Giải bài 50 trang 58 – SGK Toán lớp 8 tập 1
Thực hiện các phép tính:
a) \(\left( \dfrac{x}{x + 1} + 1 \right) : \left(1 - \dfrac{3x^2}{1 - x^2} \right)\)
b) \((x^2 - 1)\left(\dfrac{1}{x - 1} - \dfrac{1}{x + 1} - 1 \right)\)
a) \(\left( \dfrac{x}{x + 1} + 1 \right) : \left(1 - \dfrac{3x^2}{1 - x^2} \right)\)
\(= \left( \dfrac{x}{x + 1} + \dfrac{x + 1}{x + 1} \right) : \left(\dfrac{1 - x^2}{1 - x^2} - \dfrac{3x^2}{1 - x^2} \right)\)
\(= \dfrac{x + x + 1}{x + 1} : \dfrac{1 - x^2 - 3x^2}{1 - x^2}\)
\(= \dfrac{2x + 1}{x + 1} : \dfrac{1 - 4x^2 }{1 - x^2}\)
\(= \dfrac{2x + 1}{x + 1} : \dfrac{-(4x^2 - 1) }{-(x^2 - 1)}\)
\(= \dfrac{2x + 1}{x + 1} : \dfrac{4x^2 - 1 }{x^2 - 1}\)
\(= \dfrac{2x + 1}{x + 1}. \dfrac{ x^2 - 1}{4x^2 - 1}\)
\(= \dfrac{(2x + 1)(x^2 - 1)}{(x + 1)(4x^2 - 1)}\)
\(= \dfrac{(2x + 1)(x + 1)(x - 1)}{(x + 1)(2x - 1)(2x + 1)}\)
\(= \dfrac{x - 1}{2x - 1}\)
b) \((x^2 - 1)\left(\dfrac{1}{x - 1} - \dfrac{1}{x + 1} - 1 \right)\)
\(= (x^2 - 1)\left(\dfrac{x + 1}{(x - 1)(x + 1)} - \dfrac{x - 1}{(x - 1)(x + 1)} - \dfrac{(x - 1)(x + 1)}{(x - 1)(x + 1)} \right)\)
\(= (x^2 - 1)\left(\dfrac{x + 1}{(x - 1)(x + 1)} - \dfrac{x - 1}{(x - 1)(x + 1)} - \dfrac{x^2 - 1}{(x - 1)(x + 1)} \right)\)
\(= (x^2 - 1)\left(\dfrac{x + 1 - x + 1 - x^2 + 1}{(x - 1)(x + 1)} \right)\)
\(= (x^2 - 1) \dfrac{-x^2 + 3}{(x - 1)(x + 1)}\)
\(= \dfrac{(-x^2 + 3)(x^2 - 1)}{(x - 1)(x + 1)}\)
\(= \dfrac{(-x^2 + 3)(x - 1)(x + 1)}{(x - 1)(x + 1)}\)
\(= -x^2 + 3\)
Lưu ý: \( \dfrac{A}{B}:\dfrac{C}{D} = \dfrac{A}{B}.\dfrac{D}{C}\)
