46. Решите уравнение: a) \frac{x² - 1}{2} - 11x = 11; б) \frac{x² + x}{2} = \frac{8x - 7}{3}; в) \frac{4x² - 1}{3} = x (10x - 9); г) \frac{3}{4} x² - \frac{2}{5}x = \frac{4}{5} x² + \frac{3}{4}.

а) Решим уравнение:

\begin{aligned} \frac{x^2 - 1}{2} - 11x &= 11 \\ \frac{x^2 - 1}{2} &= 11x + 11 \\ x^2 - 1 &= 2(11x + 11) \\ x^2 - 1 &= 22x + 22 \\ x^2 - 22x - 1 - 22 &= 0 \\ x^2 - 22x - 23 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= (-22)^2 - 4 \cdot 1 \cdot (-23) = 484 + 92 = 576 \\ \sqrt{D} &= \sqrt{576} = 24 \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-(-22) + 24}{2 \cdot 1} = \frac{22 + 24}{2} = \frac{46}{2} = 23 \\ x_2 &= \frac{-(-22) - 24}{2 \cdot 1} = \frac{22 - 24}{2} = \frac{-2}{2} = -1 \end{aligned}

Ответ: x₁ = 23, x₂ = -1

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б) Решим уравнение:

\begin{aligned} \frac{x^2 + x}{2} &= \frac{8x - 7}{3} \\ 3(x^2 + x) &= 2(8x - 7) \\ 3x^2 + 3x &= 16x - 14 \\ 3x^2 + 3x - 16x + 14 &= 0 \\ 3x^2 - 13x + 14 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= (-13)^2 - 4 \cdot 3 \cdot 14 = 169 - 168 = 1 \\ \sqrt{D} &= \sqrt{1} = 1 \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-(-13) + 1}{2 \cdot 3} = \frac{13 + 1}{6} = \frac{14}{6} = \frac{7}{3} \\ x_2 &= \frac{-(-13) - 1}{2 \cdot 3} = \frac{13 - 1}{6} = \frac{12}{6} = 2 \end{aligned}

Ответ: x_1 = \frac{7}{3}, x_2 = 2

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в) Решим уравнение:

\begin{aligned} \frac{4x^2 - 1}{3} &= x(10x - 9) \\ 4x^2 - 1 &= 3(10x^2 - 9x) \\ 4x^2 - 1 &= 30x^2 - 27x \\ 4x^2 - 30x^2 + 27x - 1 &= 0 \\ -26x^2 + 27x - 1 &= 0 \\ 26x^2 - 27x + 1 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= (-27)^2 - 4 \cdot 26 \cdot 1 = 729 - 104 = 625 \\ \sqrt{D} &= \sqrt{625} = 25 \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-(-27) + 25}{2 \cdot 26} = \frac{27 + 25}{52} = \frac{52}{52} = 1 \\ x_2 &= \frac{-(-27) - 25}{2 \cdot 26} = \frac{27 - 25}{52} = \frac{2}{52} = \frac{1}{26} \end{aligned}

Ответ: x_1 = 1, x_2 = \frac{1}{26}

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г) Решим уравнение:

\begin{aligned} \frac{3}{4}x^2 - \frac{2}{5}x &= \frac{4}{5}x^2 + \frac{3}{4} \\ \frac{3}{4}x^2 - \frac{4}{5}x^2 - \frac{2}{5}x - \frac{3}{4} &= 0 \\ \frac{15x^2 - 16x^2}{20} - \frac{2}{5}x - \frac{3}{4} &= 0 \\ -\frac{1}{20}x^2 - \frac{2}{5}x - \frac{3}{4} &= 0 \\ \frac{1}{20}x^2 + \frac{2}{5}x + \frac{3}{4} &= 0 \\ x^2 + 8x + 15 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= 8^2 - 4 \cdot 1 \cdot 15 = 64 - 60 = 4 \\ \sqrt{D} &= \sqrt{4} = 2 \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-8 + 2}{2 \cdot 1} = \frac{-6}{2} = -3 \\ x_2 &= \frac{-8 - 2}{2 \cdot 1} = \frac{-10}{2} = -5 \end{aligned}

Ответ: x₁ = -3, x₂ = -5