1.If the equations ax^2+bx+c=0 and px^2+qx+r=0 have one root in common, prove that (br-cq)(aq+bp)=(cp-ar)^2
2.If the equations x^2-ax+8=0 and x^2-5x+a=0 have one root in common, prove that a^3-4a^2-56a+264=0
about sum and product of roots
2009-06-27 3:00 am
回答 (3)
2009-06-27 3:38 am
✔ 最佳答案
As follows:圖片參考：http://i707.photobucket.com/albums/ww74/stevieg90/01-44.gif
圖片參考：http://i707.photobucket.com/albums/ww74/stevieg90/02-40.gif
2009-06-27 3:48 am
Let α be the common root
aα^2+bα+c=0
pα^2+qα+r =0
α=(ar-cp) / (bp-aq)
a [(ar - cp) / (bp - aq) ]^2 + b[(ar - cp) / (bp - aq )] +c=0
a(ar-cp)^2+b(ar-cp)(bp-aq)+c (bp-aq)^2=0
(ar-cp)^2= -(bp-aq) (br-cq) ... (*)
2 Sub a=1,b=-a,c=8,p=1,q=-5,r=a in (*)
(a-8)^2= -(-a+5) (-a^2+40)
a^3-4a^2-56a+264=0
aα^2+bα+c=0
pα^2+qα+r =0
α=(ar-cp) / (bp-aq)
a [(ar - cp) / (bp - aq) ]^2 + b[(ar - cp) / (bp - aq )] +c=0
a(ar-cp)^2+b(ar-cp)(bp-aq)+c (bp-aq)^2=0
(ar-cp)^2= -(bp-aq) (br-cq) ... (*)
2 Sub a=1,b=-a,c=8,p=1,q=-5,r=a in (*)
(a-8)^2= -(-a+5) (-a^2+40)
a^3-4a^2-56a+264=0
2009-06-27 3:47 am
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