537. Решите уравнение, используя формулу (II): a) 3x² - 14x + 16 = 0; б) 5p² - 16p + 3 = 0; в) d² + 2d - 80 = 0; г) х² - 22x - 23 = 0; д) 4t² - 36t + 77 = 0; е) 15y² - 22y - 37 = 0; ж) 7z² - 20z + 14 = 0; з) у² - 10у - 25 = 0.

Решим каждое уравнение, используя формулу дискриминанта. a) 3x² - 14x + 16 = 0; D = (-14)² - 4 * 3 * 16 = 196 - 192 = 4; x₁ = (14 + √4) / (2 * 3) = (14 + 2) / 6 = 16 / 6 = 8 / 3; x₂ = (14 - √4) / (2 * 3) = (14 - 2) / 6 = 12 / 6 = 2. Ответ: x₁ = 8/3, x₂ = 2 б) 5p² - 16p + 3 = 0; D = (-16)² - 4 * 5 * 3 = 256 - 60 = 196; p₁ = (16 + √196) / (2 * 5) = (16 + 14) / 10 = 30 / 10 = 3; p₂ = (16 - √196) / (2 * 5) = (16 - 14) / 10 = 2 / 10 = 1 / 5. Ответ: p₁ = 3, p₂ = 1/5 в) d² + 2d - 80 = 0; D = 2² - 4 * 1 * (-80) = 4 + 320 = 324; d₁ = (-2 + √324) / (2 * 1) = (-2 + 18) / 2 = 16 / 2 = 8; d₂ = (-2 - √324) / (2 * 1) = (-2 - 18) / 2 = -20 / 2 = -10. Ответ: d₁ = 8, d₂ = -10 г) x² - 22x - 23 = 0; D = (-22)² - 4 * 1 * (-23) = 484 + 92 = 576; x₁ = (22 + √576) / (2 * 1) = (22 + 24) / 2 = 46 / 2 = 23; x₂ = (22 - √576) / (2 * 1) = (22 - 24) / 2 = -2 / 2 = -1. Ответ: x₁ = 23, x₂ = -1 д) 4t² - 36t + 77 = 0; D = (-36)² - 4 * 4 * 77 = 1296 - 1232 = 64; t₁ = (36 + √64) / (2 * 4) = (36 + 8) / 8 = 44 / 8 = 11 / 2; t₂ = (36 - √64) / (2 * 4) = (36 - 8) / 8 = 28 / 8 = 7 / 2. Ответ: t₁ = 11/2, t₂ = 7/2 е) 15y² - 22y - 37 = 0; D = (-22)² - 4 * 15 * (-37) = 484 + 2220 = 2704; y₁ = (22 + √2704) / (2 * 15) = (22 + 52) / 30 = 74 / 30 = 37 / 15; y₂ = (22 - √2704) / (2 * 15) = (22 - 52) / 30 = -30 / 30 = -1. Ответ: y₁ = 37/15, y₂ = -1 ж) 7z² - 20z + 14 = 0; D = (-20)² - 4 * 7 * 14 = 400 - 392 = 8; z₁ = (20 + √8) / (2 * 7) = (20 + 2√2) / 14 = (10 + √2) / 7; z₂ = (20 - √8) / (2 * 7) = (20 - 2√2) / 14 = (10 - √2) / 7. Ответ: z₁ = (10 + √2) / 7, z₂ = (10 - √2) / 7 з) y² - 10y - 25 = 0; D = (-10)² - 4 * 1 * (-25) = 100 + 100 = 200; y₁ = (10 + √200) / (2 * 1) = (10 + 10√2) / 2 = 5 + 5√2; y₂ = (10 - √200) / (2 * 1) = (10 - 10√2) / 2 = 5 - 5√2. Ответ: y₁ = 5 + 5√2, y₂ = 5 - 5√2