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san 寫到:

1.(x^3+y^3)/(x^3-y^3)=-234/109

求(x^2+y^2)/(x^2-y^2)=?

 

2.2x^4-3x^3+5x^2-3x+2=0

求x+  1/x=?

 

1. (x³+y³)/(x³-y³) = -(y³+x³)/(y³-x³) = -(y³-x³+2x³)/(y³-x³) = -[1+ 2x³/(y³-x³)] = -234/109 = - (1+ 125/109)
=> 2x³/(y³-x³) = 125/109 => 218x³ = 125y³-125x³ => 343x³=125y³ => 7x=5y => y = (7/5)x
(x²+y²)/(x²-y²) = (x²+(7/5)²x²)/(x²-(7/5)²x²) = (1+49/25)x²/(1-49/25)x² = (74/25)/(-24/5) = -37/12

2. 設 y= x+1/x，y² = (x+1/x)² = x²+2+1/x²;
0 = 2x⁴-3x³+5x²-3x+2，若x=0，則0=2 =>故x=/=0，將兩邊除以x²得：
=> 0 = 2x²-3x+5-3/x+2/x² = 2x²+4+2/x² - 3x-3/x + 1 = 2(x²+2+1/x²) - 3(x+1/x) +1 = 2y²-3y+1 = (2y-1)(y-1) => y = 1/2 or 1
驗證： 1/2=x+1/x or 1=x+1/x => x/2=x²+1 or x=x²+1 => 2x²-x+2=0 or x²-x+1=0
b²-4ac = (-1)²-4(2)(2) or (-1)²-4(1)(1)=-15or-3<0，故若x為實數，則此題無解；若x能為虛數，則 x+1/x = 1/2 或 1。

san 寫到:

1.(x^3+y^3)/(x^3-y^3)=-234/109

求(x^2+y^2)/(x^2-y^2)=?