534. 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;\(\ne\)) 16x² - 8x + 1 = 0.

Question

Accepted Answer

Краткое пояснение: Корни квадратного уравнения ax² + bx + c = 0 находятся по формулам x₁,₂ = (-b ± √D) / 2a, где D = b² - 4ac.

a) 5x² - 11x + 2 = 0
D = (-11)² - 4⋅5⋅2 = 121 - 40 = 81
√D = 9
x₁ = (11 - 9) / (2⋅5) = 2 / 10 = 1/5
x₂ = (11 + 9) / (2⋅5) = 20 / 10 = 2
б) 2p² + 7p - 30 = 0
D = 7² - 4⋅2⋅(-30) = 49 + 240 = 289
√D = 17
p₁ = (-7 - 17) / (2⋅2) = -24 / 4 = -6
p₂ = (-7 + 17) / (2⋅2) = 10 / 4 = 5/2
в) 9y² - 30y + 25 = 0
D = (-30)² - 4⋅9⋅25 = 900 - 900 = 0
√D = 0
y = (30 - 0) / (2⋅9) = 30 / 18 = 5/3
г) 35x² + 2x - 1 = 0
D = 2² - 4⋅35⋅(-1) = 4 + 140 = 144
√D = 12
x₁ = (-2 - 12) / (2⋅35) = -14 / 70 = -1/5
x₂ = (-2 + 12) / (2⋅35) = 10 / 70 = 1/7
д) 2y² - y - 5 = 0
D = (-1)² - 4⋅2⋅(-5) = 1 + 40 = 41
√D = √41
y₁ = (1 - √41) / (2⋅2) = (1 - √41) / 4
y₂ = (1 + √41) / (2⋅2) = (1 + √41) / 4
e) 16x² - 8x + 1 = 0
D = (-8)² - 4⋅16⋅1 = 64 - 64 = 0
√D = 0
x = (8 - 0) / (2⋅16) = 8 / 32 = 1/4

Ответ:
a) x₁ = 1/5, x₂ = 2
б) p₁ = -6, p₂ = 5/2
в) y = 5/3
г) x₁ = -1/5, x₂ = 1/7
д) y₁ = (1 - √41) / 4, y₂ = (1 + √41) / 4
e) x = 1/4