Question

Сложение и вычитание рациональных дробей
1. x-3/3x+6 - x-6/x+2
2. x+5/x-5 - x-1/x+5
3.a²+b²/a²-b² - b/a+b

Answer 1

$frac{x - 3}{3x + 6} - frac{x - 6}{x + 2} = frac{x - 3}{3(x - 2)} - frac{x - 6}{x + 2} = frac{x - 3 - 3(x - 6)}{3(x + 2)} = frac{x - 3 - 3x + 18}{3x + 6} = frac{15 - 2x}{3x + 6}$

$frac{x + 5}{x - 5} - frac{x - 1}{x + 5} = frac{(x + 5) {}^{2} - (x - 1)(x - 5) }{(x - 5)(x + 5)} = frac{ {x}^{2} + 10x + 25 - ( {x}^{2} - x - 5x + 5)}{ {x}^{2} - 25 } = frac{ {x}^{2} + 10x + 25 - {x}^{2} + 6x - 5 }{ {x}^{2} - 25} = frac{16x + 20}{ {x}^{2} - 25}$

$frac{ {a}^{2} + {b}^{2} }{ {a}^{2} - {b}^{2} } - frac{b}{a + b} = frac{ {a}^{2} + {b}^{2} - b(a - b) }{(a - b)(a + b)} = frac{ {a}^{2} + {b}^{2} - ab + {b}^{2} }{ {a}^{2} - {b}^{2} } = frac{ {a}^{2} - ab + 2 {b}^{2} }{ {a}^{2} - {b}^{2} }$