F4 math Algebraic Method 急!!

1.
(2/x + 1) (x + 6) = 1
x (2/x + 1) (x + 6) = x
(2 + x) (x + 6) = x
x2 + 8x + 12 = x
x2 + 7x + 12 = 0
(x + 4) (x + 3) = 0
x = -4 ooro x = -3

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2.
1/(4 - x) + 1/(2 + x) = 3/4

Multiply all terms on the both sides by 4(4 - x)(2 + x):
4(2 + x) + 4(4 - x) = 3(2 + x)(4 - x)
8 + 4x + 16 - 4x = 24 + 6x - 3x2
3x2 - 6x = 0
3x(x - 2) = 0
x = 0 ooro x = 2

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3.
2x4 - x2 - 1 =0
2(x)2 - (x2) - 1 = 0
(x2 - 1)(2x2 + 1) = 0
(x + 1)(x - 1)(2x2 + 1) = 0
x = -1 ooro x = 1 ooro 2x2 + 1 = 0 (the unreal roots rejected)

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4.
x5 - 8x3 - 9x = 0
x[x4 - 8x2 - 9] = 0
x[(x2)2 - 8(x2) - 9] = 0
x(x2 - 9)(x2 + 1) = 0
x(x - 3)(x + 3)(x2 + 1) = 0
x = 0 ooro x = 3 ooro x = -3 ooro x2 + 1 = 0 (the unreal roots rejcted)

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5.
x = 10 - 2√(x - 2)
x - 10 = -2√(x - 2)
(x - 10)2 = [-2√(x - 2)]2
x2 - 20x + 100 = 4(x - 2)
x2 - 20x + 100 = 4x - 8
x2 - 24x + 108 = 0
(x - 6)(x - 18) = 0
x = 6 ooro x = 18 (rejected)

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6.
√(2x - 3) - 2/√(2x - 3) = 1

Multiply all terms on the both sides by √(2x - 3):
(2x - 3) - 2 = √(2x - 3)
2x - 5 = √(2x - 3)
(2x - 5)2 = 2x - 3
4x2 - 20x + 25 = 2x - 3
4x2 - 22x + 28 = 0
2x2 - 11x + 14 = 0
(2x - 7)(x - 2) = 0
x = 7/2 ooro x = 2 (rejected)

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7.
72x - 56(7x) + 343 =0
(7x)2 - 56(7x) + 343 = 0
(7x - 7)(7x - 49) = 0
7x = 7 ooro 7x = 49
7x = 71 ooro 7x = 72
x = 1 ooro x = 2

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8.
(x2 - 2x)2 - 2x2 + 4x - 3 = 0
(x2 - 2x)2 - 2(x2 - 2x) - 3 = 0
[(x2 - 2x) - 3][(x2 - 2x) + 1] = 0
(x2 - 2x - 3)(x2 - 2x + 1) = 0
(x - 3)(x + 1)(x - 1)2 = 0
x = 3 ooro x = -1 ooro x = 1ooro x = 1
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