591. Решите уравнение: д) $$\frac{8}{x}=3x+2$$;

Решим уравнение:

$$\frac{8}{x}=3x+2$$

$$8 = x(3x+2)$$

$$8 = 3x^2 + 2x$$

$$3x^2 + 2x - 8 = 0$$

По теореме Виета:

D = $$b^2 - 4ac = 4 - 4 \times 3 \times (-8) = 4 + 96 = 100$$

x1 = $$\frac{-b + \sqrt{D}}{2a} = \frac{-2 + \sqrt{100}}{6} = \frac{-2 + 10}{6} = \frac{8}{6} = \frac{4}{3}$$

x2 = $$\frac{-b - \sqrt{D}}{2a} = \frac{-2 - \sqrt{100}}{6} = \frac{-2 - 10}{6} = \frac{-12}{6} = -2$$

Проверим:

$$\frac{8}{\frac{4}{3}} = 8 \times \frac{3}{4} = 6$$

$$3 \times \frac{4}{3} + 2 = 4 + 2 = 6$$

$$\frac{8}{-2} = -4$$

$$3 \times (-2) + 2 = -6 + 2 = -4$$

Ответ: $$\frac{4}{3}$$; -2