續二次方程(1-10題)

1. 32x^2 - x^6 = 4x^4
==> x^6 + 4x^4 - 32x^2 = 0
==> x^2 (x^4 + 4 x^2 - 32) = 0
==> x^2 (x^2 - 4)(x^2 + 8) = 0
==> x = 0, -2, 2 (if x is real number)

2. 17x^2 = 4(1 + x^4)
==> 4x^4 - 17x^2 + 4 = 0
==> (4x^2 - 1)(x^2 - 4) = 0
==> x = 1/2, -1/2, 2, -2

3. (x - 3)^2 - 6(x - 3) + 5 = 0
==> (x - 3 - 1)(x - 3 - 5) = 0
==> x = 4, 8

4. (x^2 + 2x)^2 - 7(x^2 + 2x) - 8 = 0
==> (x^2 + 2x - 8)(x^2 + 2x + 1) = 0
==> (x + 4)(x - 2)(x + 1)^2 = 0
==> x = -4, 2, -1

5. (x^2 - 4x)^2 - 2x(x - 4) = 15
==> (x^2 - 4x)^2 - 2(x^2 - 4x) - 15 = 0
==> (x^2 - 4x - 5)(x^2 - 4x + 3) = 0
==> (x - 5)(x + 1)((x - 1)(x - 3) = 0
==> x = 5, -1, 1, 3

2012-10-14 12:22:24 補充：
6. x^3 = 9 - 8/x^3
==> x^6 - 9x^3 + 8 = 0
==> (x^3 - 1)(x^3 - 8) = 0
==> (x - 1)(x^2 + x + 1)(x - 2)(x^2 + 2x + 4) = 0
==> x = 1, 2

7. 4/x^4 - 65/x^2 + 16 = 0
==> 16x^4 - 65x^2 + 4 = 0
==> (16x^2 - 1)(x^2 - 4) = 0
==> x = 1/4, -1/4, 2, -2

2012-10-14 18:46:32 補充：
8. 1/(x + 1)^2 - 1/(x + 1) - 12 = 0
==> 12(x + 1)^2 + (x + 1) - 1 = 0
==> [4(x + 1) - 1][3(x + 1) + 1] = 0
==> (4x + 3)(3x + 4) = 0
==> x = -3/4, -4/3

2012-10-14 18:55:53 補充：
9. 3/(x^2 - 2x)^2 = 1 - 2/(x^2 - 2x)
==> (x^2 - 2x)^2 - 2(x^2 - 2x) - 3 = 0
==> (x^2 - 2x - 3)(x^2 - 2x + 1) = 0
==> (x - 3)(x + 1)(x - 1)^2 = 0
==> x = 3, -1, 1

10. 2(x + 1) - 5根號(x + 1) + 3 = 0
==> [2根號(x + 1) - 3][根號(x + 1) - 1] = 0
==> 2根號(x + 1) = 3 or 根號(x + 1) = 1
==> x = 1.25, 0