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. (x
3+y
3)/(x
3-y
3) = -(y
3+x
3)/(y
3-x
3) = -(y
3-x
3+2x
3)/(y
3-x
3) = -[1+ 2x
3/(y
3-x
3)] = -234/109 = - (1+ 125/109)
=> 2x
3/(y
3-x
3) = 125/109 => 218x
3 = 125y
3-125x
3 => 343x
3=125y
3 => 7x=5y => y = (7/5)x
(x
2+y
2)/(x
2-y
2) = (x
2+(7/5)
2x
2)/(x
2-(7/5)
2x
2) = (1+49/25)x
2/(1-49/25)x
2 = (74/25)/(-24/5) = -37/12
2. 設 y= x+1/x,y
2 = (x+1/x)
2 = x
2+2+1/x
2;
0 = 2x
4-3x
3+5x
2-3x+2,若x=0,則0=2 =>故x=/=0,將兩邊除以x
2得:
=> 0 = 2x
2-3x+5-3/x+2/x
2 = 2x
2+4+2/x
2 - 3x-3/x + 1 = 2(x
2+2+1/x
2) - 3(x+1/x) +1 = 2y
2-3y+1 = (2y-1)(y-1) => y = 1/2 or 1
驗證: 1/2=x+1/x or 1=x+1/x => x/2=x
2+1 or x=x
2+1 => 2x
2-x+2=0 or x
2-x+1=0
b
2-4ac = (-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。