34. Найдите корни уравнения: a) 5x² - 11x + 2 = 0; б) 2p² + 7p - 30 = 0; в) 9y² - 30y + 25 = 0; г) 35x² + 2x - 1 = 0; д) 2y² - y - 5 = 0; e) 16x² - 8x + 1 = 0.

Решим каждое квадратное уравнение:

a) $$5x^2 - 11x + 2 = 0$$

• $$D = (-11)^2 - 4 \cdot 5 \cdot 2 = 121 - 40 = 81$$
• $$x_1 = \frac{11 + \sqrt{81}}{2 \cdot 5} = \frac{11 + 9}{10} = \frac{20}{10} = 2$$
• $$x_2 = \frac{11 - \sqrt{81}}{2 \cdot 5} = \frac{11 - 9}{10} = \frac{2}{10} = \frac{1}{5}$$

б) $$2p^2 + 7p - 30 = 0$$

• $$D = 7^2 - 4 \cdot 2 \cdot (-30) = 49 + 240 = 289$$
• $$p_1 = \frac{-7 + \sqrt{289}}{2 \cdot 2} = \frac{-7 + 17}{4} = \frac{10}{4} = \frac{5}{2}$$
• $$p_2 = \frac{-7 - \sqrt{289}}{2 \cdot 2} = \frac{-7 - 17}{4} = \frac{-24}{4} = -6$$

в) $$9y^2 - 30y + 25 = 0$$

• $$D = (-30)^2 - 4 \cdot 9 \cdot 25 = 900 - 900 = 0$$
• $$y = \frac{30}{2 \cdot 9} = \frac{30}{18} = \frac{5}{3}$$

г) $$35x^2 + 2x - 1 = 0$$

• $$D = 2^2 - 4 \cdot 35 \cdot (-1) = 4 + 140 = 144$$
• $$x_1 = \frac{-2 + \sqrt{144}}{2 \cdot 35} = \frac{-2 + 12}{70} = \frac{10}{70} = \frac{1}{7}$$
• $$x_2 = \frac{-2 - \sqrt{144}}{2 \cdot 35} = \frac{-2 - 12}{70} = \frac{-14}{70} = -\frac{1}{5}$$

д) $$2y^2 - y - 5 = 0$$

• $$D = (-1)^2 - 4 \cdot 2 \cdot (-5) = 1 + 40 = 41$$
• $$y_1 = \frac{1 + \sqrt{41}}{2 \cdot 2} = \frac{1 + \sqrt{41}}{4}$$
• $$y_2 = \frac{1 - \sqrt{41}}{2 \cdot 2} = \frac{1 - \sqrt{41}}{4}$$

e) $$16x^2 - 8x + 1 = 0$$

• $$D = (-8)^2 - 4 \cdot 16 \cdot 1 = 64 - 64 = 0$$
• $$x = \frac{8}{2 \cdot 16} = \frac{8}{32} = \frac{1}{4}$$

Ответ:

• a) $$x_1 = 2$$, $$x_2 = \frac{1}{5}$$
• б) $$p_1 = \frac{5}{2}$$, $$p_2 = -6$$
• в) $$y = \frac{5}{3}$$
• г) $$x_1 = \frac{1}{7}$$, $$x_2 = -\frac{1}{5}$$
• д) $$y_1 = \frac{1 + \sqrt{41}}{4}$$, $$y_2 = \frac{1 - \sqrt{41}}{4}$$
• e) $$x = \frac{1}{4}$$