14. Решите уравнения:

Решение:

1. \((x-6)^2 - x(x+8) = 2\)
   
   \(x^2 - 12x + 36 - x^2 - 8x = 2\)
   
   \(-20x = 2 - 36\)
   
   \(-20x = -34\)
   
   \(x = \frac{-34}{-20} = \frac{17}{10} = 1.7\)
2. \(9x(x+6) - (3x+1)^2 = 1\)
   
   \(9x^2 + 54x - (9x^2 + 6x + 1) = 1\)
   
   \(9x^2 + 54x - 9x^2 - 6x - 1 = 1\)
   
   \(48x = 2\)
   
   \(x = \frac{2}{48} = \frac{1}{24}\)
3. \(y(y-1) - (y-5)^2 = 2\)
   
   \(y^2 - y - (y^2 - 10y + 25) = 2\)
   
   \(y^2 - y - y^2 + 10y - 25 = 2\)
   
   \(9y = 27\)
   
   \(y = 3\)
4. \(16y(2-y) + (4y-5)^2 = 0\)
   
   \(32y - 16y^2 + 16y^2 - 40y + 25 = 0\)
   
   \(-8y + 25 = 0\)
   
   \(8y = 25\)
   
   \(y = \frac{25}{8} = 3.125\)
5. \((x-5)^2 - x^2 = 3\)
   
   \(x^2 - 10x + 25 - x^2 = 3\)
   
   \(-10x = 3 - 25\)
   
   \(-10x = -22\)
   
   \(x = \frac{-22}{-10} = 2.2\)
6. \((2y+1)^2 - 4y^2 = 5\)
   
   \(4y^2 + 4y + 1 - 4y^2 = 5\)
   
   \(4y + 1 = 5\)
   
   \(4y = 4\)
   
   \(y = 1\)
7. \(9x^2 - 1 - (3x-2)^2 = 0\)
   
   \(9x^2 - 1 - (9x^2 - 12x + 4) = 0\)
   
   \(9x^2 - 1 - 9x^2 + 12x - 4 = 0\)
   
   \(12x - 5 = 0\)
   
   \(12x = 5\)
   
   \(x = \frac{5}{12}\)
8. \(x + (5x+2)^2 = 25(1+x^2)\)
   
   \(x + 25x^2 + 20x + 4 = 25 + 25x^2\)
   
   \(26x + 4 = 25\)
   
   \(26x = 21\)
   
   \(x = \frac{21}{26}\)

Ответ: 1) 1.7; 2) 1/24; 3) 3; 4) 3.125; 5) 2.2; 6) 1; 7) 5/12; 8) 21/26.