543. Решите уравнение: a) $$(x + 4)^{2} = 3x + 40$$; б) $$(2p - 3)^{2} = 11p - 19$$; в) $$3(x + 4)^{2} = 10x + 32$$; г) $$15y^{2} + 17 = 15(y + 1)^{2}$$; д) $$(х + 1)^{2} = 7918 - 2x$$; е) $$(m + 2)^{2} = 3131 - 2m$$; ж) $$(x + 1)^{2} = (2x - 1)^{2}$$; з) $$(n - 2)^{2} + 48 = (2-3n)^{2}$$.

Решим уравнения:

а) $$(x + 4)^2 = 3x + 40$$

$$x^2 + 8x + 16 = 3x + 40$$

$$x^2 + 5x - 24 = 0$$

$$D = 5^2 - 4 \cdot 1 \cdot (-24) = 25 + 96 = 121$$

$$x_1 = \frac{-5 + \sqrt{121}}{2} = \frac{-5 + 11}{2} = \frac{6}{2} = 3$$

$$x_2 = \frac{-5 - \sqrt{121}}{2} = \frac{-5 - 11}{2} = \frac{-16}{2} = -8$$

Ответ: $$x_1 = 3, x_2 = -8$$

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б) $$(2p - 3)^2 = 11p - 19$$

$$4p^2 - 12p + 9 = 11p - 19$$

$$4p^2 - 23p + 28 = 0$$

$$D = (-23)^2 - 4 \cdot 4 \cdot 28 = 529 - 448 = 81$$

$$p_1 = \frac{23 + \sqrt{81}}{8} = \frac{23 + 9}{8} = \frac{32}{8} = 4$$

$$p_2 = \frac{23 - \sqrt{81}}{8} = \frac{23 - 9}{8} = \frac{14}{8} = \frac{7}{4} = 1.75$$

Ответ: $$p_1 = 4, p_2 = 1.75$$

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в) $$3(x + 4)^2 = 10x + 32$$

$$3(x^2 + 8x + 16) = 10x + 32$$

$$3x^2 + 24x + 48 = 10x + 32$$

$$3x^2 + 14x + 16 = 0$$

$$D = (14)^2 - 4 \cdot 3 \cdot 16 = 196 - 192 = 4$$

$$x_1 = \frac{-14 + \sqrt{4}}{6} = \frac{-14 + 2}{6} = \frac{-12}{6} = -2$$

$$x_2 = \frac{-14 - \sqrt{4}}{6} = \frac{-14 - 2}{6} = \frac{-16}{6} = -\frac{8}{3}$$

Ответ: $$x_1 = -2, x_2 = -\frac{8}{3}$$

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г) $$15y^2 + 17 = 15(y + 1)^2$$

$$15y^2 + 17 = 15(y^2 + 2y + 1)$$

$$15y^2 + 17 = 15y^2 + 30y + 15$$

$$30y = 2$$

$$y = \frac{2}{30} = \frac{1}{15}$$

Ответ: $$y = \frac{1}{15}$$

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д) $$(x + 1)^2 = 7918 - 2x$$

$$x^2 + 2x + 1 = 7918 - 2x$$

$$x^2 + 4x - 7917 = 0$$

$$D = 4^2 - 4 \cdot 1 \cdot (-7917) = 16 + 31668 = 31684$$

$$x_1 = \frac{-4 + \sqrt{31684}}{2} = \frac{-4 + 178}{2} = \frac{174}{2} = 87$$

$$x_2 = \frac{-4 - \sqrt{31684}}{2} = \frac{-4 - 178}{2} = \frac{-182}{2} = -91$$

Ответ: $$x_1 = 87, x_2 = -91$$

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е) $$(m + 2)^2 = 3131 - 2m$$

$$m^2 + 4m + 4 = 3131 - 2m$$

$$m^2 + 6m - 3127 = 0$$

$$D = 6^2 - 4 \cdot 1 \cdot (-3127) = 36 + 12508 = 12544$$

$$m_1 = \frac{-6 + \sqrt{12544}}{2} = \frac{-6 + 112}{2} = \frac{106}{2} = 53$$

$$m_2 = \frac{-6 - \sqrt{12544}}{2} = \frac{-6 - 112}{2} = \frac{-118}{2} = -59$$

Ответ: $$m_1 = 53, m_2 = -59$$

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ж) $$(x + 1)^2 = (2x - 1)^2$$

$$x^2 + 2x + 1 = 4x^2 - 4x + 1$$

$$3x^2 - 6x = 0$$

$$3x(x - 2) = 0$$

$$x_1 = 0, x_2 = 2$$

Ответ: $$x_1 = 0, x_2 = 2$$

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з) $$(n - 2)^2 + 48 = (2-3n)^2$$

$$n^2 - 4n + 4 + 48 = 4 - 12n + 9n^2$$

$$8n^2 - 8n - 48 = 0$$

$$n^2 - n - 6 = 0$$

$$D = (-1)^2 - 4 \cdot 1 \cdot (-6) = 1 + 24 = 25$$

$$n_1 = \frac{1 + \sqrt{25}}{2} = \frac{1 + 5}{2} = \frac{6}{2} = 3$$

$$n_2 = \frac{1 - \sqrt{25}}{2} = \frac{1 - 5}{2} = \frac{-4}{2} = -2$$

Ответ: $$n_1 = 3, n_2 = -2$$