1) x/x-3-5/x+3=18/x^2-92) 70/x^2-16-17/x-4=3x/x+43) 3 / (2-x) ^2-5 / (x+2) ^2=14/x^2-4

Question

Фёдор Иванов

Математика

17 сентября, 10:55

1) x/x-3-5/x+3=18/x^2-92) 70/x^2-16-17/x-4=3x/x+43) 3 / (2-x) ^2-5 / (x+2) ^2=14/x^2-4

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Mambl · Answer 1

1) x / (x - 3) - 5 / (x + 3) = 18 / (x² - 9);

x / (x - 3) - 5 / (x + 3) = 18 / (x - 3) (x + 3);

x / (x - 3) - 5 / (x + 3) - 18 / (x - 3) (x + 3) = 0;

(x (x + 3) - 5 (x - 3) - 18) / (x - 3) (x + 3) = 0;

(x² + 3x - 5x + 15 - 18) / (x - 3) (x + 3) = 0;

(x² - 2x - 3) / (x - 3) (x + 3) = 0;

x² - 2x - 3 = 0;

D = 4 - 4 * 1 * ( - 3) = 4 + 12 = 16>0;

x₁ = (2 - 4) / 2 = - 2/2 = - 1;

x₂ = (2 + 4) / 2 = 3.

x - 3 ≠ 0;

x ≠ 3.

x + 3 ≠ 0;

x ≠ - 3.

Ответ: х = - 1.

2) 70 / (x² - 16) - 17 / (x - 4) = 3x / (x + 4);

70 / (x - 4) (x + 4) - 17 / (x - 4) - 3x / (x + 4) = 0;

(70 - 17 (x + 4) - 3x (x - 4)) / (x - 4) (x + 4) = 0;

x ≠ 4, x ≠ - 4.

70 - 17 (x + 4) - 3x (x - 4) = 0;

70 - 17x - 68 - 3x² + 12x = 0;

- 3x² - 5x + 2 = 0;

3x² + 5x - 2 = 0;

D = 25 - 4 * 3 * ( - 2) = 25 + 24 = 49>0;

x₁ = ( - 5 - 7) / 6 = - 2;

x₂ = ( - 5 + 7) / 6 = 2/6 = 1/3.

Ответ: x₁ = - 2; x₂ = 1/3.

3) 3 / (2 - x) ² - 5 / (x + 2) ² = 14 / (x² - 4);

3 / (x - 2) (x - 2) - 5 / (x + 2) (x + 2) = 14 / (x - 2) (x + 2);

(3 (x + 2) ² - 5 (x - 2) ² - 14 (x² - 4)) / (x - 2) (x - 2) (x + 2) (x + 2) = 0;

(3 (x² + 4x + 4) - 5 (x² - 4x + 4) - 14x² + 56) / (x - 2) (x - 2) (x + 2) (x + 2) = 0;

(3x² + 12x + 12 - 5x² + 20x - 20 - 14x² + 56) / (x - 2) (x - 2) (x + 2) (x + 2) = 0;

( - 16x² + 32x + 48) / (x - 2) (x - 2) (x + 2) (x + 2) = 0;

x ≠ 2;

x ≠ - 2;

- 16x² + 32x + 48 = 0;

16x² - 32x - 48 = 0;

4x² - 8x - 12 = 0;

D = 64 - 4 * 4 * ( - 12) = 64 + 192 = 256>0;

x₁ = (8 - 16) / 8 = - 8/8 = - 1;

x₂ = (8 + 16) / 8 = 24/8 = 3.

Ответ: x₁ = - 1; x₂ = 3.