\arraystretch}{1.7}\begin{array}\text{Область определения: }\left[\frac{1}{2}, \infty\right)\\\text{Область значений: }\left(-\infty, - \frac{\sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}}}{2} - \frac{1}{8} + \sqrt{- \frac{3}{4} + \sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}} + \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \sqrt{\frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}}}{2} + \frac{\sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}}{2} + \frac{9}{8}}\right]\\\text{Точки экстремума: }\left\{\frac{1}{8} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}}}{2} + \frac{\sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}}{2}\right\}\\\text{Максимальное значение: }- \frac{\sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}}}{2} - \frac{1}{8} + \sqrt{- \frac{3}{4} + \sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}} + \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \sqrt{\frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} - \frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + \frac{3}{4 \sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}} + \frac{73}{24}}}{2} + \frac{\sqrt{\frac{4417}{4608 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}}} + 2 \sqrt[3]{\frac{\sqrt{667}}{512} + \frac{296929}{884736}} + \frac{73}{48}}}{2} + \frac{9}{8}}\\\text{Минимальное значение: }-\infty\\\text{Нули функции: }\left[ \frac{3}{2} + \frac{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + \frac{56}{\sqrt{- \frac{23}{18 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}}} + 2 \sqrt[3]{\frac{1909}{216} + \frac{46 \sqrt{3}}{9}} + \frac{25}{3}}} + \frac{50}{3}}}{2}\right]\\\text{Чётность: }\mathtt{\text{Ни чётная, ни нечётная}}\end{array}$$
Разложение на множители:
$$\renewcommand{\arraystretch}{1.7}- x + \sqrt{x + 1} + \sqrt{2 x - 1}$$
Пошаговое решение