8. Решите уравнения: a)3x² +13x -10 = 0; б)2x² = 3x в)16х2 = 49 г)x²-2x-35 = 0

а) Решим уравнение $$ 3x^2 + 13x - 10 = 0 $$.

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

$$ x_{1,2} = \frac{-13 \pm \sqrt{289}}{2 \cdot 3} = \frac{-13 \pm 17}{6} $$.

$$ x_1 = \frac{-13 + 17}{6} = \frac{4}{6} = \frac{2}{3} $$.

$$ x_2 = \frac{-13 - 17}{6} = \frac{-30}{6} = -5 $$.

б) Решим уравнение $$ 2x^2 = 3x $$.

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

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

$$ x_1 = 0 $$, $$ 2x - 3 = 0 $$, $$ 2x = 3 $$, $$ x_2 = \frac{3}{2} = 1.5 $$.

в) Решим уравнение $$ 16x^2 = 49 $$.

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

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

г) Решим уравнение $$ x^2 - 2x - 35 = 0 $$.

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

$$ x_{1,2} = \frac{2 \pm \sqrt{144}}{2 \cdot 1} = \frac{2 \pm 12}{2} $$.

$$ x_1 = \frac{2 + 12}{2} = \frac{14}{2} = 7 $$.

$$ x_2 = \frac{2 - 12}{2} = \frac{-10}{2} = -5 $$.

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