1. x²-x-56=0 2. x²+x-20=0 3. x²-5x-14=0 4. x²+6x-16=0 5. x²+7x+12=0 6. x²-10x+9=0 7. x²+22x+120=0 8. x²+25x+156=0 9. x²-6x+5=0 10. x²-6x-7=0 11. 5x²+4x-1=0 12. 6x²-5x-1=0

Привет! Разберем решение этих квадратных уравнений. Уравнения будем решать через дискриминант.

Напоминаю формулы:

• Дискриминант: \[D = b^2 - 4ac\]
• Корни квадратного уравнения: \[x_{1,2} = \frac{-b \pm \sqrt{D}}{2a}\]

где a, b и c – коэффициенты квадратного уравнения ax² + bx + c = 0.

1. x² - x - 56 = 0

   a = 1, b = -1, c = -56

   \[D = (-1)^2 - 4 \cdot 1 \cdot (-56) = 1 + 224 = 225\]

   \[x_{1} = \frac{1 + \sqrt{225}}{2} = \frac{1 + 15}{2} = 8\]

   \[x_{2} = \frac{1 - \sqrt{225}}{2} = \frac{1 - 15}{2} = -7\]

   Ответ: x₁ = 8, x₂ = -7

2. x² + x - 20 = 0

   a = 1, b = 1, c = -20

   \[D = 1^2 - 4 \cdot 1 \cdot (-20) = 1 + 80 = 81\]

   \[x_{1} = \frac{-1 + \sqrt{81}}{2} = \frac{-1 + 9}{2} = 4\]

   \[x_{2} = \frac{-1 - \sqrt{81}}{2} = \frac{-1 - 9}{2} = -5\]

   Ответ: x₁ = 4, x₂ = -5

3. x² - 5x - 14 = 0

   a = 1, b = -5, c = -14

   \[D = (-5)^2 - 4 \cdot 1 \cdot (-14) = 25 + 56 = 81\]

   \[x_{1} = \frac{5 + \sqrt{81}}{2} = \frac{5 + 9}{2} = 7\]

   \[x_{2} = \frac{5 - \sqrt{81}}{2} = \frac{5 - 9}{2} = -2\]

   Ответ: x₁ = 7, x₂ = -2

4. x² + 6x - 16 = 0

   a = 1, b = 6, c = -16

   \[D = 6^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100\]

   \[x_{1} = \frac{-6 + \sqrt{100}}{2} = \frac{-6 + 10}{2} = 2\]

   \[x_{2} = \frac{-6 - \sqrt{100}}{2} = \frac{-6 - 10}{2} = -8\]

   Ответ: x₁ = 2, x₂ = -8

5. x² + 7x + 12 = 0

   a = 1, b = 7, c = 12

   \[D = 7^2 - 4 \cdot 1 \cdot 12 = 49 - 48 = 1\]

   \[x_{1} = \frac{-7 + \sqrt{1}}{2} = \frac{-7 + 1}{2} = -3\]

   \[x_{2} = \frac{-7 - \sqrt{1}}{2} = \frac{-7 - 1}{2} = -4\]

   Ответ: x₁ = -3, x₂ = -4

6. x² - 10x + 9 = 0

   a = 1, b = -10, c = 9

   \[D = (-10)^2 - 4 \cdot 1 \cdot 9 = 100 - 36 = 64\]

   \[x_{1} = \frac{10 + \sqrt{64}}{2} = \frac{10 + 8}{2} = 9\]

   \[x_{2} = \frac{10 - \sqrt{64}}{2} = \frac{10 - 8}{2} = 1\]

   Ответ: x₁ = 9, x₂ = 1

7. x² + 22x + 120 = 0

   a = 1, b = 22, c = 120

   \[D = 22^2 - 4 \cdot 1 \cdot 120 = 484 - 480 = 4\]

   \[x_{1} = \frac{-22 + \sqrt{4}}{2} = \frac{-22 + 2}{2} = -10\]

   \[x_{2} = \frac{-22 - \sqrt{4}}{2} = \frac{-22 - 2}{2} = -12\]

   Ответ: x₁ = -10, x₂ = -12

8. x² + 25x + 156 = 0

   a = 1, b = 25, c = 156

   \[D = 25^2 - 4 \cdot 1 \cdot 156 = 625 - 624 = 1\]

   \[x_{1} = \frac{-25 + \sqrt{1}}{2} = \frac{-25 + 1}{2} = -12\]

   \[x_{2} = \frac{-25 - \sqrt{1}}{2} = \frac{-25 - 1}{2} = -13\]

   Ответ: x₁ = -12, x₂ = -13

9. x² - 6x + 5 = 0

   a = 1, b = -6, c = 5

   \[D = (-6)^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16\]

   \[x_{1} = \frac{6 + \sqrt{16}}{2} = \frac{6 + 4}{2} = 5\]

   \[x_{2} = \frac{6 - \sqrt{16}}{2} = \frac{6 - 4}{2} = 1\]

   Ответ: x₁ = 5, x₂ = 1

10. x² - 6x - 7 = 0

    a = 1, b = -6, c = -7

    \[D = (-6)^2 - 4 \cdot 1 \cdot (-7) = 36 + 28 = 64\]

    \[x_{1} = \frac{6 + \sqrt{64}}{2} = \frac{6 + 8}{2} = 7\]

    \[x_{2} = \frac{6 - \sqrt{64}}{2} = \frac{6 - 8}{2} = -1\]

    Ответ: x₁ = 7, x₂ = -1

11. 5x² + 4x - 1 = 0

    a = 5, b = 4, c = -1

    \[D = 4^2 - 4 \cdot 5 \cdot (-1) = 16 + 20 = 36\]

    \[x_{1} = \frac{-4 + \sqrt{36}}{10} = \frac{-4 + 6}{10} = 0.2\]

    \[x_{2} = \frac{-4 - \sqrt{36}}{10} = \frac{-4 - 6}{10} = -1\]

    Ответ: x₁ = 0.2, x₂ = -1

12. 6x² - 5x - 1 = 0

    a = 6, b = -5, c = -1

    \[D = (-5)^2 - 4 \cdot 6 \cdot (-1) = 25 + 24 = 49\]

    \[x_{1} = \frac{5 + \sqrt{49}}{12} = \frac{5 + 7}{12} = 1\]

    \[x_{2} = \frac{5 - \sqrt{49}}{12} = \frac{5 - 7}{12} = -\frac{1}{6}\]

    Ответ: x₁ = 1, x₂ = -1/6