534. Найдите корни уравнения: a) 5x² - 11x + 2 = 0; б) 2p² + 7p - 30 = 0; в) 9у² - 30у + 25 = 0; г) 35х² + 2x - 1 = 0; д) 2у² - y - 5 = 0;\(\ne\)) 16x² - 8x + 1 = 0.

Решение:

Решим каждое квадратное уравнение, найдя дискриминант \( D = b^2 - 4ac \) и корни по формуле \( x = \frac{-b ± √{D}}{2a} \).

1. \( 5x^2 - 11x + 2 = 0 \). \( D = (-11)^2 - 4 · 5 · 2 = 121 - 40 = 81 \). \( x_1 = \frac{11+9}{10} = 2 \). \( x_2 = \frac{11-9}{10} = 0.2 \).
2. \( 2p^2 + 7p - 30 = 0 \). \( D = 7^2 - 4 · 2 · (-30) = 49 + 240 = 289 \). \( p_1 = \frac{-7+17}{4} = \frac{10}{4} = 2.5 \). \( p_2 = \frac{-7-17}{4} = \frac{-24}{4} = -6 \).
3. \( 9y^2 - 30y + 25 = 0 \). \( D = (-30)^2 - 4 · 9 · 25 = 900 - 900 = 0 \). \( y = \frac{30}{18} = \frac{5}{3} \).
4. \( 35x^2 + 2x - 1 = 0 \). \( D = 2^2 - 4 · 35 · (-1) = 4 + 140 = 144 \). \( x_1 = \frac{-2+12}{70} = \frac{10}{70} = \frac{1}{7} \). \( x_2 = \frac{-2-12}{70} = \frac{-14}{70} = -0.2 \).
5. \( 2y^2 - y - 5 = 0 \). \( D = (-1)^2 - 4 · 2 · (-5) = 1 + 40 = 41 \). \( y_1 = \frac{1+√{41}}{4} \). \( y_2 = \frac{1-√{41}}{4} \).
6. \( 16x^2 - 8x + 1 = 0 \). \( D = (-8)^2 - 4 · 16 · 1 = 64 - 64 = 0 \). \( x = \frac{8}{32} = 0.25 \).

Ответ: а) \( x_1 = 2, x_2 = 0.2 \); б) \( p_1 = 2.5, p_2 = -6 \); в) \( y = \frac{5}{3} \); г) \( x_1 = \frac{1}{7}, x_2 = -0.2 \); д) \( y_1 = \frac{1+√{41}}{4}, y_2 = \frac{1-√{41}}{4} \); е) \( x = 0.25 \).