α and β are the roots of the quadratic equation 2 p (x ^ 2) - ( 3 p - 1 ) x + 5 = 0.
(a) If α = - 1, find p.
(b) Hence find (α ^ 2) / β + (β ^ 2 ) /α .
(c) Find the values of m and n if (α- 1 ) and ( β - 1 ) are the roots of the equation 8 (x ^ 2) + m x + n = 0 .
15分 1題 F.4 quadratic equation
2009-10-03 8:22 am
回答 (1)
2009-10-04 2:39 am
✔ 最佳答案
a) if α= -1 , then 2p(-1)^2-(3p-1)(-1)+5=0 => 2p+2p-1+5 =0 => p=-1//b) sum of roots =α+β= (3p-1)/2p = 2
produce of roots = αβ= 5/(2)(-1)=-5/2
(α ^ 2) / β + (β ^ 2 ) /α
=(α^3+β^3)/αβ
=(α+β)(α^2-αβ+β^2)/αβ
=(α+β)[(α+β)^2-3αβ]/αβ
=2(2^2+15/2)/(-5/2)
=-46/5//
c)(α-1)+(β-1)=α+β-2=2-2=0
(α-1)(β-1)=αβ-(α+β)+1=-5/2-2+1=-5/2-1=-7/2//h
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