Quadratic Equation

👨🏻‍🏫 學術台數學

[匿名]

2014-07-12 3:51 pm

α and β are roots of the equation x^2 + 2x + 10 = 0. Find the value of
α^3 + β. ( |α| > |β| ).

更新1:

Sorry, you may disregard |α| > |β|, the key is to find α^3 + β using simplest method.

更新2:

Simplest method means not to solve the equation to find α and β and then calculate α^3 + β.

回答 (3)

螞蟻雄兵

2014-07-12 10:47 pm

✔ 最佳答案

α and βare roots of the equation x^2+2x+10=0，Find thevalue ofα^3+β
Sol
x^2+2x+10=0
x=[－2+/-√(4－40)]/2=－1+/-3i
α+β=－2
x^3+2x^2+10x=0
x^3=－2x^2－10x=－2(－2x－10)－10x=－6x+20
α^3=－6α+20
α^3+β
=－6α+β+20
=(α+β)－7α+20
=18－7α
(1) α=－1+3i
α^3+β=18+7+21i=25+21i
(1) α=－1－3i
α^3+β=18+7－21i=25－21i
So
α^3+β=25+/-21i
*******************
α^3+6β=20

or
x^2+2x+10=0
x=[－2+/-√(4－40)]/2=－1+/-3i
(1) α=－1+3i， β=－1－3i
α^3=(－1+3i)^3=(－8－6i)(－1+3i)=(8－24i+6i+18)=26－18i
α^3+β=(26－18i)+(－1－3i)=25－21i
(2) α=－1－3，, β=－1+3i
α^3=(－1－3i)^3=(－8+6i)(－1－3i)=(8+24i－6i+18)=26+18i
α^3+β=(26+18i)+(－1+3i)=25+21i
So
α^3+β=25+/-21i

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YA HOO！知識＋管理員

2014-07-12 8:07 pm

What is the meaning of 'simplest method' ?...

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知足常樂

2014-07-12 4:12 pm

不是吧？！？！？！？

2014-07-12 08:45:03 補充：
Complex roots means |α| = |β|.

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