maths(解方程 二 : 代數方法)

18. (x + 3) / (3x - 5) + 1 = 3(x - 2) / (x - 1)
(x + 3)(x - 1) + (3x - 5)(x - 1) = 3(x - 2)(3x - 5)
x2 + 2x - 3 + 3x2 - 8x + 5 = 9x2 - 33x + 30
5x2 - 27x + 28 = 0
(5x - 7)(x - 4) = 0
5x - 7 = 0 或x - 4 = 0
x = 7/5 或x = 4
30. (x - 2/x)2 + (x - 2/x) - 2 = 0
Let u = x - 2/x
u2 + u - 2 = 0
(u + 2)(u - 1) = 0
(x - 2/x + 2)(x - 2/x - 1) = 0
x - 2/x + 2 = 0
=> x2 + 2x - 2 = 0
=> (x + 1)2 = 3
=> x = -1 +/- √3
x - 2/x - 1 = 0
=> x2 - x - 2 = 0
=> (x - 2)(x + 1) = 0
=> x = 2 或x = -1
31. (x2 - 2x)2 - 2x2 + 4x - 3 = 0
= (x2 - 2x)2 - 2(2x2 - 2x) - 3 = 0
Let u = x2 - 2x
u2 - 2u - 3 = 0
(u - 3)(u + 1) = 0
(x2 - 2x - 3)(x2 - 2x + 1) = 0
(x - 3)(x + 1)(x - 1)2 = 0
x = 1, 3 或-1
{ x + y = 10 ... (i)
{ xy = 24 ... (ii)
在一次方程 (i) 中,以 x 表示 y ,可得 y = 10 - x ... (iii)
將 (iii) 代入二次方程 (ii) 中 , 可得 x(10 - x) = 24 ... (iv)
1.方程 (iv) 有1個未知數. 該方程是1元2次方程
2. x(10 - x) = 24
10x - x2 = 24
x2 - 10x + 24 = 0
(x - 6)(x - 4) = 0
x = 6 或x = 4
x = 6 => y = 4
x = 4 => y = 6

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