Решите уравнения: 15. x² - 9x + 18 = 0 16. x² - 10x + 24 = 0 17. x² + 3x = 10 18. x² + 7x = 18 19. x² + 2x = 15 20. x² - 6x = 16 21. x² + 18 = 9x 22. x² + 7 = 8x 23. x² - 20 = x 24. x² - 35 = 2x 25. 2x² - 3x + 1 = 0 26. 5x² + 4x - 1 = 0 27. 2x² + 5x - 7 = 0 28. 5x² - 12x + 7 = 0

Решение:

Для решения квадратных уравнений будем использовать формулу дискриминанта \( D = b^2 - 4ac \) и корни \( x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} \).

1. Уравнение 15: \( x^2 - 9x + 18 = 0 \)
   \( a = 1, b = -9, c = 18 \)
   \( D = (-9)^2 - 4 \cdot 1 \cdot 18 = 81 - 72 = 9 \)
   \( \sqrt{D} = 3 \)
   \( x_1 = \frac{9 + 3}{2} = 6 \)
   \( x_2 = \frac{9 - 3}{2} = 3 \)
2. Уравнение 16: \( x^2 - 10x + 24 = 0 \)
   \( a = 1, b = -10, c = 24 \)
   \( D = (-10)^2 - 4 \cdot 1 \cdot 24 = 100 - 96 = 4 \)
   \( \sqrt{D} = 2 \)
   \( x_1 = \frac{10 + 2}{2} = 6 \)
   \( x_2 = \frac{10 - 2}{2} = 4 \)
3. Уравнение 17: \( x^2 + 3x - 10 = 0 \)
   \( a = 1, b = 3, c = -10 \)
   \( D = 3^2 - 4 \cdot 1 \cdot (-10) = 9 + 40 = 49 \)
   \( \sqrt{D} = 7 \)
   \( x_1 = \frac{-3 + 7}{2} = 2 \)
   \( x_2 = \frac{-3 - 7}{2} = -5 \)
4. Уравнение 18: \( x^2 + 7x - 18 = 0 \)
   \( a = 1, b = 7, c = -18 \)
   \( D = 7^2 - 4 \cdot 1 \cdot (-18) = 49 + 72 = 121 \)
   \( \sqrt{D} = 11 \)
   \( x_1 = \frac{-7 + 11}{2} = 2 \)
   \( x_2 = \frac{-7 - 11}{2} = -9 \)
5. Уравнение 19: \( x^2 + 2x - 15 = 0 \)
   \( a = 1, b = 2, c = -15 \)
   \( D = 2^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64 \)
   \( \sqrt{D} = 8 \)
   \( x_1 = \frac{-2 + 8}{2} = 3 \)
   \( x_2 = \frac{-2 - 8}{2} = -5 \)
6. Уравнение 20: \( x^2 - 6x - 16 = 0 \)
   \( a = 1, b = -6, c = -16 \)
   \( D = (-6)^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 \)
   \( \sqrt{D} = 10 \)
   \( x_1 = \frac{6 + 10}{2} = 8 \)
   \( x_2 = \frac{6 - 10}{2} = -2 \)
7. Уравнение 21: \( x^2 - 9x + 18 = 0 \)
   \( x^2 - 9x + 18 = 0 \)
   \( a = 1, b = -9, c = 18 \)
   \( D = (-9)^2 - 4 \cdot 1 \cdot 18 = 81 - 72 = 9 \)
   \( \sqrt{D} = 3 \)
   \( x_1 = \frac{9 + 3}{2} = 6 \)
   \( x_2 = \frac{9 - 3}{2} = 3 \)
8. Уравнение 22: \( x^2 - 8x + 7 = 0 \)
   \( a = 1, b = -8, c = 7 \)
   \( D = (-8)^2 - 4 \cdot 1 \cdot 7 = 64 - 28 = 36 \)
   \( \sqrt{D} = 6 \)
   \( x_1 = \frac{8 + 6}{2} = 7 \)
   \( x_2 = \frac{8 - 6}{2} = 1 \)
9. Уравнение 23: \( x^2 - x - 20 = 0 \)
   \( a = 1, b = -1, c = -20 \)
   \( D = (-1)^2 - 4 \cdot 1 \cdot (-20) = 1 + 80 = 81 \)
   \( \sqrt{D} = 9 \)
   \( x_1 = \frac{1 + 9}{2} = 5 \)
   \( x_2 = \frac{1 - 9}{2} = -4 \)
10. Уравнение 24: \( x^2 - 2x - 35 = 0 \)
    \( a = 1, b = -2, c = -35 \)
    \( D = (-2)^2 - 4 \cdot 1 \cdot (-35) = 4 + 140 = 144 \)
    \( \sqrt{D} = 12 \)
    \( x_1 = \frac{2 + 12}{2} = 7 \)
    \( x_2 = \frac{2 - 12}{2} = -5 \)
11. Уравнение 25: \( 2x^2 - 3x + 1 = 0 \)
    \( a = 2, b = -3, c = 1 \)
    \( D = (-3)^2 - 4 \cdot 2 \cdot 1 = 9 - 8 = 1 \)
    \( \sqrt{D} = 1 \)
    \( x_1 = \frac{3 + 1}{2 \cdot 2} = 1 \)
    \( x_2 = \frac{3 - 1}{2 \cdot 2} = \frac{2}{4} = 0.5 \)
12. Уравнение 26: \( 5x^2 + 4x - 1 = 0 \)
    \( a = 5, b = 4, c = -1 \)
    \( D = 4^2 - 4 \cdot 5 \cdot (-1) = 16 + 20 = 36 \)
    \( \sqrt{D} = 6 \)
    \( x_1 = \frac{-4 + 6}{2 \cdot 5} = \frac{2}{10} = 0.2 \)
    \( x_2 = \frac{-4 - 6}{2 \cdot 5} = \frac{-10}{10} = -1 \)
13. Уравнение 27: \( 2x^2 + 5x - 7 = 0 \)
    \( a = 2, b = 5, c = -7 \)
    \( D = 5^2 - 4 \cdot 2 \cdot (-7) = 25 + 56 = 81 \)
    \( \sqrt{D} = 9 \)
    \( x_1 = \frac{-5 + 9}{2 \cdot 2} = \frac{4}{4} = 1 \)
    \( x_2 = \frac{-5 - 9}{2 \cdot 2} = \frac{-14}{4} = -3.5 \)
14. Уравнение 28: \( 5x^2 - 12x + 7 = 0 \)
    \( a = 5, b = -12, c = 7 \)
    \( D = (-12)^2 - 4 \cdot 5 \cdot 7 = 144 - 140 = 4 \)
    \( \sqrt{D} = 2 \)
    \( x_1 = \frac{12 + 2}{2 \cdot 5} = \frac{14}{10} = 1.4 \)
    \( x_2 = \frac{12 - 2}{2 \cdot 5} = \frac{10}{10} = 1 \)

Ответ:

• 15: \( x_1 = 6, x_2 = 3 \)
• 16: \( x_1 = 6, x_2 = 4 \)
• 17: \( x_1 = 2, x_2 = -5 \)
• 18: \( x_1 = 2, x_2 = -9 \)
• 19: \( x_1 = 3, x_2 = -5 \)
• 20: \( x_1 = 8, x_2 = -2 \)
• 21: \( x_1 = 6, x_2 = 3 \)
• 22: \( x_1 = 7, x_2 = 1 \)
• 23: \( x_1 = 5, x_2 = -4 \)
• 24: \( x_1 = 7, x_2 = -5 \)
• 25: \( x_1 = 1, x_2 = 0.5 \)
• 26: \( x_1 = 0.2, x_2 = -1 \)
• 27: \( x_1 = 1, x_2 = -3.5 \)
• 28: \( x_1 = 1.4, x_2 = 1 \)