544. Найдите корни уравнения: a) (2x – 3)(5x + 1) = 2x+\frac{2}{5}; б) (3x - 1)(x + 3) = x (1 + 6x); в) (x - 1)(x + 1) = 2(5x-10.) г) -x (x + 7) = (x - 2)(x+2)

a) Решим уравнение:

\begin{aligned} (2x - 3)(5x + 1) &= 2x + \frac{2}{5} \\ 10x^2 + 2x - 15x - 3 &= 2x + \frac{2}{5} \\ 10x^2 - 13x - 3 &= 2x + \frac{2}{5} \\ 10x^2 - 15x - 3 - \frac{2}{5} &= 0 \\ 10x^2 - 15x - \frac{17}{5} &= 0 \\ 50x^2 - 75x - 17 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= (-75)^2 - 4 \cdot 50 \cdot (-17) = 5625 + 3400 = 9025 \\ \sqrt{D} &= \sqrt{9025} = 95 \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-(-75) + 95}{2 \cdot 50} = \frac{75 + 95}{100} = \frac{170}{100} = 1.7 \\ x_2 &= \frac{-(-75) - 95}{2 \cdot 50} = \frac{75 - 95}{100} = \frac{-20}{100} = -0.2 \end{aligned}

Ответ: x₁ = 1.7, x₂ = -0.2

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б) Решим уравнение:

\begin{aligned} (3x - 1)(x + 3) &= x(1 + 6x) \\ 3x^2 + 9x - x - 3 &= x + 6x^2 \\ 3x^2 + 8x - 3 &= x + 6x^2 \\ 3x^2 - 6x^2 + 8x - x - 3 &= 0 \\ -3x^2 + 7x - 3 &= 0 \\ 3x^2 - 7x + 3 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= (-7)^2 - 4 \cdot 3 \cdot 3 = 49 - 36 = 13 \\ \sqrt{D} &= \sqrt{13} \end{aligned}

Найдем корни:

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

Ответ: x_1 = \frac{7 + \sqrt{13}}{6}, x_2 = \frac{7 - \sqrt{13}}{6}

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в) Решим уравнение:

\begin{aligned} (x - 1)(x + 1) &= 2(5x - 10) \\ x^2 - 1 &= 10x - 20 \\ x^2 - 10x - 1 + 20 &= 0 \\ x^2 - 10x + 19 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= (-10)^2 - 4 \cdot 1 \cdot 19 = 100 - 76 = 24 \\ \sqrt{D} &= \sqrt{24} = 2\sqrt{6} \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-(-10) + 2\sqrt{6}}{2 \cdot 1} = \frac{10 + 2\sqrt{6}}{2} = 5 + \sqrt{6} \\ x_2 &= \frac{-(-10) - 2\sqrt{6}}{2 \cdot 1} = \frac{10 - 2\sqrt{6}}{2} = 5 - \sqrt{6} \end{aligned}

Ответ: x_1 = 5 + \sqrt{6}, x_2 = 5 - \sqrt{6}

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г) Решим уравнение:

\begin{aligned} -x(x + 7) &= (x - 2)(x + 2) \\ -x^2 - 7x &= x^2 - 4 \\ -x^2 - x^2 - 7x + 4 &= 0 \\ -2x^2 - 7x + 4 &= 0 \\ 2x^2 + 7x - 4 &= 0 \end{aligned}

Найдем дискриминант:

\begin{aligned} D &= 7^2 - 4 \cdot 2 \cdot (-4) = 49 + 32 = 81 \\ \sqrt{D} &= \sqrt{81} = 9 \end{aligned}

Найдем корни:

\begin{aligned} x_1 &= \frac{-7 + 9}{2 \cdot 2} = \frac{2}{4} = 0.5 \\ x_2 &= \frac{-7 - 9}{2 \cdot 2} = \frac{-16}{4} = -4 \end{aligned}

Ответ: x₁ = 0.5, x₂ = -4