748. При каких значениях х вер a) (5x + 3)² = 5(x + 3); б) (3x + 10)² = 3(x + 10); в) (3x - 8)² = 3x² - 8x; г) (4x + 5)² = 5x² + 4x;

Решим данные уравнения.

a) $$(5x + 3)^2 = 5(x + 3)$$

$$25x^2 + 30x + 9 = 5x + 15$$

$$25x^2 + 25x - 6 = 0$$

Вычислим дискриминант:

$$D = b^2 - 4ac = 25^2 - 4 \cdot 25 \cdot (-6) = 625 + 600 = 1225$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-25 + \sqrt{1225}}{2 \cdot 25} = \frac{-25 + 35}{50} = \frac{10}{50} = \frac{1}{5} = 0.2$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-25 - \sqrt{1225}}{2 \cdot 25} = \frac{-25 - 35}{50} = \frac{-60}{50} = -\frac{6}{5} = -1.2$$

Ответ: $$x_1 = 0.2$$, $$x_2 = -1.2$$

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б) $$(3x + 10)^2 = 3(x + 10)$$

$$9x^2 + 60x + 100 = 3x + 30$$

$$9x^2 + 57x + 70 = 0$$

Вычислим дискриминант:

$$D = b^2 - 4ac = 57^2 - 4 \cdot 9 \cdot 70 = 3249 - 2520 = 729$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-57 + \sqrt{729}}{2 \cdot 9} = \frac{-57 + 27}{18} = \frac{-30}{18} = -\frac{5}{3} = -1\frac{2}{3}$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-57 - \sqrt{729}}{2 \cdot 9} = \frac{-57 - 27}{18} = \frac{-84}{18} = -\frac{14}{3} = -4\frac{2}{3}$$

Ответ: $$x_1 = -1\frac{2}{3}$$, $$x_2 = -4\frac{2}{3}$$

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в) $$(3x - 8)^2 = 3x^2 - 8x$$

$$9x^2 - 48x + 64 = 3x^2 - 8x$$

$$6x^2 - 40x + 64 = 0$$

$$3x^2 - 20x + 32 = 0$$

Вычислим дискриминант:

$$D = b^2 - 4ac = (-20)^2 - 4 \cdot 3 \cdot 32 = 400 - 384 = 16$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{20 + \sqrt{16}}{2 \cdot 3} = \frac{20 + 4}{6} = \frac{24}{6} = 4$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{20 - \sqrt{16}}{2 \cdot 3} = \frac{20 - 4}{6} = \frac{16}{6} = \frac{8}{3} = 2\frac{2}{3}$$

Ответ: $$x_1 = 4$$, $$x_2 = 2\frac{2}{3}$$

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г) $$(4x + 5)^2 = 5x^2 + 4x$$

$$16x^2 + 40x + 25 = 5x^2 + 4x$$

$$11x^2 + 36x + 25 = 0$$

Вычислим дискриминант:

$$D = b^2 - 4ac = 36^2 - 4 \cdot 11 \cdot 25 = 1296 - 1100 = 196$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-36 + \sqrt{196}}{2 \cdot 11} = \frac{-36 + 14}{22} = \frac{-22}{22} = -1$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-36 - \sqrt{196}}{2 \cdot 11} = \frac{-36 - 14}{22} = \frac{-50}{22} = -\frac{25}{11} = -2\frac{3}{11}$$

Ответ: $$x_1 = -1$$, $$x_2 = -2\frac{3}{11}$$