活用公式变形求值 请详解已知x^2-4x+1=0,求x的4次方+1/x^4的值?
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活用公式变形求值 请详解已知x^2-4x+1=0,求x的4次方+1/x^4的值?
活用公式变形求值 请详解
已知x^2-4x+1=0,求x的4次方+1/x^4的值?
活用公式变形求值 请详解已知x^2-4x+1=0,求x的4次方+1/x^4的值?
x^2-4x+1=0
x-4+1/x=0
x+1/x=4
(x+1/x)^2=16
x^2+1/x^2=14
(x^2+1/x^2)^2=196
x^4+1/x^4=196-2
x^4+1/x^4=194
x^2-4x+1=0
x-4+1/x=0
x+1/x=4
x^4+1/x^4=(x^2+1/x^2)^2-2=[(x+1/x)^2-2]^2-2=194
x^2-4x+1=0
x1=(4+(16-4)^0.5)/2=2+2*(3^0.5)=2*(1+3^0.5)
x1=(4-(16-4)^0.5)/2=2-2*(3^0.5)=2*(1-3^0.5)
以x1代入:
x^4=16*(1+3^0.5)^4=16*(1+2*(3^0.5)+3)^2=16*(4+2*(3^0.5))^2
=64*(2+3^0.5...
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x^2-4x+1=0
x1=(4+(16-4)^0.5)/2=2+2*(3^0.5)=2*(1+3^0.5)
x1=(4-(16-4)^0.5)/2=2-2*(3^0.5)=2*(1-3^0.5)
以x1代入:
x^4=16*(1+3^0.5)^4=16*(1+2*(3^0.5)+3)^2=16*(4+2*(3^0.5))^2
=64*(2+3^0.5)^2=64*(4+4*(3^0.5)+3)=64*(7+4*(3^0.5))
1/x^4=1/(64*(7+4*(3^0.5)))
x^4+1/x^4=64*(7+4*(3^0.5))+1/(64*(7+4*(3^0.5)))
=((64*(7+4*(3^0.5)))^2+1)/(64*(7+4*(3^0.5)))
=((64^2)*(49+56(3^0.5)+48)+1)/(64*(7+4*(3^0.5)))
=(4096*(97+56(3^0.5))+1)/(64*(7+4*(3^0.5)))
=(397312+229276*(3^0.5)+1)/(64*(7+4*(3^0.5)))
=(397313+229276*(3^0.5))/(64*(7+4*(3^0.5)))
以x2代入:
x^4=16*(1-3^0.5)^4=16*(1-2*(3^0.5)+3)^2=16*(4-2*(3^0.5))^2
=64*(2-3^0.5)^2=64*(4-4*(3^0.5)+3)=64*(7-4*(3^0.5))
1/x^4=1/(64*(7-4*(3^0.5)))
x^4+1/x^4=64*(7-4*(3^0.5))+1/(64*(7-4*(3^0.5)))
=((64*(7-4*(3^0.5)))^2+1)/(64*(7-4*(3^0.5)))
=((64^2)*(49-56(3^0.5)+48)+1)/(64*(7-4*(3^0.5)))
=(4096*(97-56(3^0.5))+1)/(64*(7-4*(3^0.5)))
=(397312-229276*(3^0.5)+1)/(64*(7-4*(3^0.5)))
=(397313-229276*(3^0.5))/(64*(7-4*(3^0.5)))
x^4+1/x^4=
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