КВАДРАТНЫЕ УРАВНЕНИЯ Решите уравнения (1-32):

Решение:

1. \( x^2 - 9 = 0 \)

1. \( x^2 = 9 \)
2. \( x = \pm \sqrt{9} \)
3. \( x = \pm 3 \)

2. \( x^2 - 64 = 0 \)

1. \( x^2 = 64 \)
2. \( x = \pm \sqrt{64} \)
3. \( x = \pm 8 \)

3. \( x^2 - 49 = 0 \)

1. \( x^2 = 49 \)
2. \( x = \pm \sqrt{49} \)
3. \( x = \pm 7 \)

4. \( x^2 - 25 = 0 \)

1. \( x^2 = 25 \)
2. \( x = \pm \sqrt{25} \)
3. \( x = \pm 5 \)

5. \( 25x^2 - 1 = 0 \)

1. \( 25x^2 = 1 \)
2. \( x^2 = \frac{1}{25} \)
3. \( x = \pm \sqrt{\frac{1}{25}} \)
4. \( x = \pm \frac{1}{5} \)

6. \( 100x^2 - 1 = 0 \)

1. \( 100x^2 = 1 \)
2. \( x^2 = \frac{1}{100} \)
3. \( x = \pm \sqrt{\frac{1}{100}} \)
4. \( x = \pm \frac{1}{10} \)

7. \( (-x-5)(2x+4) = 0 \)

1. \( -x - 5 = 0 \) или \( 2x + 4 = 0 \)
2. \( -x = 5 \) или \( 2x = -4 \)
3. \( x = -5 \) или \( x = -2 \)

8. \( (6x-3)(-x+3) = 0 \)

1. \( 6x - 3 = 0 \) или \( -x + 3 = 0 \)
2. \( 6x = 3 \) или \( -x = -3 \)
3. \( x = \frac{3}{6} = \frac{1}{2} \) или \( x = 3 \)

9. \( (-x-4)(3x+3) = 0 \)

1. \( -x - 4 = 0 \) или \( 3x + 3 = 0 \)
2. \( -x = 4 \) или \( 3x = -3 \)
3. \( x = -4 \) или \( x = -1 \)

10. \( (5x+2)(x-4) = 0 \)

1. \( 5x + 2 = 0 \) или \( x - 4 = 0 \)
2. \( 5x = -2 \) или \( x = 4 \)
3. \( x = -\frac{2}{5} \) или \( x = 4 \)

11. \( 3x^2 - 9x = 0 \)

1. \( 3x(x - 3) = 0 \)
2. \( 3x = 0 \) или \( x - 3 = 0 \)
3. \( x = 0 \) или \( x = 3 \)

12. \( 5x^2 - 10x = 0 \)

1. \( 5x(x - 2) = 0 \)
2. \( 5x = 0 \) или \( x - 2 = 0 \)
3. \( x = 0 \) или \( x = 2 \)

13. \( 9x^2 = 54x \)

1. \( 9x^2 - 54x = 0 \)
2. \( 9x(x - 6) = 0 \)
3. \( 9x = 0 \) или \( x - 6 = 0 \)
4. \( x = 0 \) или \( x = 6 \)

14. \( 3x^2 = 27x \)

1. \( 3x^2 - 27x = 0 \)
2. \( 3x(x - 9) = 0 \)
3. \( 3x = 0 \) или \( x - 9 = 0 \)
4. \( x = 0 \) или \( x = 9 \)

15. \( x^2 - 8x + 12 = 0 \)

1. \( D = b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16 \)
2. \( x_1 = \frac{-(-8) + \sqrt{16}}{2 \cdot 1} = \frac{8 + 4}{2} = 6 \)
3. \( x_2 = \frac{-(-8) - \sqrt{16}}{2 \cdot 1} = \frac{8 - 4}{2} = 2 \)

16. \( x^2 - 10x + 21 = 0 \)

1. \( D = b^2 - 4ac = (-10)^2 - 4 \cdot 1 \cdot 21 = 100 - 84 = 16 \)
2. \( x_1 = \frac{-(-10) + \sqrt{16}}{2 \cdot 1} = \frac{10 + 4}{2} = 7 \)
3. \( x_2 = \frac{-(-10) - \sqrt{16}}{2 \cdot 1} = \frac{10 - 4}{2} = 3 \)

17. \( 5x^2 + 9x + 4 = 0 \)

1. \( D = b^2 - 4ac = 9^2 - 4 \cdot 5 \cdot 4 = 81 - 80 = 1 \)
2. \( x_1 = \frac{-9 + \sqrt{1}}{2 \cdot 5} = \frac{-9 + 1}{10} = -\frac{8}{10} = -\frac{4}{5} \)
3. \( x_2 = \frac{-9 - \sqrt{1}}{2 \cdot 5} = \frac{-9 - 1}{10} = -\frac{10}{10} = -1 \)

18. \( 5x^2 + 4x - 1 = 0 \)

1. \( D = b^2 - 4ac = 4^2 - 4 \cdot 5 \cdot (-1) = 16 + 20 = 36 \)
2. \( x_1 = \frac{-4 + \sqrt{36}}{2 \cdot 5} = \frac{-4 + 6}{10} = \frac{2}{10} = \frac{1}{5} \)
3. \( x_2 = \frac{-4 - \sqrt{36}}{2 \cdot 5} = \frac{-4 - 6}{10} = -\frac{10}{10} = -1 \)

19. \( x^2 = 2x + 15 \)

1. \( x^2 - 2x - 15 = 0 \)
2. \( D = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64 \)
3. \( x_1 = \frac{-(-2) + \sqrt{64}}{2 \cdot 1} = \frac{2 + 8}{2} = 5 \)
4. \( x_2 = \frac{-(-2) - \sqrt{64}}{2 \cdot 1} = \frac{2 - 8}{2} = -3 \)

20. \( x^2 = 8x - 7 \)

1. \( x^2 - 8x + 7 = 0 \)
2. \( D = b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 7 = 64 - 28 = 36 \)
3. \( x_1 = \frac{-(-8) + \sqrt{36}}{2 \cdot 1} = \frac{8 + 6}{2} = 7 \)
4. \( x_2 = \frac{-(-8) - \sqrt{36}}{2 \cdot 1} = \frac{8 - 6}{2} = 1 \)

21. \( x^2 - 4x = 21 \)

1. \( x^2 - 4x - 21 = 0 \)
2. \( D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-21) = 16 + 84 = 100 \)
3. \( x_1 = \frac{-(-4) + \sqrt{100}}{2 \cdot 1} = \frac{4 + 10}{2} = 7 \)
4. \( x_2 = \frac{-(-4) - \sqrt{100}}{2 \cdot 1} = \frac{4 - 10}{2} = -3 \)

22. \( x^2 - 6x = 16 \)

1. \( x^2 - 6x - 16 = 0 \)
2. \( D = b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100 \)
3. \( x_1 = \frac{-(-6) + \sqrt{100}}{2 \cdot 1} = \frac{6 + 10}{2} = 8 \)
4. \( x_2 = \frac{-(-6) - \sqrt{100}}{2 \cdot 1} = \frac{6 - 10}{2} = -2 \)

23. \( -6x = x^2 + 5 \)

1. \( x^2 + 6x + 5 = 0 \)
2. \( D = b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16 \)
3. \( x_1 = \frac{-6 + \sqrt{16}}{2 \cdot 1} = \frac{-6 + 4}{2} = -1 \)
4. \( x_2 = \frac{-6 - \sqrt{16}}{2 \cdot 1} = \frac{-6 - 4}{2} = -5 \)

24. \( 9x = -x^2 - 18 \)

1. \( x^2 + 9x + 18 = 0 \)
2. \( D = b^2 - 4ac = 9^2 - 4 \cdot 1 \cdot 18 = 81 - 72 = 9 \)
3. \( x_1 = \frac{-9 + \sqrt{9}}{2 \cdot 1} = \frac{-9 + 3}{2} = -3 \)
4. \( x_2 = \frac{-9 - \sqrt{9}}{2 \cdot 1} = \frac{-9 - 3}{2} = -6 \)

25. \( (x+1)^2 + (x-6)^2 = 2x^2 \)

1. \( (x^2 + 2x + 1) + (x^2 - 12x + 36) = 2x^2 \)
2. \( 2x^2 - 10x + 37 = 2x^2 \)
3. \( -10x + 37 = 0 \)
4. \( -10x = -37 \)
5. \( x = \frac{-37}{-10} = 3.7 \)

26. \( (x-2)^2 + (x-8)^2 = 2x^2 \)

1. \( (x^2 - 4x + 4) + (x^2 - 16x + 64) = 2x^2 \)
2. \( 2x^2 - 20x + 68 = 2x^2 \)
3. \( -20x + 68 = 0 \)
4. \( -20x = -68 \)
5. \( x = \frac{-68}{-20} = \frac{17}{5} = 3.4 \)

27. \( (x-6)^2 + (x+8)^2 = 2x^2 \)

1. \( (x^2 - 12x + 36) + (x^2 + 16x + 64) = 2x^2 \)
2. \( 2x^2 + 4x + 100 = 2x^2 \)
3. \( 4x + 100 = 0 \)
4. \( 4x = -100 \)
5. \( x = -25 \)

28. \( (x-2)^2 + (x-3)^2 = 2x^2 \)

1. \( (x^2 - 4x + 4) + (x^2 - 6x + 9) = 2x^2 \)
2. \( 2x^2 - 10x + 13 = 2x^2 \)
3. \( -10x + 13 = 0 \)
4. \( -10x = -13 \)
5. \( x = \frac{-13}{-10} = 1.3 \)

29. \( x^2 + x + 6 = -x^2 - 3x + (-2+2x^2) \)

1. \( x^2 + x + 6 = -x^2 - 3x - 2 + 2x^2 \)
2. \( x^2 + x + 6 = x^2 - 3x - 2 \)
3. \( x + 6 = -3x - 2 \)
4. \( x + 3x = -2 - 6 \)
5. \( 4x = -8 \)
6. \( x = -2 \)

30. \( -3x^2 + 5x - 3 = -x^2 + 3x + (2-2x^2) \)

1. \( -3x^2 + 5x - 3 = -x^2 + 3x + 2 - 2x^2 \)
2. \( -3x^2 + 5x - 3 = -3x^2 + 3x + 2 \)
3. \( 5x - 3 = 3x + 2 \)
4. \( 5x - 3x = 2 + 3 \)
5. \( 2x = 5 \)
6. \( x = \frac{5}{2} = 2.5 \)

31. \( 3x^2 - 4x + 7 = x^2 - 5x + (-1+2x^2) \)

1. \( 3x^2 - 4x + 7 = x^2 - 5x - 1 + 2x^2 \)
2. \( 3x^2 - 4x + 7 = 3x^2 - 5x - 1 \)
3. \( -4x + 7 = -5x - 1 \)
4. \( -4x + 5x = -1 - 7 \)
5. \( x = -8 \)

32. \( 2x - 4x^2 + 6 = 3x - (2x^2 - 3) - 2 \)

1. \( 2x - 4x^2 + 6 = 3x - 2x^2 + 3 - 2 \)
2. \( 2x - 4x^2 + 6 = 3x - 2x^2 + 1 \)
3. \( -4x^2 + 2x + 6 = -2x^2 + 3x + 1 \)
4. \( -4x^2 + 2x^2 + 2x - 3x + 6 - 1 = 0 \)
5. \( -2x^2 - x + 5 = 0 \)
6. \( 2x^2 + x - 5 = 0 \)
7. \( D = b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot (-5) = 1 + 40 = 41 \)
8. \( x_1 = \frac{-1 + \sqrt{41}}{2 \cdot 2} = \frac{-1 + \sqrt{41}}{4} \)
9. \( x_2 = \frac{-1 - \sqrt{41}}{2 \cdot 2} = \frac{-1 - \sqrt{41}}{4} \)