8. Решите уравнения: a)3x²+13x-10= 0; б)2x² = 3x в)16x² = 49 г)x²-2x-35= 0

8. Решите уравнения:

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

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

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-13 + \sqrt{289}}{2 \\cdot 3} = \frac{-13 + 17}{6} = \frac{4}{6} = \frac{2}{3}$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-13 - \sqrt{289}}{2 \\cdot 3} = \frac{-13 - 17}{6} = \frac{-30}{6} = -5$$

б) $$2x^2 = 3x$$

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

$$x(2x - 3) = 0$$

$$x_1 = 0$$

$$2x - 3 = 0$$

$$2x = 3$$

$$x_2 = \frac{3}{2} = 1.5$$

в) $$16x^2 = 49$$

$$x^2 = \frac{49}{16}$$

$$x = \pm \sqrt{\frac{49}{16}} = \pm \frac{7}{4} = \pm 1.75$$

г) $$x^2 - 2x - 35 = 0$$

$$D = b^2 - 4ac = (-2)^2 - 4 \\cdot 1 \\cdot (-35) = 4 + 140 = 144$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{2 + \sqrt{144}}{2 \\cdot 1} = \frac{2 + 12}{2} = \frac{14}{2} = 7$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{2 - \sqrt{144}}{2 \\cdot 1} = \frac{2 - 12}{2} = \frac{-10}{2} = -5$$

Ответ: а) $$x_1 = \frac{2}{3}$$, $$x_2 = -5$$, б) $$x_1 = 0$$, $$x_2 = 1.5$$, в) $$x = \pm 1.75$$, г) $$x_1 = 7$$, $$x_2 = -5$$