F.3 Factorization

x + y = a
xy = b
x2 + y2
= x2 + 2xy + y2 - 2xy
= (x + y)2 - 2xy
= a2 - 2b
(x - y)2
= x2 - 2xy + y2
= x2 + 2xy + y2 - 4xy
= (x + y)2 - 4xy
= a2 - 4b
x2 - y2
= (x + y)(x - y)
= a√(x - y)2
= +/-a√(a2 - 4b)

1: x^2+y^2 = (x+y)^2-2xy=a^2-2b
2:(x-y)^2 = (x+y)^2-4xy=a^2-4b
3:x^2-y^2 = (x+y)(x-y) = (x+y)((x-y)^2)^(1/2)
=a(a^2-4b)^1/2
註: x^1/2 = 開方x

1. x^2+y^2
= (x+y)^2 - 2xy << (A+B)^2 = A^2 + 2AB + B^2
= a^2 - 2b
　　
2. (x-y)^2
= x^2 - 2xy + y^2
= x^2 + y^2 - 2b
Using the same way in Q1, we get
= a^2 - 2b - 2b
= a^2 - 4b
　　
3. x^2-y^2
= (x+y)(x-y)
= a(x-y)
in Q2, since (x-y)^2 = a^2 - 4b
so x-y = a - 2開方b
so we get
= a(a - 2開方b)
　　
我沒consider 開方的正負
如有需要你可以做埋
如有錯漏請指教

2009-09-17 22:07:25 補充：
對示起第三條有錯
x-y = 開方(a^2 - 4b) or 負開方(a^2 - 4b)
答案是 = a開方(a^2 - 4b) or 負a開方(a^2 - 4b)
(consider左正負)