Solve the quadratic equations:

Решение:

Для решения приведенных квадратных уравнений, воспользуемся формулой дискриминанта: D = b² - 4ac, и формулами для корней: x_1,2 = (-b ± √D) / 2a.

• 4. x² + 14x = 0
  x(x + 14) = 0
  x₁ = 0, x₂ = -14
• 5. x² - x - 20 = 0
  D = (-1)² - 4(1)(-20) = 1 + 80 = 81
  x_1,2 = (1 ± √81) / 2 = (1 ± 9) / 2
  x₁ = 10/2 = 5, x₂ = -8/2 = -4
• 6. x² + x - 30 = 0
  D = 1² - 4(1)(-30) = 1 + 120 = 121
  x_1,2 = (-1 ± √121) / 2 = (-1 ± 11) / 2
  x₁ = 10/2 = 5, x₂ = -12/2 = -6
• 7. x² - x - 12 = 0
  D = (-1)² - 4(1)(-12) = 1 + 48 = 49
  x_1,2 = (1 ± √49) / 2 = (1 ± 7) / 2
  x₁ = 8/2 = 4, x₂ = -6/2 = -3
• 8. x² - 7x + 12 = 0
  D = (-7)² - 4(1)(12) = 49 - 48 = 1
  x_1,2 = (7 ± √1) / 2 = (7 ± 1) / 2
  x₁ = 8/2 = 4, x₂ = 6/2 = 3
• 9. x² - 2,4x - 13 = 0
  D = (-2,4)² - 4(1)(-13) = 5,76 + 52 = 57,76
  √D ≈ 7,6
  x_1,2 = (2,4 ± 7,6) / 2
  x₁ = 10/2 = 5, x₂ = -5,2/2 = -2,6
• 10. x² - 5,6x + 6,4 = 0
  D = (-5,6)² - 4(1)(6,4) = 31,36 - 25,6 = 5,76
  √D = 2,4
  x_1,2 = (5,6 ± 2,4) / 2
  x₁ = 8/2 = 4, x₂ = 3,2/2 = 1,6
• 11. x² + 2,5x + 1 = 0
  D = (2,5)² - 4(1)(1) = 6,25 - 4 = 2,25
  √D = 1,5
  x_1,2 = (-2,5 ± 1,5) / 2
  x₁ = -1/2 = -0,5, x₂ = -4/2 = -2
• 12. x² - 4,5x + 4,5 = 0
  D = (-4,5)² - 4(1)(4,5) = 20,25 - 18 = 2,25
  √D = 1,5
  x_1,2 = (4,5 ± 1,5) / 2
  x₁ = 6/2 = 3, x₂ = 3/2 = 1,5
• 13. -x² + 2x + 8 = 0
  x² - 2x - 8 = 0
  D = (-2)² - 4(1)(-8) = 4 + 32 = 36
  x_1,2 = (2 ± √36) / 2 = (2 ± 6) / 2
  x₁ = 8/2 = 4, x₂ = -4/2 = -2
• 14. -x² + 7x - 10 = 0
  x² - 7x + 10 = 0
  D = (-7)² - 4(1)(10) = 49 - 40 = 9
  x_1,2 = (7 ± √9) / 2 = (7 ± 3) / 2
  x₁ = 10/2 = 5, x₂ = 4/2 = 2
• 15. -x² + 7x + 8 = 0
  x² - 7x - 8 = 0
  D = (-7)² - 4(1)(-8) = 49 + 32 = 81
  x_1,2 = (7 ± √81) / 2 = (7 ± 9) / 2
  x₁ = 16/2 = 8, x₂ = -2/2 = -1
• 16. -x² - 2x + 15 = 0
  x² + 2x - 15 = 0
  D = 2² - 4(1)(-15) = 4 + 60 = 64
  x_1,2 = (-2 ± √64) / 2 = (-2 ± 8) / 2
  x₁ = 6/2 = 3, x₂ = -10/2 = -5
• 17. 3x² - 5x - 2 = 0
  D = (-5)² - 4(3)(-2) = 25 + 24 = 49
  x_1,2 = (5 ± √49) / (2*3) = (5 ± 7) / 6
  x₁ = 12/6 = 2, x₂ = -2/6 = -1/3
• 18. 4x² + x - 3 = 0
  D = 1² - 4(4)(-3) = 1 + 48 = 49
  x_1,2 = (-1 ± √49) / (2*4) = (-1 ± 7) / 8
  x₁ = 6/8 = 3/4, x₂ = -8/8 = -1
• 19. 2x² - 7x + 6 = 0
  D = (-7)² - 4(2)(6) = 49 - 48 = 1
  x_1,2 = (7 ± √1) / (2*2) = (7 ± 1) / 4
  x₁ = 8/4 = 2, x₂ = 6/4 = 3/2
• 20. 5x² - 8x + 3 = 0
  D = (-8)² - 4(5)(3) = 64 - 60 = 4
  x_1,2 = (8 ± √4) / (2*5) = (8 ± 2) / 10
  x₁ = 10/10 = 1, x₂ = 6/10 = 3/5

Ответ:

• 4. x₁ = 0, x₂ = -14
• 5. x₁ = 5, x₂ = -4
• 6. x₁ = 5, x₂ = -6
• 7. x₁ = 4, x₂ = -3
• 8. x₁ = 4, x₂ = 3
• 9. x₁ = 5, x₂ = -2,6
• 10. x₁ = 4, x₂ = 1,6
• 11. x₁ = -0,5, x₂ = -2
• 12. x₁ = 3, x₂ = 1,5
• 13. x₁ = 4, x₂ = -2
• 14. x₁ = 5, x₂ = 2
• 15. x₁ = 8, x₂ = -1
• 16. x₁ = 3, x₂ = -5
• 17. x₁ = 2, x₂ = -1/3
• 18. x₁ = 3/4, x₂ = -1
• 19. x₁ = 2, x₂ = 3/2
• 20. x₁ = 1, x₂ = 3/5