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二次不等式不會

Jun 24th 2013, 00:27

1. 2x²+4x-3無法十字交乘，先令2x²+4x-3=0

用公式解得x= -1±√10/2

[x-(-1-√10/2)][x-(-1+√10/2) > 0

x < -1-√10/2 or x > -1+√10/2

2. 先將-x²+3x+5 > 0改為x²-3x-5 < 0

用上一題的方法找出x=(3±√29)/2

[x-(3-√29)/2][x-(3+√29)/2] < 0

(3-√29)/2 < x < (3+√29)/2

3. 4x²-4x+1 > 0

(2x-1)² > 0

x為任意實數，但x≠1/2

4. 3x²-5x-2 < 0

(x-2)(3x+1) < 0

-1/3 < x < 2

5. 4x²-4x+5 < 0

此為開口向上的拋物線

但判別式b²-4ac=16-80< 0，表示4x²-4x+5=0無解，表示拋物線碰不到x軸(和x軸無交點)。拋物線開口向上又和x軸無交點，所以4x²-4x+5恆正，

因此4x²-4x+5 <0無解

6. x²+2x+4 > 0

也是開口向上的拋物線且判別式小於0為恆正，

因此x²+2x+4 >0，x可為任意實數

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