537. Решите уравнение, используя формулу (II): a) 3x² - 14x + 16 = 0; б) 5p² - 16p + 3 = 0; в) d² + 2d - 80 = 0; г) 4t² - 36t + 77 = 0; д) 15y² - 22y - 3 = 0;\(\ne\)) 7z² - 20z + 1 = 0; ж) y² - 10y - 25 = 0.

Краткое пояснение:

Решение:

• a) 3x² - 14x + 16 = 0
  D = (-14)² - 4⋅3⋅16 = 196 - 192 = 4
  √D = 2
  x₁ = (14 - 2) / (2⋅3) = 12 / 6 = 2
  x₂ = (14 + 2) / (2⋅3) = 16 / 6 = 8/3
• б) 5p² - 16p + 3 = 0
  D = (-16)² - 4⋅5⋅3 = 256 - 60 = 196
  √D = 14
  p₁ = (16 - 14) / (2⋅5) = 2 / 10 = 1/5
  p₂ = (16 + 14) / (2⋅5) = 30 / 10 = 3
• в) d² + 2d - 80 = 0
  D = 2² - 4⋅1⋅(-80) = 4 + 320 = 324
  √D = 18
  d₁ = (-2 - 18) / 2 = -20 / 2 = -10
  d₂ = (-2 + 18) / 2 = 16 / 2 = 8
• г) 4t² - 36t + 77 = 0
  D = (-36)² - 4⋅4⋅77 = 1296 - 1232 = 64
  √D = 8
  t₁ = (36 - 8) / (2⋅4) = 28 / 8 = 7/2
  t₂ = (36 + 8) / (2⋅4) = 44 / 8 = 11/2
• д) 15y² - 22y - 3 = 0
  D = (-22)² - 4⋅15⋅(-3) = 484 + 180 = 664
  √D = √664 = 2√166
  y₁ = (22 - 2√166) / (2⋅15) = (11 - √166) / 15
  y₂ = (22 + 2√166) / (2⋅15) = (11 + √166) / 15
• e) 7z² - 20z + 1 = 0
  D = (-20)² - 4⋅7⋅1 = 400 - 28 = 372
  √D = √372 = 2√93
  z₁ = (20 - 2√93) / (2⋅7) = (10 - √93) / 7
  z₂ = (20 + 2√93) / (2⋅7) = (10 + √93) / 7
• ж) y² - 10y - 25 = 0
  D = (-10)² - 4⋅1⋅(-25) = 100 + 100 = 200
  √D = √200 = 10√2
  y₁ = (10 - 10√2) / 2 = 5 - 5√2
  y₂ = (10 + 10√2) / 2 = 5 + 5√2

Ответ:
a) x₁ = 2, x₂ = 8/3
б) p₁ = 1/5, p₂ = 3
в) d₁ = -10, d₂ = 8
г) t₁ = 7/2, t₂ = 11/2
д) y₁ = (11 - √166) / 15, y₂ = (11 + √166) / 15
e) z₁ = (10 - √93) / 7, z₂ = (10 + √93) / 7
ж) y₁ = 5 - 5√2, y₂ = 5 + 5√2