Question

Решите уравнение х^-2/3-x^-1/3-2=0

Answer 1

${x}^{ - \frac{2}{3} } - {x}^{ - \frac{1}{3} } - 2 = 0$

Пусть

${x}^{ - \frac{1}{3} } = y$

Тогда

${y}^{2} - y - 2 = 0 \\ d = {b}^{2} - 4ac = {( - 1)}^{2} - 4 \times 1 \times ( - 2) = 1 + 8 = 9 \\ y1 = \frac{ - b - \sqrt{d} }{2a} = \frac{ - ( - 1) - \sqrt{9} }{2 \times 1} = \frac{1 - 3}{2} = \frac{ - 2}{2} = - 1 \\ y2 = \frac{ - b + \sqrt{d} }{2a} = \frac{ - ( - 1) + \sqrt{9} }{2 \times 1} = \frac{1 + 3}{2} = \frac{4}{2} = 2$

${x}^{ - \frac{1}{3} } = - 1 \\$

Нет решений

${x}^{ - \frac{1}{3} } = 2 \\ \frac{1}{ \sqrt[3]{x} } = 2 \\ 2 \sqrt[3]{x} = 1 \\ \sqrt[3]{x} = \frac{1}{2} \\ x = \frac{1}{8}$

ответ: x = 1/8