$\frac{60}{x} - \frac{60}{x + 10} = \frac{1}{5}$
x ≠ 0
и
x + 10 ≠ 0
x ≠ −10
$\frac{60}{x} - \frac{60}{x + 10} - \frac{1}{5} = 0$ | * 5x(x + 10)
300(x + 10) − 300x − x(x + 10) = 0
$300x + 3000 - 300x - x^2 - 10x = 0$
$-x^2 - 10x + 3000 = 0$ | * (−1)
$x^2 + 10x - 3000 = 0$
$D = b^2 - 4ac = 10^2 - 4 * 1 * (-3000) = 100 + 12000 = 12100 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-10 + \sqrt{12100}}{2 * 1} = \frac{-10 + 110}{2} = \frac{100}{2} = 50$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-10 - \sqrt{12100}}{2 * 1} = \frac{-10 - 110}{2} = \frac{-120}{2} = -60$
Ответ: −60 и 50

$\frac{x}{x + 2} + \frac{x + 2}{x - 2} = \frac{16}{x^2 - 4}$
$\frac{x}{x + 2} + \frac{x + 2}{x - 2} = \frac{16}{(x - 2)(x + 2)}$
x + 2 ≠ 0
x ≠ −2
и
x − 2 ≠ 0
x ≠ 2
$\frac{x}{x + 2} + \frac{x + 2}{x - 2} - \frac{16}{(x - 2)(x + 2)} = 0$ | * (x − 2)(x + 2)
$x(x - 2) + (x + 2)^2 - 16 = 0$
$x^2 - 2x + x^2 + 4x + 4 - 16 = 0$
$2x^2 + 2x - 12 = 0$ | : 2
$x^2 + x - 6 = 0$
$D = b^2 - 4ac = 1^2 - 4 * 1 * (-6) = 1 + 24 = 25 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-1 + \sqrt{25}}{2 * 1} = \frac{-1 + 5}{2} = \frac{4}{2} = 2$ − не удовлетворяет условию, так как x ≠ 2.
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-1 - \sqrt{25}}{2 * 1} = \frac{-1 - 5}{2} = \frac{-6}{2} = -3$
Ответ: −3

$\frac{9}{x + 3} + \frac{14}{x - 3} = \frac{24}{x}$
x + 3 ≠ 0
x ≠ −3
и
x − 3 ≠ 0
x ≠ 3
и
x ≠ 0
$\frac{9}{x + 3} + \frac{14}{x - 3} - \frac{24}{x} = 0$ | * x(x + 3)(x − 3)
$9x(x - 3) + 14x(x + 3) - 24(x^2 - 9) = 0$
$9x^2 - 27x + 14x^2 + 42x - 24x^2 + 216 = 0$
$-x^2 + 15x + 216 = 0$ | * (−1)
$x^2 - 15x - 216 = 0$
$D = b^2 - 4ac = (-15)^2 - 4 * 1 * (-216) = 225 + 864 = 1089 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{15 + \sqrt{1089}}{2 * 1} = \frac{15 + 33}{2} = \frac{48}{2} = 24$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{15 - \sqrt{1089}}{2 * 1} = \frac{15 - 33}{2} = \frac{-18}{2} = -9$
Ответ: −9 и 24

$\frac{2y + 3}{2y + 2} - \frac{y + 1}{2y - 2} + \frac{1}{y^2 - 1} = 0$
$\frac{2y + 3}{2(y + 1)} - \frac{y + 1}{2(y - 1)} + \frac{1}{(y - 1)(y + 1)} = 0$
y + 1 ≠ 0
y ≠ −1
и
y − 1 ≠ 0
y ≠ 1
$\frac{2y + 3}{2(y + 1)} - \frac{y + 1}{2(y - 1)} + \frac{1}{(y - 1)(y + 1)} = 0$ | * 2(y − 1)(y + 1)
$(2y + 3)(y - 1) - (y + 1)^2 + 2 = 0$
$2y^2 + 3y - 2y - 3 - (y^2 + 2y + 1) + 2 = 0$
$2y^2 + y - 1 - y^2 - 2y - 1 = 0$
$y^2 - y - 2 = 0$
$D = b^2 - 4ac = (-1)^2 - 4 * 1 * (-2) = 1 + 8 = 9 > 0$
$y_1 = \frac{-b + \sqrt{D}}{2a} = \frac{1 + \sqrt{9}}{2 * 1} = \frac{1 + 3}{2} = \frac{4}{2} = 2$
$y_2 = \frac{-b - \sqrt{D}}{2a} = \frac{1 - \sqrt{9}}{2 * 1} = \frac{1 - 3}{2} = \frac{-2}{2} = -1$ − не удовлетворяет условию, так как y ≠ −1.
Ответ: 2

$\frac{3x}{x^2 - 10x + 25} - \frac{x - 3}{x^2 - 5x} = \frac{1}{x}$
$\frac{3x}{(x - 5)^2} - \frac{x - 3}{x(x - 5)} = \frac{1}{x}$
$(x - 5)^2 ≠ 0$
x − 5 ≠ 0
x ≠ 5
и
x ≠ 0
$\frac{3x}{(x - 5)^2} - \frac{x - 3}{x(x - 5)} - \frac{1}{x} = 0$ | * $x(x - 5)^2$
$3x^2 - (x - 3)(x - 5) - (x - 5)^2 = 0$
$3x^2 - (x^2 - 3x - 5x + 15) - (x^2 - 10x + 25) = 0$
$3x^2 - x^2 + 8x - 15 - x^2 + 10x - 25 = 0$
$x^2 + 18x - 40 = 0$
$D = b^2 - 4ac = 18^2 - 4 * 1 * (-40) = 324 + 160 = 484 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-18 + \sqrt{484}}{2 * 1} = \frac{-18 + 22}{2} = \frac{4}{2} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-18 - \sqrt{484}}{2 * 1} = \frac{-18 - 22}{2} = \frac{-40}{2} = -20$
Ответ: −20 и 2

$\frac{x - 20}{x^2 + 10x} + \frac{10}{x^2 - 100} - \frac{5}{x^2 - 10x} = 0$
$\frac{x - 20}{x(x + 10)} + \frac{10}{(x - 10)(x + 10)} - \frac{5}{x(x - 10)} = 0$
x − 10 ≠ 0
x ≠ 10
и
x + 10 ≠ 0
x ≠ −10
и
x ≠ 0
$\frac{x - 20}{x(x + 10)} + \frac{10}{(x - 10)(x + 10)} - \frac{5}{x(x - 10)} = 0$ | * x(x − 10)(x + 10)
(x − 20)(x − 10) + 10x − 5(x + 10) = 0
$x^2 - 20x - 10x + 200 + 10x - 5x - 50 = 0$
$x^2 - 25x + 150 = 0$
$D = b^2 - 4ac = (-25)^2 - 4 * 1 * 150 = 625 + 600 = 25 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{25 + \sqrt{25}}{2 * 1} = \frac{25 + 5}{2} = \frac{30}{2} = 15$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{25 - \sqrt{25}}{2 * 1} = \frac{25 - 5}{2} = \frac{20}{2} = 10$ − не удовлетворяет условию, так как x ≠ 10.
Ответ: 15