544. Решите уравнение: a) $$\frac{x^{2}-1}{2}-11x = 11$$; б) $$\frac{x^{2}+x}{2}=\frac{8x-7}{3}$$; в) $$\frac{4x^{2}-1}{3} = x(10x - 9)$$; г) $$\frac{3}{4}x^{2}-\frac{2}{5}x = \frac{4}{5}x^{2}+\frac{3}{4}$$.

Решим уравнения:

a) $$\frac{x^2 - 1}{2} - 11x = 11$$

$$x^2 - 1 - 22x = 22$$

$$x^2 - 22x - 23 = 0$$

$$D = (-22)^2 - 4 \cdot 1 \cdot (-23) = 484 + 92 = 576 = 24^2$$

$$x_1 = \frac{22 + \sqrt{576}}{2} = \frac{22 + 24}{2} = \frac{46}{2} = 23$$

$$x_2 = \frac{22 - \sqrt{576}}{2} = \frac{22 - 24}{2} = \frac{-2}{2} = -1$$

Ответ: $$x_1 = 23, x_2 = -1$$

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

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

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

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

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

$$x_1 = \frac{13 + \sqrt{1}}{6} = \frac{13 + 1}{6} = \frac{14}{6} = \frac{7}{3}$$

$$x_2 = \frac{13 - \sqrt{1}}{6} = \frac{13 - 1}{6} = \frac{12}{6} = 2$$

Ответ: $$x_1 = \frac{7}{3}, x_2 = 2$$

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в) $$\frac{4x^2 - 1}{3} = x(10x - 9)$$

$$4x^2 - 1 = 3x(10x - 9)$$

$$4x^2 - 1 = 30x^2 - 27x$$

$$26x^2 - 27x + 1 = 0$$

$$D = (-27)^2 - 4 \cdot 26 \cdot 1 = 729 - 104 = 625 = 25^2$$

$$x_1 = \frac{27 + \sqrt{625}}{52} = \frac{27 + 25}{52} = \frac{52}{52} = 1$$

$$x_2 = \frac{27 - \sqrt{625}}{52} = \frac{27 - 25}{52} = \frac{2}{52} = \frac{1}{26}$$

Ответ: $$x_1 = 1, x_2 = \frac{1}{26}$$

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г) $$\frac{3}{4}x^2 - \frac{2}{5}x = \frac{4}{5}x^2 + \frac{3}{4}$$

$$15x^2 - 8x = 16x^2 + 15$$

$$x^2 + 8x + 15 = 0$$

$$D = 8^2 - 4 \cdot 1 \cdot 15 = 64 - 60 = 4$$

$$x_1 = \frac{-8 + \sqrt{4}}{2} = \frac{-8 + 2}{2} = \frac{-6}{2} = -3$$

$$x_2 = \frac{-8 - \sqrt{4}}{2} = \frac{-8 - 2}{2} = \frac{-10}{2} = -5$$

Ответ: $$x_1 = -3, x_2 = -5$$