用换元法解方程3x/x^2-1+x^2-1/ 2x=5/2

用换元法解方程3x/x^2-1+x^2-1/ 2x=5/2
用换元法解方程3x/x^2-1+x^2-1/ 2x=5/2

用换元法解方程3x/x^2-1+x^2-1/ 2x=5/2
令t=x/(x²-1)；则有：
3t+1/(2t)=5/2;
6t²+1-5t=0;
(2t-1)(3t-1)=0;
t=1/2或t=1/3;
∴x/x²-1=1/2;
x²-1-2x=0;
(x-1)²=2;
x-1=±√2;
x=1±√2;
或x/x²-1=1/3;
x²-3x-1=0;
(x-3/2)²=13/4;
x-3/2=±√13/2;
x=(3±√13)/2;
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3x/X^2-1+X^2-1/3x=5/2
令 t = 3x/(x² - 1)
则 t + 1/t = 5/2
2t² - 5t + 2 = 0
(2t - 1)(t - 2) = 0
t = 1/2 或 t = 2
①当 t = 1/2 时，
3x/(x² - 1) = 1/2
x = 3 + √10...

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3x/X^2-1+X^2-1/3x=5/2
令 t = 3x/(x² - 1)
则 t + 1/t = 5/2
2t² - 5t + 2 = 0
(2t - 1)(t - 2) = 0
t = 1/2 或 t = 2
①当 t = 1/2 时，
3x/(x² - 1) = 1/2
x = 3 + √10 或 x = 3 - √10
②当 t = 2 时，
3x/(x² - 1) = 2
x = 2 或 x = -1/2
综上： x = 3 + √10 或 x = 3 - √10 或 x = 2 或 x = -1/2

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