x^2+[x/(x+1)]^2=3
来源:百度知道 编辑:UC知道 时间:2024/06/21 19:53:44
用换元法解这个方程
x²+x²/(x+1)² = 3
去分母
x²(x+1)² + x² = 3x² + 6x + 3
x^4 + 2x³ + 2x² = 3x² + 6x + 3
x^4 + 2x³ - x² - 6x - 3 = 0
四次方程求根公式比较复杂
这里不好打出来
令x = y - 1/2
代入化简,消去x³项
得
y^4 - 2.5y² - 4y - 7/16 = 0
(y² - 5/4)² = 4y + 2
(y² - 5/4)² + 2a(y² - 5/4) + a² = 2a(y² - 5/4) + a² + 4y + 2 = 2ay² + 4y + (a² - 2.5a + 2)
因等号左边为完全平方
故取适当的a值使右边也为完全平方,即
4² - 4 * 2a * (a² - 2.5a + 2) = 0
把a解出并代入上式,可解除出y值
x = y - 1/2,也能解出
解关于a的三次方程
令a = b + 5/6
代入方程并化简得
b³ - b/12 - 161/108 = 0
再令b = m^(1/3) + n^(1/3)
b³ = m + n + 3(nm)^(1/3) * [m^(1/3) + n^(1/3)]
即b³ = m + n + 3(nm)^(1/3) * b
代入关于b的3次方程得
m + n + 3(nm)^(1/3) * b - b/12 -161/108 = 0
故得m + n - 161/108 = 0
3(nm)^(1/3) - 1/12 = 0
即m + n - 161/108 = 0
mn = 1/36³
再解
x^2+x+1=2/(x^2+x)
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