345. Решите уравнение наиболее рациональным способом: 1) = x² = 3x-4; 1x2 2 2) 2x(12x + 5) = 8; 3) 11x2 = 18x + 511; 4) 0,7x2 = 1,3x + 2; 5) 81x2 - 13 = 0; 6) 9x²+2x-8 = 0; 7) 5 + 13x² - 16x = 0; x2 4 7 8) - - = 0; 3 12 9)-13x² - 25x = 0; 2 1 10)x-5+x² = 0; 3 2 11) 8 - 90x + 43x² = 0; x2 + x 8x-7 12) = .

345. Решите уравнение наиболее рациональным способом:

1) $$\frac{1}{2}x^2 = 3x - 4$$

$$x^2 = 6x - 8$$

$$x^2 - 6x + 8 = 0$$

По теореме Виета:

$$\begin{cases}x_1 + x_2 = 6\\x_1 \cdot x_2 = 8\end{cases}$$ Подбором находим, что $$x_1 = 2, x_2 = 4$$.

2) $$2x(12x + 5) = 8$$

$$24x^2 + 10x - 8 = 0$$

$$12x^2 + 5x - 4 = 0$$

$$D = 5^2 - 4 \cdot 12 \cdot (-4) = 25 + 192 = 217$$

$$x_1 = \frac{-5 + \sqrt{217}}{24}, x_2 = \frac{-5 - \sqrt{217}}{24}$$

3) $$11x^2 = 18x + 511$$

$$11x^2 - 18x - 511 = 0$$

$$D = (-18)^2 - 4 \cdot 11 \cdot (-511) = 324 + 22484 = 22808$$

$$x_1 = \frac{18 + \sqrt{22808}}{22}, x_2 = \frac{18 - \sqrt{22808}}{22}$$

4) $$0.7x^2 = 1.3x + 2$$

$$0.7x^2 - 1.3x - 2 = 0$$

$$7x^2 - 13x - 20 = 0$$

$$D = (-13)^2 - 4 \cdot 7 \cdot (-20) = 169 + 560 = 729 = 27^2$$

$$x_1 = \frac{13 + 27}{14} = \frac{40}{14} = \frac{20}{7}, x_2 = \frac{13 - 27}{14} = -1$$

5) $$81x^2 - 13 = 0$$

$$81x^2 = 13$$

$$x^2 = \frac{13}{81}$$

$$x_1 = \frac{\sqrt{13}}{9}, x_2 = -\frac{\sqrt{13}}{9}$$

6) $$9x^2 + 2x - 8 = 0$$

$$D = 2^2 - 4 \cdot 9 \cdot (-8) = 4 + 288 = 292$$

$$x_1 = \frac{-2 + \sqrt{292}}{18}, x_2 = \frac{-2 - \sqrt{292}}{18}$$

7) $$13x^2 - 16x + 5 = 0$$

$$D_1 = (-8)^2 - 13 \cdot 5 = 64 - 65 = -1$$

$$x_1 = \frac{8 + \sqrt{-1}}{13}, x_2 = \frac{8 - \sqrt{-1}}{13}$$

8) $$\frac{x^2}{4} - \frac{x}{3} - \frac{7}{12} = 0$$

$$3x^2 - 4x - 7 = 0$$

$$D_1 = (-2)^2 - 3 \cdot (-7) = 4 + 21 = 25$$

$$x_1 = \frac{2 + 5}{3} = \frac{7}{3}, x_2 = \frac{2 - 5}{3} = -1$$

9) $$-13x^2 - 25x = 0$$

$$x(-13x - 25) = 0$$

$$x_1 = 0, x_2 = -\frac{25}{13}$$

10) $$\frac{2}{3}x - 5 + \frac{1}{2}x^2 = 0$$

$$3x^2 + 4x - 30 = 0$$

$$D_1 = 2^2 - 3 \cdot (-30) = 4 + 90 = 94$$

$$x_1 = \frac{-2 + \sqrt{94}}{3}, x_2 = \frac{-2 - \sqrt{94}}{3}$$

11) $$43x^2 - 90x + 8 = 0$$

$$D_1 = (-45)^2 - 43 \cdot 8 = 2025 - 344 = 1681 = 41^2$$

$$x_1 = \frac{45 + 41}{43} = \frac{86}{43} = 2, x_2 = \frac{45 - 41}{43} = \frac{4}{43}$$

12) $$\frac{x^2 + x}{2} = \frac{8x - 7}{3}$$

$$3(x^2 + x) = 2(8x - 7)$$

$$3x^2 + 3x = 16x - 14$$

$$3x^2 - 13x + 14 = 0$$

$$D = (-13)^2 - 4 \cdot 3 \cdot 14 = 169 - 168 = 1$$

$$x_1 = \frac{13 + 1}{6} = \frac{14}{6} = \frac{7}{3}, x_2 = \frac{13 - 1}{6} = 2$$

Ответ: 1) $$x_1 = 2, x_2 = 4$$; 2) $$x_1 = \frac{-5 + \sqrt{217}}{24}, x_2 = \frac{-5 - \sqrt{217}}{24}$$; 3) $$x_1 = \frac{18 + \sqrt{22808}}{22}, x_2 = \frac{18 - \sqrt{22808}}{22}$$; 4) $$x_1 = \frac{20}{7}, x_2 = -1$$; 5) $$x_1 = \frac{\sqrt{13}}{9}, x_2 = -\frac{\sqrt{13}}{9}$$; 6) $$x_1 = \frac{-2 + \sqrt{292}}{18}, x_2 = \frac{-2 - \sqrt{292}}{18}$$; 7) $$x_1 = \frac{8 + \sqrt{-1}}{13}, x_2 = \frac{8 - \sqrt{-1}}{13}$$; 8) $$x_1 = \frac{7}{3}, x_2 = -1$$; 9) $$x_1 = 0, x_2 = -\frac{25}{13}$$; 10) $$x_1 = \frac{-2 + \sqrt{94}}{3}, x_2 = \frac{-2 - \sqrt{94}}{3}$$; 11) $$x_1 = 2, x_2 = \frac{4}{43}$$; 12) $$x_1 = \frac{7}{3}, x_2 = 2$$