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Thực hiện các phép tính sau: (2x+1)/(2x-1) - (2x-1)/(2x+1)

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Giải Toán 8 Bài Ôn tập chương 2

Video Giải Bài 58 trang 62 Toán 8 Tập 1

Bài 58 trang 62 Toán 8 Tập 1:

a) $(\frac{2 x + 1}{2 x - 1} - \frac{2 x - 1}{2 x + 1}) : \frac{4 x}{10 x - 5};$

b) $(\frac{1}{x^{2} + x} - \frac{2 - x}{x + 1}) : (\frac{1}{x} + x - 2);$

c) $\frac{1}{x - 1} - \frac{x^{3} - x}{x^{2} + 1} . (\frac{1}{x^{2} - 2 x + 1} + \frac{1}{1 - x^{2}}) .$

Lời giải

a) $(\frac{2 x + 1}{2 x - 1} - \frac{2 x - 1}{2 x + 1}) : \frac{4 x}{10 x - 5}$

$\begin{array}{l} = \frac{{(2 x + 1)}^{2}}{(2 x - 1) (2 x + 1)} - \frac{{(2 x - 1)}^{2}}{(2 x - 1) (2 x + 1)}] : \frac{4 x}{5 (2 x - 1)} \\ = \frac{{(2 x + 1)}^{2} - {(2 x - 1)}^{2}}{(2 x - 1) (2 x + 1)}] : \frac{4 x}{5 (2 x - 1)} \\ = \frac{(2 x + 1 - 2 x + 1) (2 x + 1 + 2 x - 1)}{(2 x - 1) (2 x + 1)}] : \frac{4 x}{5 (2 x - 1)} \\ = \frac{2.4 x}{(2 x - 1) (2 x + 1)}] : \frac{4 x}{5 (2 x - 1)} \\ = \frac{8 x}{(2 x - 1) (2 x + 1)} . \frac{5 (2 x - 1)}{4 x} \\ = \frac{8 x .5 (2 x - 1)}{(2 x - 1) (2 x + 1) .4 x} \\ = \frac{10}{2 x + 1} . \end{array}$

b) $(\frac{1}{x^{2} + x} - \frac{2 - x}{x + 1}) : (\frac{1}{x} + x - 2)$

$\begin{array}{l} = (\frac{1}{x (x + 1)} - \frac{x (2 - x)}{x (x + 1)}) : (\frac{1}{x} + \frac{x^{2}}{x} - \frac{2 x}{x}) \\ = \frac{1}{x (x + 1)} - \frac{2 x - x^{2}}{x (x + 1)}] : (\frac{x^{2} - 2 x + 1}{x}) \\ = \frac{1 - 2 x + x^{2}}{x (x + 1)}] : (\frac{x^{2} - 2 x + 1}{x}) \\ = \frac{x^{2} - 2 x + 1}{x (x + 1)} : \frac{x^{2} - 2 x + 1}{x} \\ = \frac{x^{2} - 2 x + 1}{x (x + 1)} . \frac{x}{x^{2} - 2 x + 1} \\ = \frac{(x^{2} - 2 x + 1) . x}{x (x + 1) . (x^{2} - 2 x + 1)} \\ = \frac{1}{x + 1} . \end{array}$

c) $\frac{1}{x - 1} - \frac{x^{3} - x}{x^{2} + 1} . (\frac{1}{x^{2} - 2 x + 1} + \frac{1}{1 - x^{2}})$

$\begin{array}{l} \frac{1}{x - 1} - \frac{x^{3} - x}{x^{2} + 1} . (\frac{1}{x^{2} - 2 x + 1} - \frac{1}{x^{2} - 1}) \\ = \frac{1}{x - 1} - \frac{x^{3} - x}{x^{2} + 1} . \frac{1}{{(x - 1)}^{2}} - \frac{1}{(x - 1) (x + 1)}] \\ = \frac{1}{x - 1} - \frac{x^{3} - x}{x^{2} + 1} . \frac{x + 1}{{(x - 1)}^{2} (x + 1)} - \frac{x - 1}{{(x - 1)}^{2} (x + 1)}] \\ = \frac{1}{x - 1} - \frac{x (x^{2} - 1)}{x^{2} + 1} . \frac{x + 1 - x + 1}{{(x - 1)}^{2} (x + 1)}] \\ = \frac{1}{x - 1} - \frac{x (x - 1) (x + 1)}{x^{2} + 1} . \frac{2}{{(x - 1)}^{2} (x + 1)} \\ = \frac{1}{x - 1} - \frac{2 x}{(x^{2} + 1) (x - 1)} \\ = \frac{x^{2} + 1}{(x - 1) (x^{2} + 1)} - \frac{2 x}{(x^{2} + 1) (x - 1)} \\ = \frac{x^{2} + 1 - 2 x}{(x - 1) (x^{2} + 1)} = \frac{x^{2} - 2 x + 1}{(x - 1) (x^{2} + 1)} \\ = \frac{{(x - 1)}^{2}}{(x - 1) (x^{2} + 1)} = \frac{x - 1}{x^{2} + 1} . \end{array}$