477. x - \frac{60}{x} = 4. 478. x + \frac{48}{x} = 14. 479. \frac{6}{x} + \frac{6}{x+1} = 5. 480. \frac{3}{x} + \frac{3}{x+2} = 4. 481. \frac{1}{x} + \frac{2}{x+2} = 1.

Решим уравнения:

477. x - \frac{60}{x} = 4\[x^2 - 60 = 4x\]\[x^2 - 4x - 60 = 0\]\[D = (-4)^2 - 4 \cdot 1 \cdot (-60) = 16 + 240 = 256\]\[x_1 = \frac{-(-4) + \sqrt{256}}{2 \cdot 1} = \frac{4 + 16}{2} = \frac{20}{2} = 10\]\[x_2 = \frac{-(-4) - \sqrt{256}}{2 \cdot 1} = \frac{4 - 16}{2} = \frac{-12}{2} = -6\]

Ответ: x = 10, x = -6

478. x + \frac{48}{x} = 14\[x^2 + 48 = 14x\]\[x^2 - 14x + 48 = 0\]\[D = (-14)^2 - 4 \cdot 1 \cdot 48 = 196 - 192 = 4\]\[x_1 = \frac{-(-14) + \sqrt{4}}{2 \cdot 1} = \frac{14 + 2}{2} = \frac{16}{2} = 8\]\[x_2 = \frac{-(-14) - \sqrt{4}}{2 \cdot 1} = \frac{14 - 2}{2} = \frac{12}{2} = 6\]

Ответ: x = 8, x = 6

479. \frac{6}{x} + \frac{6}{x+1} = 5\[\frac{6(x+1) + 6x}{x(x+1)} = 5\]\[6x + 6 + 6x = 5x(x+1)\]\[12x + 6 = 5x^2 + 5x\]\[5x^2 - 7x - 6 = 0\]\[D = (-7)^2 - 4 \cdot 5 \cdot (-6) = 49 + 120 = 169\]\[x_1 = \frac{-(-7) + \sqrt{169}}{2 \cdot 5} = \frac{7 + 13}{10} = \frac{20}{10} = 2\]\[x_2 = \frac{-(-7) - \sqrt{169}}{2 \cdot 5} = \frac{7 - 13}{10} = \frac{-6}{10} = -0.6\]

Ответ: x = 2, x = -0.6

480. \frac{3}{x} + \frac{3}{x+2} = 4\[\frac{3(x+2) + 3x}{x(x+2)} = 4\]\[3x + 6 + 3x = 4x(x+2)\]\[6x + 6 = 4x^2 + 8x\]\[4x^2 + 2x - 6 = 0\]\[2x^2 + x - 3 = 0\]\[D = 1^2 - 4 \cdot 2 \cdot (-3) = 1 + 24 = 25\]\[x_1 = \frac{-1 + \sqrt{25}}{2 \cdot 2} = \frac{-1 + 5}{4} = \frac{4}{4} = 1\]\[x_2 = \frac{-1 - \sqrt{25}}{2 \cdot 2} = \frac{-1 - 5}{4} = \frac{-6}{4} = -1.5\]

Ответ: x = 1, x = -1.5

481. \frac{1}{x} + \frac{2}{x+2} = 1\[\frac{x+2 + 2x}{x(x+2)} = 1\]\[3x + 2 = x(x+2)\]\[3x + 2 = x^2 + 2x\]\[x^2 - x - 2 = 0\]\[D = (-1)^2 - 4 \cdot 1 \cdot (-2) = 1 + 8 = 9\]\[x_1 = \frac{-(-1) + \sqrt{9}}{2 \cdot 1} = \frac{1 + 3}{2} = \frac{4}{2} = 2\]\[x_2 = \frac{-(-1) - \sqrt{9}}{2 \cdot 1} = \frac{1 - 3}{2} = \frac{-2}{2} = -1\]

Ответ: x = 2, x = -1

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