2 Сократите дробь 3x² + 1 6 x + 10 10 - 13 X - 3 X Z

Сократим дробь:

$$\frac{3x^2 + 16x + 10}{10 - 13x - 3x^2}$$

Разложим числитель и знаменатель на множители.

$$3x^2 + 16x + 10 = 0$$

$$D = 16^2 - 4 \cdot 3 \cdot 10 = 256 - 120 = 136$$

$$x_1 = \frac{-16 + \sqrt{136}}{2 \cdot 3} = \frac{-16 + 2\sqrt{34}}{6} = \frac{-8 + \sqrt{34}}{3}$$

$$x_2 = \frac{-16 - \sqrt{136}}{2 \cdot 3} = \frac{-16 - 2\sqrt{34}}{6} = \frac{-8 - \sqrt{34}}{3}$$

$$3x^2 + 16x + 10 = 3(x - \frac{-8 + \sqrt{34}}{3})(x - \frac{-8 - \sqrt{34}}{3})$$

$$-3x^2 - 13x + 10 = 0$$

$$D = (-13)^2 - 4 \cdot (-3) \cdot 10 = 169 + 120 = 289$$

$$x_1 = \frac{13 + \sqrt{289}}{2 \cdot (-3)} = \frac{13 + 17}{-6} = \frac{30}{-6} = -5$$

$$x_2 = \frac{13 - \sqrt{289}}{2 \cdot (-3)} = \frac{13 - 17}{-6} = \frac{-4}{-6} = \frac{2}{3}$$

$$-3x^2 - 13x + 10 = -3(x + 5)(x - \frac{2}{3}) = -(x+5)(3x-2)$$.

$$\frac{3x^2 + 16x + 10}{10 - 13x - 3x^2} = \frac{3(x - \frac{-8 + \sqrt{34}}{3})(x - \frac{-8 - \sqrt{34}}{3})}{-(x+5)(3x-2)}$$

Ответ: $$\frac{3(x - \frac{-8 + \sqrt{34}}{3})(x - \frac{-8 - \sqrt{34}}{3})}{-(x+5)(3x-2)}$$