Решите уравнения (1-32):

Решение:

Решим каждое уравнение по очереди.

1. \(x^2 - 9 = 0\)
   \(x^2 = 9\)
   \(x = \pm 3\)
2. \(x^2 - 64 = 0\)
   \(x^2 = 64\)
   \(x = \pm 8\)
3. \(x^2 - 49 = 0\)
   \(x^2 = 49\)
   \(x = \pm 7\)
4. \(x^2 - 25 = 0\)
   \(x^2 = 25\)
   \(x = \pm 5\)
5. \(25x^2 - 1 = 0\)
   \(25x^2 = 1\)
   \(x^2 = \frac{1}{25}\)
   \(x = \pm \frac{1}{5}\)
6. \(100x^2 - 1 = 0\)
   \(100x^2 = 1\)
   \(x^2 = \frac{1}{100}\)
   \(x = \pm \frac{1}{10}\)
7. \((-x-5)(2x+4) = 0\)
   \(-x-5 = 0\) или \(2x+4 = 0\)
   \(-x = 5\) или \(2x = -4\)
   \(x = -5\) или \(x = -2\)
8. \((6x-3)(-x+3) = 0\)
   \(6x-3 = 0\) или \(-x+3 = 0\)
   \(6x = 3\) или \(-x = -3\)
   \(x = \frac{1}{2}\) или \(x = 3\)
9. \((-x-4)(3x+3) = 0\)
   \(-x-4 = 0\) или \(3x+3 = 0\)
   \(-x = 4\) или \(3x = -3\)
   \(x = -4\) или \(x = -1\)
10. \((5x+2)(x-4) = 0\)
    \(5x+2 = 0\) или \(x-4 = 0\)
    \(5x = -2\) или \(x = 4\)
    \(x = -\frac{2}{5}\) или \(x = 4\)
11. \(3x^2 - 9x = 0\)
    \(3x(x-3) = 0\)
    \(3x = 0\) или \(x-3 = 0\)
    \(x = 0\) или \(x = 3\)
12. \(5x^2 - 10x = 0\)
    \(5x(x-2) = 0\)
    \(5x = 0\) или \(x-2 = 0\)
    \(x = 0\) или \(x = 2\)
13. \(9x^2 = 54x\)
    \(9x^2 - 54x = 0\)
    \(9x(x-6) = 0\)
    \(9x = 0\) или \(x-6 = 0\)
    \(x = 0\) или \(x = 6\)
14. \(3x^2 = 27x\)
    \(3x^2 - 27x = 0\)
    \(3x(x-9) = 0\)
    \(3x = 0\) или \(x-9 = 0\)
    \(x = 0\) или \(x = 9\)
15. \(x^2 - 8x + 12 = 0\)
    \(D = (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16\)
    \(x = \frac{8 \pm \sqrt{16}}{2} = \frac{8 \pm 4}{2}\)
    \(x_1 = \frac{12}{2} = 6\), \(x_2 = \frac{4}{2} = 2\)
16. \(x^2 - 10x + 21 = 0\)
    \(D = (-10)^2 - 4 \cdot 1 \cdot 21 = 100 - 84 = 16\)
    \(x = \frac{10 \pm \sqrt{16}}{2} = \frac{10 \pm 4}{2}\)
    \(x_1 = \frac{14}{2} = 7\), \(x_2 = \frac{6}{2} = 3\)
17. \(5x^2 + 9x + 4 = 0\)
    \(D = 9^2 - 4 \cdot 5 \cdot 4 = 81 - 80 = 1\)
    \(x = \frac{-9 \pm \sqrt{1}}{2 \cdot 5} = \frac{-9 \pm 1}{10}\)
    \(x_1 = \frac{-8}{10} = -0.8\), \(x_2 = \frac{-10}{10} = -1\)
18. \(5x^2 + 4x - 1 = 0\)
    \(D = 4^2 - 4 \cdot 5 \cdot (-1) = 16 + 20 = 36\)
    \(x = \frac{-4 \pm \sqrt{36}}{2 \cdot 5} = \frac{-4 \pm 6}{10}\)
    \(x_1 = \frac{2}{10} = 0.2\), \(x_2 = \frac{-10}{10} = -1\)
19. \(x^2 = 2x + 15\)
    \(x^2 - 2x - 15 = 0\)
    \(D = (-2)^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64\)
    \(x = \frac{2 \pm \sqrt{64}}{2} = \frac{2 \pm 8}{2}\)
    \(x_1 = \frac{10}{2} = 5\), \(x_2 = \frac{-6}{2} = -3\)
20. \(x^2 = 8x - 7\)
    \(x^2 - 8x + 7 = 0\)
    \(D = (-8)^2 - 4 \cdot 1 \cdot 7 = 64 - 28 = 36\)
    \(x = \frac{8 \pm \sqrt{36}}{2} = \frac{8 \pm 6}{2}\)
    \(x_1 = \frac{14}{2} = 7\), \(x_2 = \frac{2}{2} = 1\)
21. \(x^2 - 4x = 21\)
    \(x^2 - 4x - 21 = 0\)
    \(D = (-4)^2 - 4 \cdot 1 \cdot (-21) = 16 + 84 = 100\)
    \(x = \frac{4 \pm \sqrt{100}}{2} = \frac{4 \pm 10}{2}\)
    \(x_1 = \frac{14}{2} = 7\), \(x_2 = \frac{-6}{2} = -3\)
22. \(x^2 - 6x = 16\)
    \(x^2 - 6x - 16 = 0\)
    \(D = (-6)^2 - 4 \cdot 1 \cdot (-16) = 36 + 64 = 100\)
    \(x = \frac{6 \pm \sqrt{100}}{2} = \frac{6 \pm 10}{2}\)
    \(x_1 = \frac{16}{2} = 8\), \(x_2 = \frac{-4}{2} = -2\)
23. \(-6x = x^2 + 5\)
    \(x^2 + 6x + 5 = 0\)
    \(D = 6^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16\)
    \(x = \frac{-6 \pm \sqrt{16}}{2} = \frac{-6 \pm 4}{2}\)
    \(x_1 = \frac{-2}{2} = -1\), \(x_2 = \frac{-10}{2} = -5\)
24. \(9x = -x^2 - 18\)
    \(x^2 + 9x + 18 = 0\)
    \(D = 9^2 - 4 \cdot 1 \cdot 18 = 81 - 72 = 9\)
    \(x = \frac{-9 \pm \sqrt{9}}{2} = \frac{-9 \pm 3}{2}\)
    \(x_1 = \frac{-6}{2} = -3\), \(x_2 = \frac{-12}{2} = -6\)
25. \((x+1)^2 + (x-6)^2 = 2x^2\)
    \(x^2 + 2x + 1 + x^2 - 12x + 36 = 2x^2\)
    \(2x^2 - 10x + 37 = 2x^2\)
    \(-10x + 37 = 0\)
    \(-10x = -37\)
    \(x = 3.7\)
26. \((x-2)^2 + (x-8)^2 = 2x^2\)
    \(x^2 - 4x + 4 + x^2 - 16x + 64 = 2x^2\)
    \(2x^2 - 20x + 68 = 2x^2\)
    \(-20x + 68 = 0\)
    \(-20x = -68\)
    \(x = \frac{68}{20} = \frac{17}{5} = 3.4\)
27. \((x-6)^2 + (x+8)^2 = 2x^2\)
    \(x^2 - 12x + 36 + x^2 + 16x + 64 = 2x^2\)
    \(2x^2 + 4x + 100 = 2x^2\)
    \(4x + 100 = 0\)
    \(4x = -100\)
    \(x = -25\)
28. \((x-2)^2 + (x-3)^2 = 2x^2\)
    \(x^2 - 4x + 4 + x^2 - 6x + 9 = 2x^2\)
    \(2x^2 - 10x + 13 = 2x^2\)
    \(-10x + 13 = 0\)
    \(-10x = -13\)
    \(x = 1.3\)
29. \(x^2+x+6 = -x^2-3x+(-2+2x^2)\)
    \(x^2+x+6 = -x^2-3x-2+2x^2\)
    \(x^2+x+6 = x^2-3x-2\)
    \(x+6 = -3x-2\)
    \(4x = -8\)
    \(x = -2\)
30. \(-3x^2+5x-3 = -x^2+3x+(2-2x^2)\)
    \(-3x^2+5x-3 = -x^2+3x+2-2x^2\)
    \(-3x^2+5x-3 = -3x^2+3x+2\)
    \(5x-3 = 3x+2\)
    \(2x = 5\)
    \(x = 2.5\)
31. \(3x^2-4x+7 = x^2-5x+(-1+2x^2)\)
    \(3x^2-4x+7 = x^2-5x-1+2x^2\)
    \(3x^2-4x+7 = 3x^2-5x-1\)
    \(-4x+7 = -5x-1\)
    \(x = -8\)
32. \(2x-4x^2+6 = 3x-(2x^2-3)-2x^2\)
    \(2x-4x^2+6 = 3x-2x^2+3-2x^2\)
    \(2x-4x^2+6 = -4x^2+3x+3\)
    \(2x+6 = 3x+3\)
    \(-x = -3\)
    \(x = 3\)

Ответ: 1) \(\pm 3\); 2) \(\pm 8\); 3) \(\pm 7\); 4) \(\pm 5\); 5) \(\pm \frac{1}{5}\); 6) \(\pm \frac{1}{10}\); 7) \(-5; -2\); 8) \(\frac{1}{2}; 3\); 9) \(-4; -1\); 10) \(-\frac{2}{5}; 4\); 11) \(0; 3\); 12) \(0; 2\); 13) \(0; 6\); 14) \(0; 9\); 15) \(2; 6\); 16) \(3; 7\); 17) \(-1; -0.8\); 18) \(-1; 0.2\); 19) \(-3; 5\); 20) \(1; 7\); 21) \(-3; 7\); 22) \(-2; 8\); 23) \(-1; -5\); 24) \(-3; -6\); 25) \(3.7\); 26) \(3.4\); 27) \(-25\); 28) \(1.3\); 29) \(-2\); 30) \(2.5\); 31) \(-8\); 32) \(3\).