Решебник по алгебре 11 класс Никольский Параграф 7. Равносильность уравнений и неравенств Задание 5

\[\boxed{\mathbf{5.}}\]

\[\textbf{а)}\ \sqrt[3]{x^{3} + 3x - 15} = x\]

\[x^{3} + 3x - 15 = x^{3}\]

\[3x - 15 = 0\]

\[3x = 15\]

\[x = 5.\]

\[\textbf{б)}\ \sqrt[3]{x^{3} - 3x - 4} = x\]

\[x^{3} - 3x - 4 = x^{3}\]

\[- 3x = 4\]

\[x = - \frac{4}{3}\]

\[x = - 1\frac{1}{3}.\]

\[\textbf{в)}\ \sqrt[3]{x^{3} - 3x - 1} = x - 1\]

\[x^{3} - 3x - 1 = (x - 1)^{3}\]

\[x^{3} - 3x - 1 = x^{3} - 3x^{2} + 3x - 1\]

\[3x^{2} - 6x = 0\]

\[3x(x - 2) = 0\]

\[x = 0;\ \ x = 2.\]

\[\textbf{г)}\ \sqrt[3]{x^{3} - 3x + 1} = x + 1\]

\[x^{3} - 3x + 1 = (x + 1)^{3}\]

\[x^{3} - 3x + 1 = x^{3} + 3x^{2} + 3x + 1\]

\[3x^{2} + 6x = 0\]

\[3x(x + 2) = 0\]

\[x = 0;\ \ x = - 2.\]