利用根与系数关系求解若X1、X2是方程2X^2-3X+1=0的两个根,不解方程,求1/X1^3+1/X2^3X1^3-X2^3关键是变化的过程,怎么变成韦达定理?
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利用根与系数关系求解若X1、X2是方程2X^2-3X+1=0的两个根,不解方程,求1/X1^3+1/X2^3X1^3-X2^3关键是变化的过程,怎么变成韦达定理?
利用根与系数关系求解
若X1、X2是方程2X^2-3X+1=0的两个根,不解方程,求
1/X1^3+1/X2^3
X1^3-X2^3
关键是变化的过程,怎么变成韦达定理?
利用根与系数关系求解若X1、X2是方程2X^2-3X+1=0的两个根,不解方程,求1/X1^3+1/X2^3X1^3-X2^3关键是变化的过程,怎么变成韦达定理?
由韦达定理:x1+x2=3/2,x1x2=1/2,
1/X1^3+1/X2^3
=(x1^3+x2^3)/x1^3x2^3
=(x1+x2)(x1^2-x1x2+x2^2)/(x1x2)^3
=(x1+x2)[(x1+x2)^2-3x1x2]/(x1x2)^3
=(3/2)(9/4-3/2)/(1/8)
=9
X1^3-X2^3
=(x1-x2)(x1^2+x1x2+x2^2)
=(x1-x2)[(x1+x2)^2-x1x2]
=±√(x1-x2)^2[(9/4)-3/2]
=±√[(x1+x2)^2-4x1x2](3/4)
=±(1/2)*(3/4)
=±3/8
x1+x2=3/2 x1x2=1/2
(1) 1/X1^3+1/X2^3=(x1^3+x2^3)/(x1x2)^3
=(x1+x2)(x1^2-x1x2+x2^2)/(x1x2)^3
=(x1+x2)[(x1+x2)^2-3x1x2)]/(x1x2)^3
带入即可
(2) X1^3-X2^3=(x1-x2)(x1^2+x1x2+x2^2)
= (x1-x2)[(x1+x2)^2-x1x2)]
带入即可
x1+x2=3/2,x1*x2=1/2 x1²+x2²=5/4
1/X1^3+1/X2^3=(1/x1+1/x2)(1/x1²-1/(x1*x2)+1/x2²)
1/X1^3-1/X2^3=(1/x1-1/x2)(1/x1²+1/(x1*x2)+1/x2²)
带入解就可以了
因为X1、X2是方程2X^2-3X+1=0
所以X1+X2=3/2，X1X2=1/2【韦达定理】
所以：
1/X1^3+1/X2^3
=(X1^3+X2^3)/[(X1X2)^3]
=(X1+X2)(X1^2-X1X2+X2^2)/[(X1X2)^3]
=(X1+X2)[(X1+X2)^2-3X1X2]/[(X1X2)^3]
=(3/2)[(...
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因为X1、X2是方程2X^2-3X+1=0
所以X1+X2=3/2，X1X2=1/2【韦达定理】
所以：
1/X1^3+1/X2^3
=(X1^3+X2^3)/[(X1X2)^3]
=(X1+X2)(X1^2-X1X2+X2^2)/[(X1X2)^3]
=(X1+X2)[(X1+X2)^2-3X1X2]/[(X1X2)^3]
=(3/2)[(3/2)^2-3*(1/2)]/(1/2)^3
=9
X1^3-X2^3
=(X1-X2)(X1^2+X1X2+X2^2)
=(X1-X2)[(X1+X2)^2-X1X2]
=(X1-X2)[(3/2)^2-1/2]
=(7/4)(X1-X2)
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