Question

[Nguyễn Thảo Linh]

Đề bài: Giải phương trình :

a) \(\frac{\left(x+1\right)}{x-2}\) - \(\frac{x-1}{x+2}\) = \(\frac{2\left(x^2+2\right)}{x^2-4}\)

b) \(\frac{x-1}{x+2}\) - \(\frac{x}{x-2}\) = \(\frac{5x-2}{4-x^2}\)

c) \(\frac{x-2}{2+x}\) - \(\frac{3}{x-2}\) = \(\frac{2\left(x-11\right)}{x^2-4}\)

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Answer 1

[Nguyễn Thành Trương]

\(a)\dfrac{{x + 1}}{{x - 2}} - \dfrac{{x - 1}}{{x + 2}} = \dfrac{{2\left( {{x^2} + 2} \right)}}{{{x^2} - 4}}\)

ĐKXĐ: \(x\ne\pm2\)

\(\Leftrightarrow \dfrac{{\left( {x + 1} \right)\left( {x + 2} \right) - \left( {x - 1} \right)\left( {x - 2} \right)}}{{{x^2} - 4}} = \dfrac{{2\left( {{x^2} + 2} \right)}}{{{x^2} - 4}}\\ \Leftrightarrow {x^2} + 3x + 2 - \left( {{x^2} - 3x + 2} \right) = 2{x^2} + 4\\ \Leftrightarrow 6x = 2{x^2} + 4\\ \Leftrightarrow - 2{x^2} + 6x - 4 = 0\\ \Leftrightarrow 2{x^2} - 6x + 4 = 0\\ \Leftrightarrow {x^2} - 3x + 2 = 0\\ \Leftrightarrow {x^2} - 2x - x + 2 = 0\\ \Leftrightarrow x\left( {x - 2} \right) - \left( {x - 2} \right) = 0\\ \Leftrightarrow \left( {x - 2} \right)\left( {x - 1} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} x - 2 = 0\\ x - 1 = 0 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = 2\left( {KTM} \right)\\ x = 1\left( {TM} \right) \end{array} \right. \)

Vậy \(x=1\)

\(b)\dfrac{{x - 1}}{{x + 2}} - \dfrac{x}{{x - 2}} = \dfrac{{5x - 2}}{{4 - {x^2}}} \)

ĐKXĐ: \(x\ne\pm2\)

\( \Leftrightarrow \dfrac{{\left( {x - 1} \right)\left( {x - 2} \right) - x\left( {x + 2} \right)}}{{{x^2} - 4}} = \dfrac{{2 - 5x}}{{{x^2} - 4}}\\ \Leftrightarrow {x^2} - 3x + 2 - {x^2} - 2x = 2 - 5x\\ \Leftrightarrow 0x = 0\left( {VSN} \right) \)

Vậy phương trình vô số nghiệm

\(c)\dfrac{{x - 2}}{{2 + x}} - \dfrac{3}{{x - 2}} = \dfrac{{2\left( {x - 11} \right)}}{{{x^2} - 4}}\)

ĐKXĐ: \(x\ne\pm2\)

\( \Leftrightarrow \dfrac{{\left( {x - 2} \right)\left( {x - 2} \right) - 3\left( {x + 2} \right)}}{{{x^2} - 4}} = \dfrac{{2x - 22}}{{{x^2} - 4}}\\ \Leftrightarrow {x^2} - 4x + 4 - 3x - 6 = 2x - 22\\ \Leftrightarrow {x^2} - 9x + 20 = 0\\ \Leftrightarrow {x^2} - 4x - 5x + 20 = 0\\ \Leftrightarrow x\left( {x - 4} \right) - 5\left( {x - 4} \right) = 0\\ \Leftrightarrow \left( {x - 4} \right)\left( {x - 5} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l} x - 4 = 0\\ x - 5 = 0 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = 4\left( {TM} \right)\\ x = 5\left( {TM} \right) \end{array} \right. \)

Vậy \(x=4,x=5\)