532. Решите уравнение: a) 3x² - 7x + 4 = 0; б) 5x² - 8x + 3 = 0; в) 3x² - 13x + 14 = 0; г) 2y² - 9y + 10 = 0; д) 5y² - 6y + 1 = 0;\(\ne\)) 4x² + x - 33 = 0; ж) y² - 10y - 24 = 0; з) p² + p - 90 = 0.

Краткое пояснение:

Решение:

• a) 3x² - 7x + 4 = 0
  D = (-7)² - 4⋅3⋅4 = 49 - 48 = 1
  √D = 1
  x₁ = (7 - 1) / (2⋅3) = 6 / 6 = 1
  x₂ = (7 + 1) / (2⋅3) = 8 / 6 = 4/3
• б) 5x² - 8x + 3 = 0
  D = (-8)² - 4⋅5⋅3 = 64 - 60 = 4
  √D = 2
  x₁ = (8 - 2) / (2⋅5) = 6 / 10 = 3/5
  x₂ = (8 + 2) / (2⋅5) = 10 / 10 = 1
• в) 3x² - 13x + 14 = 0
  D = (-13)² - 4⋅3⋅14 = 169 - 168 = 1
  √D = 1
  x₁ = (13 - 1) / (2⋅3) = 12 / 6 = 2
  x₂ = (13 + 1) / (2⋅3) = 14 / 6 = 7/3
• г) 2y² - 9y + 10 = 0
  D = (-9)² - 4⋅2⋅10 = 81 - 80 = 1
  √D = 1
  y₁ = (9 - 1) / (2⋅2) = 8 / 4 = 2
  y₂ = (9 + 1) / (2⋅2) = 10 / 4 = 5/2
• д) 5y² - 6y + 1 = 0
  D = (-6)² - 4⋅5⋅1 = 36 - 20 = 16
  √D = 4
  y₁ = (6 - 4) / (2⋅5) = 2 / 10 = 1/5
  y₂ = (6 + 4) / (2⋅5) = 10 / 10 = 1
• e) 4x² + x - 33 = 0
  D = 1² - 4⋅4⋅(-33) = 1 + 528 = 529
  √D = 23
  x₁ = (-1 - 23) / (2⋅4) = -24 / 8 = -3
  x₂ = (-1 + 23) / (2⋅4) = 22 / 8 = 11/4
• ж) y² - 10y - 24 = 0
  D = (-10)² - 4⋅1⋅(-24) = 100 + 96 = 196
  √D = 14
  y₁ = (10 - 14) / (2⋅1) = -4 / 2 = -2
  y₂ = (10 + 14) / (2⋅1) = 24 / 2 = 12
• з) p² + p - 90 = 0
  D = 1² - 4⋅1⋅(-90) = 1 + 360 = 361
  √D = 19
  p₁ = (-1 - 19) / (2⋅1) = -20 / 2 = -10
  p₂ = (-1 + 19) / (2⋅1) = 18 / 2 = 9

Ответ:
a) x₁ = 1, x₂ = 4/3
б) x₁ = 3/5, x₂ = 1
в) x₁ = 2, x₂ = 7/3
г) y₁ = 2, y₂ = 5/2
д) y₁ = 1/5, y₂ = 1
e) x₁ = -3, x₂ = 11/4
ж) y₁ = -2, y₂ = 12
з) p₁ = -10, p₂ = 9