Задание 46. Решите квадратные уравнения.

Краткое пояснение:

Для решения квадратных уравнений будем использовать формулу корней квадратного уравнения: \( x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} \), где \( D = b^2 - 4ac \) (дискриминант).

Пошаговое решение:

1. Уравнение 1: \( x^2 + 5x - 14 = 0 \)
• \( a = 1, b = 5, c = -14 \)
• \( D = 5^2 - 4 · 1 · (-14) = 25 + 56 = 81 \)
• \( √{D} = √{81} = 9 \)
• \( x_1 = \frac{-5 + 9}{2 · 1} = \frac{4}{2} = 2 \)
• \( x_2 = \frac{-5 - 9}{2 · 1} = \frac{-14}{2} = -7 \)
2. Уравнение 2: \( 12x^2 + 7x + 1 = 0 \)
• \( a = 12, b = 7, c = 1 \)
• \( D = 7^2 - 4 · 12 · 1 = 49 - 48 = 1 \)
• \( √{D} = √{1} = 1 \)
• \( x_1 = \frac{-7 + 1}{2 · 12} = \frac{-6}{24} = -rac{1}{4} \)
• \( x_2 = \frac{-7 - 1}{2 · 12} = \frac{-8}{24} = -rac{1}{3} \)
3. Уравнение 3: \( 6x^2 - x - 1 = 0 \)
• \( a = 6, b = -1, c = -1 \)
• \( D = (-1)^2 - 4 · 6 · (-1) = 1 + 24 = 25 \)
• \( √{D} = √{25} = 5 \)
• \( x_1 = \frac{-(-1) + 5}{2 · 6} = \frac{1 + 5}{12} = rac{6}{12} = rac{1}{2} \)
• \( x_2 = \frac{-(-1) - 5}{2 · 6} = \frac{1 - 5}{12} = rac{-4}{12} = -rac{1}{3} \)
4. Уравнение 4: \( 4x^2 + 4x + 1 = 0 \)
• \( a = 4, b = 4, c = 1 \)
• \( D = 4^2 - 4 · 4 · 1 = 16 - 16 = 0 \)
• \( x_1 = x_2 = \frac{-4}{2 · 4} = \frac{-4}{8} = -rac{1}{2} \)
5. Уравнение 5: \( 9x^2 - 8x - 1 = 0 \)
• \( a = 9, b = -8, c = -1 \)
• \( D = (-8)^2 - 4 · 9 · (-1) = 64 + 36 = 100 \)
• \( √{D} = √{100} = 10 \)
• \( x_1 = \frac{-(-8) + 10}{2 · 9} = \frac{8 + 10}{18} = rac{18}{18} = 1 \)
• \( x_2 = \frac{-(-8) - 10}{2 · 9} = \frac{8 - 10}{18} = rac{-2}{18} = -rac{1}{9} \)
6. Уравнение 6: \( -x^2 - 3x + 4 = 0 \)
• \( a = -1, b = -3, c = 4 \)
• \( D = (-3)^2 - 4 · (-1) · 4 = 9 + 16 = 25 \)
• \( √{D} = √{25} = 5 \)
• \( x_1 = \frac{-(-3) + 5}{2 · (-1)} = \frac{3 + 5}{-2} = rac{8}{-2} = -4 \)
• \( x_2 = \frac{-(-3) - 5}{2 · (-1)} = \frac{3 - 5}{-2} = rac{-2}{-2} = 1 \)
7. Уравнение 7: \( 2x^2 - 2x + 5 = 0 \)
• \( a = 2, b = -2, c = 5 \)
• \( D = (-2)^2 - 4 · 2 · 5 = 4 - 40 = -36 \)
• Так как \( D < 0 \), действительных корней нет.
8. Уравнение 8: \( x^2 + 7x - 8 = 0 \)
• \( a = 1, b = 7, c = -8 \)
• \( D = 7^2 - 4 · 1 · (-8) = 49 + 32 = 81 \)
• \( √{D} = √{81} = 9 \)
• \( x_1 = \frac{-7 + 9}{2 · 1} = \frac{2}{2} = 1 \)
• \( x_2 = \frac{-7 - 9}{2 · 1} = \frac{-16}{2} = -8 \)
9. Уравнение 9: \( 6x^2 + 7x + 2 = 0 \)
• \( a = 6, b = 7, c = 2 \)
• \( D = 7^2 - 4 · 6 · 2 = 49 - 48 = 1 \)
• \( √{D} = √{1} = 1 \)
• \( x_1 = \frac{-7 + 1}{2 · 6} = \frac{-6}{12} = -rac{1}{2} \)
• \( x_2 = \frac{-7 - 1}{2 · 6} = \frac{-8}{12} = -rac{2}{3} \)
10. Уравнение 10: \( -6x^2 - x + 5 = 0 \)
• \( a = -6, b = -1, c = 5 \)
• \( D = (-1)^2 - 4 · (-6) · 5 = 1 + 120 = 121 \)
• \( √{D} = √{121} = 11 \)
• \( x_1 = \frac{-(-1) + 11}{2 · (-6)} = \frac{1 + 11}{-12} = rac{12}{-12} = -1 \)
• \( x_2 = \frac{-(-1) - 11}{2 · (-6)} = \frac{1 - 11}{-12} = rac{-10}{-12} = rac{5}{6} \)

Ответ:

• 1) \( x_1 = 2, x_2 = -7 \)
• 2) \( x_1 = -1/4, x_2 = -1/3 \)
• 3) \( x_1 = 1/2, x_2 = -1/3 \)
• 4) \( x_1 = x_2 = -1/2 \)
• 5) \( x_1 = 1, x_2 = -1/9 \)
• 6) \( x_1 = -4, x_2 = 1 \)
• 7) Действительных корней нет.
• 8) \( x_1 = 1, x_2 = -8 \)
• 9) \( x_1 = -1/2, x_2 = -2/3 \)
• 10) \( x_1 = -1, x_2 = 5/6 \)